2.1 Slopes and the discriminant

Recall that for a locally free sheaf, we have

$$\begin{eqnarray}\operatorname{ch}_{0}(E)=\operatorname{rank}(E)=r(E).\end{eqnarray}$$

Using this equality, we define the slope and discriminant of $E$ to be

$$\begin{eqnarray}\unicode[STIX]{x1D707}(E)=\frac{\operatorname{ch}_{1}(E)}{\operatorname{ch}_{0}(E)}\qquad \text{and}\qquad \unicode[STIX]{x1D6E5}(E)=\frac{1}{2}\unicode[STIX]{x1D707}(E)^{2}-\frac{\operatorname{ch}_{2}(E)}{\operatorname{ch}_{0}(E)}.\end{eqnarray}$$

The slope and discriminant are more convenient than the Chern character for us as they have the property that

$$\begin{eqnarray}\unicode[STIX]{x1D707}(E\otimes F)=\unicode[STIX]{x1D707}(E)+\unicode[STIX]{x1D707}(F)\qquad \text{and}\qquad \unicode[STIX]{x1D6E5}(E\otimes F)=\unicode[STIX]{x1D6E5}(E)+\unicode[STIX]{x1D6E5}(F).\end{eqnarray}$$

We should note that these notions are easily extended to Chern characters in $K\left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)\otimes \mathbb{R}$ .

On a Picard rank one variety, the slope is a generalization of the degree of a line bundle to higher rank vector bundles. On higher Picard rank varieties, we can think of the slope as a generalization of a multi-degree that carries the information of the degree for every choice of embedding. Sometimes we would like an analog to the degree with respect to a specific embedding so we give the following definition. The slope with respect to an ample divisor $H$ is

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{H}(E)=\unicode[STIX]{x1D707}(E)\cdot H.\end{eqnarray}$$

We also call this the $H$ -Slope of $E$ . The two ample divisors that we use in this paper are ${\mathcal{O}}_{\mathbb{P}^{1}\times \mathbb{P}^{1}}(1,1)$ and ${\mathcal{O}}_{\mathbb{P}^{1}\times \mathbb{P}^{1}}(1,2)$ for which the slopes are denoted $\unicode[STIX]{x1D707}_{1,1}$ and $\unicode[STIX]{x1D707}_{1,2}$ , respectively.

We want to write the Riemann–Roch formula in terms of these invariants so let us recall the definition of the (reduced) Hilbert polynomial.

Definition 2.1. The Hilbert polynomial of a sheaf ${\mathcal{F}}$ with respect to an ample line bundle $H$ is

$$\begin{eqnarray}P_{{\mathcal{F}}}(k)=\unicode[STIX]{x1D712}\left({\mathcal{F}}(k)\right)=\frac{\unicode[STIX]{x1D6FC}_{d}}{d!}k^{d}+\cdots +\unicode[STIX]{x1D6FC}_{1}k+\unicode[STIX]{x1D6FC}_{0}\end{eqnarray}$$

where ${\mathcal{F}}(k)={\mathcal{F}}\otimes H^{\otimes k}$ , where the dimension of ${\mathcal{F}}$ is $d$ , and where we think of this as a polynomial in the variable $k$ .

Definition 2.2. The reduced Hilbert polynomial of a sheaf ${\mathcal{F}}$ with respect to an ample line bundle $H$ is defined to be

$$\begin{eqnarray}p_{{\mathcal{F}}}(k)=\frac{P_{{\mathcal{F}}}(k)}{\unicode[STIX]{x1D6FC}_{d}}\end{eqnarray}$$

where $d$ is the dimension of ${\mathcal{F}}$ .

The Hilbert polynomial, unlike the individual cohomology groups it sums over, is a numerical object that is entirely determined by the Chern character of the sheaf and the Chern character of the ample line bundle.

When $\operatorname{ch}_{0}(E)>0$ , we can write the Riemann–Roch formula as

$$\begin{eqnarray}\unicode[STIX]{x1D712}(E)=r\left(P(\unicode[STIX]{x1D707})-\unicode[STIX]{x1D6E5}\right),\end{eqnarray}$$

where $P(\unicode[STIX]{x1D707})$ is the Hilbert polynomial of ${\mathcal{O}}$ ,

$$\begin{eqnarray}\displaystyle P(m,n) & = & \displaystyle {\textstyle \frac{1}{2}}(m,n)^{2}+(1,1)\cdot (m,n)+1\nonumber\\ \displaystyle & = & \displaystyle mn+m+n+1=(m+1)(n+1).\nonumber\end{eqnarray}$$

2.1.1 Classical stability conditions

Using our notations of slope and the reduced Hilbert polynomial, we can now define the different classical notions of stability that we need.

Definition 2.3. A sheaf ${\mathcal{F}}$ is slope (semi-)stable with respect to an ample line bundle $H$ if for all proper subsheaves ${\mathcal{F}}^{\prime }\subset {\mathcal{F}}$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{H}({\mathcal{F}}^{\prime })\left({\leqslant}\right)<\unicode[STIX]{x1D707}_{H}({\mathcal{F}}).\end{eqnarray}$$

A stronger notation of stability is the notion of Gieseker stability.

Definition 2.4. A sheaf ${\mathcal{F}}$ is Gieseker (semi-)stable with respect to an ample divisor $H$ if for all proper subsheaves ${\mathcal{F}}^{\prime }\subset {\mathcal{F}}$ , $p_{{\mathcal{F}}^{\prime }}\left({\leqslant}\right)<p_{{\mathcal{F}}}$ where the polynomials are compared for all sufficiently large input values. We also call this $\unicode[STIX]{x1D6FE}$ (semi-)stability.

The ordering on the polynomials could also have been phrased as the lexicographic ordering on their coefficients starting from the highest degree term’s coefficient and working down. The condition on the Hilbert polynomial is equivalent on surfaces to requiring $\unicode[STIX]{x1D707}_{H}({\mathcal{F}}^{\prime })\leqslant \unicode[STIX]{x1D707}_{H}({\mathcal{F}})$ and, in the case of equality, $\unicode[STIX]{x1D6E5}({\mathcal{F}}^{\prime })\left({\geqslant}\right)>\unicode[STIX]{x1D6E5}({\mathcal{F}})$ .

These two notations of stability and semistability are related by a string of implications that seems slightly odd at first, but becomes clear using this last equivalence:

$$\begin{eqnarray}\text{slope stable}\rightarrow \text{Gieseker stable}\rightarrow \text{Gieseker semistable}\rightarrow \text{slope semistable}.\end{eqnarray}$$

As we are focused on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ , one additional notion of stability is relevant.

Definition 2.5. A sheaf ${\mathcal{F}}$ is $\overline{\unicode[STIX]{x1D6FE}}$ (semi-)stable if it is Gieseker semistable with respect to ${\mathcal{O}}(1,1)$ and for all proper subsheaves ${\mathcal{F}}^{\prime }\subset {\mathcal{F}}$ , if $\unicode[STIX]{x1D707}_{1,1}\left({\mathcal{F}}^{\prime }\right)=\unicode[STIX]{x1D707}_{1,1}\left({\mathcal{F}}\right)$ and $\unicode[STIX]{x1D6E5}\left({\mathcal{F}}^{\prime }\right)=\unicode[STIX]{x1D6E5}\left({\mathcal{F}}\right)$ , then $\unicode[STIX]{x1D707}_{1,2}\left({\mathcal{F}}^{\prime }\right)\left({\leqslant}\right)<\unicode[STIX]{x1D707}_{1,2}\left({\mathcal{F}}\right)$ .

Again we have implications:

$$\begin{eqnarray}\displaystyle \text{slope stable} & \rightarrow & \displaystyle \text{Gieseker stable}\rightarrow \overline{\unicode[STIX]{x1D6FE}}\text{stable}\nonumber\\ \displaystyle & \rightarrow & \displaystyle \overline{\unicode[STIX]{x1D6FE}}\text{semistable}\rightarrow \text{Gieseker semistable}\rightarrow \text{slope semistable}.\nonumber\end{eqnarray}$$

Because Gieseker stability generalizes to all varieties, it might seem odd to add a third, very variety specific, condition. However, adding this third condition allows $\overline{\unicode[STIX]{x1D6FE}}$ stability to order all Chern characters in a way that was not possible for slope or Gieseker stability. This order is possible with $\overline{\unicode[STIX]{x1D6FE}}$ stability because it has three conditions that can distinguish the three variables for a Chern character on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ : the two coordinates of $c_{1}$ and the coordinate of $c_{2}$ .

2.2 Exceptional bundles

In this subsection, we start with a relative version of Euler characteristic. We eventually see that sheaves $E$ with relative Euler characteristic $\unicode[STIX]{x1D712}(E,E)=1$ control the geometry of our moduli spaces.

2.2.1 The relative Euler characteristic

The relative Euler characteristic of two sheaves on a variety $X$ of dimension $n$ is

$$\begin{eqnarray}\unicode[STIX]{x1D712}(E,F)=\mathop{\sum }_{i=0}^{n}(-1)^{i}\operatorname{ext}^{i}(E,F)\end{eqnarray}$$

where $\operatorname{ext}^{i}(E,F)=\dim (\operatorname{Ext}^{i}(E,F))$ . For locally free sheaves, we can equivalently define it by the formula

$$\begin{eqnarray}\unicode[STIX]{x1D712}(E,F)=\unicode[STIX]{x1D712}(E^{\ast }\otimes F)=\mathop{\sum }_{i=0}^{n}(-1)^{i}h^{i}(E^{\ast }\otimes F).\end{eqnarray}$$

Restricting to the case where $n=2$ , we again write out Hirzebruch–Riemann–Roch in order to explicitly compute this as

$$\begin{eqnarray}\unicode[STIX]{x1D712}(E,F)=r(E)r(F)\left(P\left(\unicode[STIX]{x1D707}(E)-\unicode[STIX]{x1D707}(F)\right)-\unicode[STIX]{x1D6E5}(E)-\unicode[STIX]{x1D6E5}(F)\right)\end{eqnarray}$$

where $P(x,y)=(x+1)(y+1)$ is the Euler characteristic of ${\mathcal{O}}_{\mathbb{P}^{1}\times \mathbb{P}^{1}}(x,y)$ .

Using the alternate definition of relative Euler characteristic, we define a bilinear pairing on $K\left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)$ by

$$\begin{eqnarray}\left(E,F\right)=\unicode[STIX]{x1D712}(E^{\ast },F)=\unicode[STIX]{x1D712}(F^{\ast },E)=\unicode[STIX]{x1D712}(E\otimes F).\end{eqnarray}$$

Then we define the orthogonal complement of $E$ , denoted $\operatorname{ch}(E)^{\bot }$ , to be all bundles $F$ such that $(E,F)=0$ . Note that the pairing is symmetric so the orthogonal complement does not depend on whether $E$ is the first or second element in the pairing.

The $\operatorname{Ext}$ groups we used to define the relative Euler characteristic also allow us to describe a set of vector bundles that control the geometry of $\mathbb{P}^{1}\times \mathbb{P}^{1}$ .

Definition 2.6. A sheaf $E$ is exceptional if $\operatorname{Hom}(E,E)=\mathbb{C}$ and $\operatorname{Ext}^{i}(E,E)=0$ for all $i>0$ .

We say a Chern character (slope) is exceptional if there is an exceptional sheaf with that character (slope). The prototypical exceptional sheaves are the line bundles, ${\mathcal{O}}(a,b)$ . Many of the properties of line bundles pass to exceptional sheaves on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ . We list some of these properties without repeating the proofs. Their proofs are in the previously mentioned paper by Gorodentsev [Reference GorodentsevGor].

Proposition 2.7. On $\mathbb{P}^{1}\times \mathbb{P}^{1}$ , we have the following results.

• ∙ Every exceptional sheaf is a vector bundle.

• ∙ There exists a unique exceptional bundle, $E$ , for each exceptional slope.

• ∙ The two coordinates of the first Chern class are coprime with the rank of  $E$ .

• ∙ The denominator of $\unicode[STIX]{x1D707}_{1,1}(E)$ is the rank of  $E$ .

• ∙ Exceptional bundles are stable with respect to ${\mathcal{O}}(1,1)$ .

Given these properties, it makes sense to define the rank of an exceptional slope $\unicode[STIX]{x1D708}$ to be the smallest $r\in \mathbb{Z}$ such that $r\unicode[STIX]{x1D707}_{1,1}(\unicode[STIX]{x1D708})\in \mathbb{Z}$ . Also, set the notation $E_{\frac{a}{r},\frac{b}{r}}$ for the unique exceptional bundle with slope $\left(\frac{a}{r},\frac{b}{r}\right)$ . Given a Chern character $\unicode[STIX]{x1D709}$ with slope $\unicode[STIX]{x1D6FC}=\left(\frac{a}{r},\frac{b}{r}\right)$ , we interchangeably write $E_{-\unicode[STIX]{x1D6FC}}=E_{\unicode[STIX]{x1D6FC}}^{\ast }=E_{\frac{-a}{r},\frac{-b}{r}}$ . Similarly, we abuse notation and write the slope and the whole Chern character interchangeably.

We can characterize exceptional bundles among all stable bundles by the Euler characteristic $\unicode[STIX]{x1D712}(E,E)$ . For stable $E$ , Serre duality implies $\operatorname{Ext}^{2}(E,E)=\operatorname{Hom}(E,E(K))=0$ . Similarly, the stability of $E$ implies that $\operatorname{Hom}(E,E)=\mathbb{C}$ . Then for stable $E$ we have

$$\begin{eqnarray}\unicode[STIX]{x1D712}(E,E)=1-\operatorname{ext}^{1}(E,E)\leqslant 1\end{eqnarray}$$

with equality precisely when $E$ is exceptional. Conversely, we use Hirzebruch–Riemann–Roch to explicitly compute

$$\begin{eqnarray}\unicode[STIX]{x1D712}(E,E)=r(E)^{2}\left(1-2\unicode[STIX]{x1D6E5}\right).\end{eqnarray}$$

Putting these together, we see that for a stable bundle

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(E)=\frac{1}{2}\left(1-\frac{\unicode[STIX]{x1D712}(E,E)}{r(E)^{2}}\right).\end{eqnarray}$$

For exceptional bundles, this reduces to

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(E)=\frac{1}{2}\left(1-\frac{1}{r(E)^{2}}\right).\end{eqnarray}$$

Since $\unicode[STIX]{x1D712}(F,F)\leqslant 0$ for all other stable bundles, we see that exceptional bundles are the only stable bundles with $\unicode[STIX]{x1D6E5}<\frac{1}{2}$ . As there is a unique exceptional bundle for an exceptional slope and there can be no other stable bundles with that discriminant, the moduli space of stable bundles with an exceptional Chern character is a single reduced point [Reference GorodentsevGor]. Giving an explicit description of the exceptional bundles analogous to the description of the exceptional bundles on $\mathbb{P}^{2}$ given in [Reference Coskun, Huizenga and WoolfCHW] and [Reference Le PotierLP] is an open question.

2.3 Exceptional collections

Exceptional bundles naturally sit inside of collections.

An exceptional collection is strong if $k=0$ for all pairs $(i,j)$ . We say the length an exceptional collection $\left(E_{0},\ldots ,E_{n}\right)$ is $n+1$ . An exceptional pair is an exceptional collection of length two. A coil is a maximal exceptional collection, which in the case of $\mathbb{P}^{1}\times \mathbb{P}^{1}$ is length four. Our stereotypical (strong) coil is $\left({\mathcal{O}},{\mathcal{O}}(1,0),{\mathcal{O}}(0,1),{\mathcal{O}}(1,1)\right)$ .

Every exceptional bundle sits inside of an exceptional collection, and every exceptional collection can be completed to a coil [Reference RudakovRud2]. Given an exceptional collection of length three $(E_{0},E_{1},E_{2})$ there are exactly four ways to complete it to a coil:

$$\begin{eqnarray}\displaystyle & \displaystyle (A,E_{0},E_{1},E_{2}),\qquad (E_{0},B,E_{1},E_{2}),\qquad (E_{0},E_{1},C,E_{2}),\qquad \text{and} & \displaystyle \nonumber\\ \displaystyle & \displaystyle (E_{0},E_{1},E_{2},D). & \displaystyle \nonumber\end{eqnarray}$$

In other words, once you pick where you would like the fourth bundle to be, there is a unique way to complete the exceptional collection to a coil. This uniqueness follows as an easy consequence of the requirement that the fourth bundle forms an exceptional pair in the correct way with the other three bundles. First, we require each bundle to be in an exceptional pair which imposes an independent condition on its rank and first Chern classes so they are determined. Then the rank and first Chern class determine its discriminant, and the bundle is uniquely determined by its Chern classes.

Before we can state how to extend an exceptional collection of length two to a coil, we must first explain a process that turns an exceptional collection into different exceptional collections called mutation or reconstruction. We first define mutation for exceptional pairs and then bootstrap this definition into a definition for all exceptional collections. The definitions of mutation that we use are equivalent on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ to the general definitions [Reference GorodentsevGor].

Definition 2.9. The left mutation of an exceptional pair $\left(E_{0},E_{1}\right)$ is the exceptional pair $L\left(E_{0},E_{1}\right)=\left(L_{E_{0}}E_{1},E_{0}\right)$ where $L_{E_{0}}E_{1}$ is determined by one of the following short exact sequences:

$$\begin{eqnarray}\displaystyle & \displaystyle \text{(regular) }0\rightarrow L_{E_{0}}E_{1}\rightarrow E_{0}\otimes \operatorname{Hom}(E_{0},E_{1})\rightarrow E_{1}\rightarrow 0, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{(rebound) }0\rightarrow E_{0}\otimes \operatorname{Hom}(E_{0},E_{1})\rightarrow E_{1}\rightarrow L_{E_{0}}E_{1}\rightarrow 0,\quad \text{or} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{(extension) }0\rightarrow E_{1}\rightarrow L_{E_{0}}E_{1}\rightarrow E_{0}\otimes \operatorname{Ext}^{1}(E_{0},E_{1})\rightarrow 0. & \displaystyle \nonumber\end{eqnarray}$$

One of these sequences exists by Gorodentsev [Reference GorodentsevGor]. By rank considerations only one of the previous short exact sequences is possible so the left mutation is unique. Rebound and extension mutations are called nonregular. Right mutation of an exceptional pair, denoted $R\left(E_{0},E_{1}\right)=\left(E_{1},R_{E_{1}}E_{0}\right)$ , is defined similarly by tensoring the $\operatorname{Hom}$ or $\operatorname{Ext}$ with $E_{1}$ rather than with $E_{0}$ . Note that left and right mutation are inverse operations in the sense that

$$\begin{eqnarray}L\left(R\left(E_{0},E_{1}\right)\right)=R\left(L\left(E_{0},E_{1}\right)\right)=\left(E_{0},E_{1}\right).\end{eqnarray}$$

We can also mutate any part of an exceptional collection. In particular, replacing any adjacent exceptional pair in an exceptional collection with any of its left or right mutations gives another exceptional collection. For example, given an exceptional collection $\left(E_{0},E_{1},E_{2},E_{3}\right)$ ,

$$\begin{eqnarray}\displaystyle & \displaystyle \left(L(E_{0},E_{1}),E_{2},E_{3}\right),\qquad \left(R(E_{0},E_{1}),E_{2},E_{3}\right),\qquad \left(E_{0},L(E_{1},E_{2}),E_{3}\right) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \left(E_{0},R(E_{1},E_{2}),E_{3}\right),\qquad \left(E_{0},E_{1},L(E_{2},E_{3})\right),\qquad \text{and} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \left(E_{0},E_{1},R(E_{2},E_{3})\right) & \displaystyle \nonumber\end{eqnarray}$$

are all exceptional collections. Rudakov proved that all possible exceptional collections can be gotten from our stereotypical collection via these pairwise reconstructions [Reference RudakovRud1].

Mutating an exceptional collection is then just mutating all of the bundles in a systematic way. We define the left(right) mutation of an exceptional collection $\left(E_{0},E_{1},E_{2},\ldots ,E_{n}\right)$ as

$$\begin{eqnarray}\displaystyle & \displaystyle \left(L_{E_{0}}\cdots L_{E_{n-1}}E_{n},\ldots ,L_{E_{0}}L_{E_{1}}E_{2},L_{E_{0}}E_{1},E_{0}\right) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \left(\left(E_{n},R_{E_{n}}E_{n-1},R_{E_{n}}R_{E_{n-1}}E_{n-2},\ldots ,R_{E_{n}}\cdots R_{E_{1}}E_{0}\right)\right). & \displaystyle \nonumber\end{eqnarray}$$

For a coil $(E_{0},E_{1},F_{0},F_{1})$ on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ , its left mutation is

$$\begin{eqnarray}(F_{1}(K),F_{2}(K),E_{-1},E_{0})\end{eqnarray}$$

and its right mutation is

$$\begin{eqnarray}(F_{1},F_{2},E_{-1}(-K),E_{0}(-K)).\end{eqnarray}$$

Now, let us return to the problem of completing an exceptional pair to a coil. Say that we could extend $(E_{0},E_{1})$ to the coil $(E_{0},E_{1},F_{0},F_{1})$ , this extension is not unique, even up to placement because $(E_{0},E_{1},L(F_{0},F_{1}))$ and $(E_{0},E_{1},R(F_{0},F_{1}))$ are also coils. To make a unique notation of extension, we need the notion of a system.

Given an exceptional pair $\left(E_{0},E_{1}\right)$ , we define the right completion system of it to be the unique system $\{F_{i}\}$ such that $\left(E_{0},E_{1},F_{i},F_{i+1}\right)$ is a coil. Left and center completion systems are defined analogously. A left (right,center) completion pair is any pair $\left(F_{i},F_{i+1}\right)$ coming from the left (right,center) completion system. By Propositions 4.5 and 4.8 of Rudakov’s paper [Reference RudakovRud1], the completion system is either a system of line bundles or has a minimally ranked completion pair where the minimally ranked completion pair is the pair in the system with the lowest sum of the two ranks of the bundles in the system.

Completing an exceptional collection of length one to a coil can be reduced to the two steps of completing it to an exceptional pair (which is highly not unique) and then completing the pair to a coil. In this paper, we start with pairs and provide a unique way to extend them to a coil.

Given a complex $W:A^{a}\rightarrow B^{b}$ of powers of an exceptional pair $(A,B)$ , we extend the idea of mutation to the complex. Define $LW$ to be the complex

$$\begin{eqnarray}LW:(L_{A}B)^{a}\rightarrow A^{b}\end{eqnarray}$$

and similarly define $RW$ to be the complex

$$\begin{eqnarray}RW:B^{a}\rightarrow (R_{B}A)^{b}.\end{eqnarray}$$

We also define the mutations relative to an exceptional bundle $C$ where $\{C,A,B\}\left(\{A,B,C\}\right)$ is an exceptional collection as follows: if $\{C,A,B\}$ is an exceptional collection, define $L_{C}W$ to be the complex

$$\begin{eqnarray}L_{C}W:(L_{C}A)^{a}\rightarrow (L_{C}B)^{b},\end{eqnarray}$$

and similarly, if $\{A,B,C\}$ is an exceptional collection, define $R_{C}W$ to be the complex

$$\begin{eqnarray}R_{C}W:(R_{C}A)^{a}\rightarrow (R_{C}B)^{b}.\end{eqnarray}$$

We can now state a theorem of Gorodentsev which establishes the spectral sequence that we need on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ .

Theorem 2.11. [Reference GorodentsevGor]

Let $U$ be a coherent sheaf on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ , and let $\left(E_{0},E_{1},F_{0},F_{1}\right)$ be a coil. Write ${\mathcal{A}}=\left(A_{0},A_{1},A_{2},A_{3}\right)=\left(E_{0},E_{1},F_{0},F_{1}\right)$ and ${\mathcal{B}}=\left(B_{-3},B_{-2},B_{-1},B_{0}\right)=\left(F_{1}(K),F_{2}(K),E_{-1},E_{0}\right)$ . There is a spectral sequence with $E_{1}^{p,q}$ -page

$$\begin{eqnarray}E_{1}^{p,q}=B_{p}\otimes \operatorname{Ext}^{q-\unicode[STIX]{x1D6E5}_{p}}\left(A_{-p},U\right)\end{eqnarray}$$

that converges to $U$ in degree 0 and to 0 in all other degrees where $\unicode[STIX]{x1D6E5}_{p}$ is the number of nonregular mutations in the string $L_{0}\ldots L_{p-1}A_{p}$ which mutates $A_{p}$ into $B_{-p}$ .

It should be clear that $\unicode[STIX]{x1D6E5}_{0}=0$ . Considering the spectral sequence converging to different bundles of the coil allows us to deduce that $\unicode[STIX]{x1D6E5}_{3}=1$ and that the other two are either 0 or 1 [Reference KarpovKar, Rmk. 1.5.2]. Also, notice that ${\mathcal{B}}$ is the left mutation of ${\mathcal{A}}$ .

2.4 Moduli spaces of sheaves

Let $E$ be exceptional with Chern character $e$ . Then, $E$ imposes a numerical condition on stable bundles with “nearby” Chern characters. To see this condition, start with an exceptional bundle $E$ and another stable bundle $F$ , we know that there are no maps from $E$ to $F$ if the $(1,1)$ -slope of $E$ is bigger than $F$ ’s by $F$ ’s stability so

$$\begin{eqnarray}\hom (E,F)=0.\end{eqnarray}$$

Similarly, there are no maps from $F$ to $E$ twisted by the canonical if the $(1,1)$ -slope of $F$ is greater than the $(1,1)$ -slope of $E(K)$ so

$$\begin{eqnarray}\hom (F,E(K))=0.\end{eqnarray}$$

By Serre duality,

$$\begin{eqnarray}\hom (F,E(K))=\operatorname{ext}^{2}(E,F)=0.\end{eqnarray}$$

Thus, if we have both of these conditions, we know that

$$\begin{eqnarray}\unicode[STIX]{x1D712}(E,F)=-\text{ext}^{1}(E,F)\leqslant 0.\end{eqnarray}$$

This is the numeric condition that $E$ imposes on nearby stable bundles. Given a fixed exceptional $E$ , we can encode this data by saying that the Chern character of $F$ must lie on or above a certain surface, $\unicode[STIX]{x1D6FF}_{E}$ , in the $(\unicode[STIX]{x1D707},\unicode[STIX]{x1D6E5})$ space. We define $\unicode[STIX]{x1D6FF}_{E}(\unicode[STIX]{x1D707})$ as follows.

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{E}(\unicode[STIX]{x1D707})=\left\{\begin{array}{@{}l@{}}(\unicode[STIX]{x1D707}_{1,0}(\unicode[STIX]{x1D707})-\unicode[STIX]{x1D707}_{1,0}(E)+1)(\unicode[STIX]{x1D707}_{1,0}(\unicode[STIX]{x1D707})-\unicode[STIX]{x1D707}_{1,0}(E)+1)-\unicode[STIX]{x1D6E5}(E)\\ \qquad \text{if }\unicode[STIX]{x1D707}_{1,1}(E)-4<\unicode[STIX]{x1D707}_{1,1}(\unicode[STIX]{x1D707})\leqslant \unicode[STIX]{x1D707}_{1,1}(E)\\ (\unicode[STIX]{x1D707}_{1,0}(E)-\unicode[STIX]{x1D707}_{1,0}(\unicode[STIX]{x1D707})+1)(\unicode[STIX]{x1D707}_{1,0}(E)-\unicode[STIX]{x1D707}_{1,0}(\unicode[STIX]{x1D707})+1)-\unicode[STIX]{x1D6E5}(E)\\ \qquad \text{if }\unicode[STIX]{x1D707}_{1,1}(E)\leqslant \unicode[STIX]{x1D707}_{1,1}(\unicode[STIX]{x1D707})<\unicode[STIX]{x1D707}_{1,1}(E)+4\\ 0\quad \text{otherwise}.\end{array}\right\}\end{eqnarray}$$

Then $F$ ’s Chern character lying on or above $\unicode[STIX]{x1D6FF}_{E}$ means that

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(F)\geqslant \unicode[STIX]{x1D6FF}_{E}(\unicode[STIX]{x1D707}(F)).\end{eqnarray}$$

Using these conditions, each exceptional bundle gives an inequality that a Chern character must satisfy in order to be stable. We combine all of these conditions into one by looking at the maximum over all of the inequalities. Formally, let ${\mathcal{E}}$ be the set of exceptional bundles and define the $\unicode[STIX]{x1D6FF}$ surface by

Definition 2.12.

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D707})=\sup _{\{E\in {\mathcal{E}}\}}\unicode[STIX]{x1D6FF}_{E}\left(\unicode[STIX]{x1D707}\right).\end{eqnarray}$$

Then saying that a stable Chern character, $\unicode[STIX]{x1D701}$ , must satisfy all of the inequalities from exceptional bundles is equivalent to

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(F)\geqslant \unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D707}(F)).\end{eqnarray}$$

Alternatively, we say that a stable Chern character must lie on or above the $\unicode[STIX]{x1D6FF}$ -surface.

Rudakov proved that lying above the $\unicode[STIX]{x1D6FF}$ surface was not only necessary but also sufficient for a non-semiexceptional Chern character to be stable.

2.5 New basic properties

With this background, we can begin our original work on $M(\unicode[STIX]{x1D709})$ . In the last subsection, we listed many properties of the nonempty moduli spaces. In this subsection, we prove that the moduli spaces are $\mathbb{Q}$ -factorial and are Mori dream spaces. In fact, we prove these results for all moduli spaces of Gieseker semistable sheaves on any del Pezzo surface, not just $\mathbb{P}^{1}\times \mathbb{P}^{1}$ .

The proof that these spaces are Mori dream spaces is slightly more involved, but similar in flavor.

2.6 Additional assumptions

There are two previously mentioned properties that we would also like our moduli spaces on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ to have. We want the complement of the stable locus to be codimension at least two, and we want them to have Picard rank 3. The first assumption allows us to ignore the strictly semistable locus when we are working with divisors, and the second assumption lets us use properties of the Picard group that we need. These assumptions are justified for a few reasons. First, they hold for the Hilbert schemes of points, which are the primary examples of such moduli spaces. Second, Yoshioka proved that the second assumption holds for $M(\unicode[STIX]{x1D709})$ where one of the slope components of $\unicode[STIX]{x1D709}$ is an integer and $\unicode[STIX]{x1D709}$ is above the $\unicode[STIX]{x1D6FF}$ surface [Reference YoshiokaYos1]. Lastly, there are no examples of $M(\unicode[STIX]{x1D709})$ where $\unicode[STIX]{x1D709}$ is above the $\unicode[STIX]{x1D6FF}$ surface for which either assumption is known to fail. In fact, both assumptions are believed to be true above the $\unicode[STIX]{x1D6FF}$ surface. Proving that they hold in this region is the focus of current research.

We also assume that $\unicode[STIX]{x1D709}$ is above the $\unicode[STIX]{x1D6FF}$ surface and is not a multiple of an exceptional Chern character.

2.7 The Picard group of $M(\unicode[STIX]{x1D709})$

We have a good description of the Picard group of $M(\unicode[STIX]{x1D709})$ if we assume that the Picard rank is three which we just added as a standing assumption. Linear and numerical equivalence coincide on $M(\unicode[STIX]{x1D709})$ , so we have

$$\begin{eqnarray}\operatorname{NS}(M(\unicode[STIX]{x1D709}))=\operatorname{Pic}(M(\unicode[STIX]{x1D709}))\otimes \mathbb{R}.\end{eqnarray}$$

As mentioned above, we work with $M^{s}(\unicode[STIX]{x1D709})$ when it is convenient. $M^{s}(\unicode[STIX]{x1D709})$ is a coarse moduli space for the stable sheaves. In contrast, $M(\unicode[STIX]{x1D709})$ is not a coarse moduli space, unless $\unicode[STIX]{x1D709}$ is a primitive character, as it identifies $S$ -equivalence classes of (strictly semistable) sheaves.

In order to understand the classes of the divisors that span the effective cone, we have to understand the isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D709}^{\bot }\cong \operatorname{NS}\left(M(\unicode[STIX]{x1D709})\right).\end{eqnarray}$$

We construct this isomorphism by uniquely defining a line bundle on families of sheaves for each element of $\unicode[STIX]{x1D709}^{\bot }$ as done in [Reference Le PotierLP] and [Reference Coskun, Huizenga and WoolfCHW] for the case of $\mathbb{P}^{2}$ .

Let ${\mathcal{U}}/S$ be a flat family of semistable sheaves with Chern character $\unicode[STIX]{x1D709}$ where $S$ is a smooth variety. Define the two projections $p:S\times \mathbb{P}^{1}\times \mathbb{P}^{1}\rightarrow S$ and $q:S\times \mathbb{P}^{1}\times \mathbb{P}^{1}\rightarrow \mathbb{P}^{1}\times \mathbb{P}^{1}$ . Then consider the composition of maps

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{{\mathcal{U}}}:K(\mathbb{P}^{1}\times \mathbb{P}^{1}) & \xrightarrow[{}]{q^{\ast }} & \displaystyle K(S\times \mathbb{P}^{1}\times \mathbb{P}^{1})\xrightarrow[{}]{\cdot [{\mathcal{U}}]}K(S\times \mathbb{P}^{1}\times \mathbb{P}^{1})\nonumber\\ \displaystyle & \xrightarrow[{}]{-p!} & \displaystyle K(S)\xrightarrow[{}]{\det }\operatorname{Pic}(S)\nonumber\end{eqnarray}$$

where $p!=\sum (-1)^{i}R^{i}p_{\ast }$ . Colloquially, we are taking a sheaf on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ , pulling it back along the projection to $S\times \mathbb{P}^{1}\times \mathbb{P}^{1}$ , restricting it to our family, pushing it forward along the projection onto $S$ , and then taking its determinant to get a line bundle on $S$ .

Tensoring our family ${\mathcal{U}}$ by $p^{\ast }L$ , where $L$ is a line bundle on $S$ , does not change the given moduli map $f:S\rightarrow M(\unicode[STIX]{x1D709})$ , but does change the $\unicode[STIX]{x1D706}$ map as follows:

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{{\mathcal{U}}\otimes p^{\ast }L}(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D706}_{{\mathcal{U}}}(\unicode[STIX]{x1D701})\otimes L^{\otimes -(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701})}.\end{eqnarray}$$

Now given a class $\unicode[STIX]{x1D701}\in \unicode[STIX]{x1D709}^{\bot }$ , we want a $\unicode[STIX]{x1D706}_{M}$ map which commutes with the moduli map in the sense that we should have

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{{\mathcal{U}}}(\unicode[STIX]{x1D701})=f^{\ast }\unicode[STIX]{x1D706}_{M}(\unicode[STIX]{x1D701})\end{eqnarray}$$

for all ${\mathcal{U}}$ . This equality determines a unique line bundle $\unicode[STIX]{x1D706}_{M}(\unicode[STIX]{x1D701})$ on $M(\unicode[STIX]{x1D709})$ and gives a linear map $\unicode[STIX]{x1D706}_{M}:\unicode[STIX]{x1D709}^{\bot }\rightarrow \operatorname{NS}(M(\unicode[STIX]{x1D709}))$ which is an isomorphism under our assumptions.

Caveat: We have normalized $\unicode[STIX]{x1D706}_{M}$ using $-p!$ as in [Reference Coskun, Huizenga and WoolfCHW] rather than $p!$ as in [Reference Huybrechts and LehnHL] and [Reference Le PotierLP] so that the positive rank Chern characters form the “primary half space” that we define later this section.

2.8 A basis for the Picard group

Using this isomorphism, we want to construct a basis for the Picard group. Following Huybrechts and Lehn [Reference Huybrechts and LehnHL] and modifying it for $\mathbb{P}^{1}\times \mathbb{P}^{1}$ , we define three Chern characters $\unicode[STIX]{x1D701}_{0}$ , $\unicode[STIX]{x1D701}_{a}$ , and $\unicode[STIX]{x1D701}_{b}$ . Bundles with these Chern characters are a basis for the Picard space. These Chern characters depend on the character $\unicode[STIX]{x1D709}$ of the moduli space. Let $a$ be the Chern character of a line of the first ruling and $b$ be the Chern character of a line of the second ruling on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ . We define our Chern characters by the equations

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}_{0} & = & \displaystyle r(\unicode[STIX]{x1D709}){\mathcal{O}}_{\mathbb{P}^{1}\times \mathbb{P}^{1}}-\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}^{\ast },{\mathcal{O}}_{\mathbb{P}^{1}\times \mathbb{P}^{1}}){\mathcal{O}}_{p},\nonumber\\ \displaystyle \unicode[STIX]{x1D701}_{a} & = & \displaystyle r(\unicode[STIX]{x1D709})a_{1}-\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}^{\ast },a){\mathcal{O}}_{p},\qquad \text{and}\nonumber\\ \displaystyle \unicode[STIX]{x1D701}_{b} & = & \displaystyle r(\unicode[STIX]{x1D709})b_{1}-\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}^{\ast },b){\mathcal{O}}_{p}\nonumber\end{eqnarray}$$

where ${\mathcal{O}}_{p}$ is the structure sheaf of a point on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ . (Note the difference from the analogous definition in [Reference Huybrechts and LehnHL] by a sign due to our convention for  $\unicode[STIX]{x1D706}_{M}$ .)

Given that $(\unicode[STIX]{x1D709},{\mathcal{O}}_{p})=r(\unicode[STIX]{x1D709})$ , it should be clear that they are all in $\unicode[STIX]{x1D709}^{\bot }$ . They can also be shown to be in $\unicode[STIX]{x1D709}^{\bot }$ by using Riemann–Roch given that the Chern characters $(\operatorname{ch}_{0},\operatorname{ch}_{1},\operatorname{ch}_{2})$ of $\unicode[STIX]{x1D701}_{0},\unicode[STIX]{x1D701}_{a},\unicode[STIX]{x1D701}_{b}$ are

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}_{0} & = & \displaystyle \left(r(\unicode[STIX]{x1D709}),(0,0),-\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709})\right),\nonumber\\ \displaystyle \unicode[STIX]{x1D701}_{a} & = & \displaystyle \left(0,\left(r(\unicode[STIX]{x1D709}),0\right),-r(\unicode[STIX]{x1D709})-c_{1}(\unicode[STIX]{x1D709})\cdot (0,1)\right),\qquad \text{and}\nonumber\\ \displaystyle \unicode[STIX]{x1D701}_{b} & = & \displaystyle \left(0,\left(0,r(\unicode[STIX]{x1D709})\right),-r(\unicode[STIX]{x1D709})-c_{1}(\unicode[STIX]{x1D709})\cdot (1,0)\right).\nonumber\end{eqnarray}$$

Using the map from the previous section, define ${\mathcal{L}}_{0}$ , ${\mathcal{L}}_{a}$ , ${\mathcal{L}}_{b}$ by the formulas ${\mathcal{L}}_{0}=\unicode[STIX]{x1D706}_{M}(\unicode[STIX]{x1D701}_{0})$ , ${\mathcal{L}}_{a}=\unicode[STIX]{x1D706}_{M}(\unicode[STIX]{x1D701}_{a})$ , and ${\mathcal{L}}_{b}=\unicode[STIX]{x1D706}_{M}(\unicode[STIX]{x1D701}_{b})$ . They are a basis for the Picard space. We know that for $n>>0$ , ${\mathcal{L}}_{0}\otimes {\mathcal{L}}_{a}^{n}$ and ${\mathcal{L}}_{0}\otimes {\mathcal{L}}_{b}^{n}$ are ample [Reference Huybrechts and LehnHL].

Now, $\unicode[STIX]{x1D701}_{a}$ and $\unicode[STIX]{x1D701}_{b}$ span the plane of rank zero sheaves in $\unicode[STIX]{x1D709}^{\bot }$ . Define the primary half space of $\unicode[STIX]{x1D709}^{\bot }$ to be the open half space of positive rank Chern characters and the secondary half space to be the closed half space of negative rank and rank zero Chern characters in $\unicode[STIX]{x1D709}^{\bot }$ . Similarly, define the primary and secondary halves of $\operatorname{NS}(M(\unicode[STIX]{x1D709}))$ as the images of these under the isomorphism $\unicode[STIX]{x1D706}_{M}$ . Every extremal ray of the effective and nef cones sits in one of our halves. Call an extremal ray of the effective or nef cone primary or secondary according to which half it lies in.

We know that the ample cone of $M(\unicode[STIX]{x1D709})$ lies in the primary half space because ${\mathcal{L}}_{0}$ lies in that half space.

2.9 Brill–Noether divisors

Now that we have constructed a basis for the Picard space, we discuss the divisors that we construct to span the effective cone. These divisors are examples of Brill–Noether divisors. A Brill–Noether locus in general is the place where the rank of some cohomology group jumps.

Proposition 2.16. Suppose $V\in M(\unicode[STIX]{x1D701})$ is a stable vector bundle and is cohomologically orthogonal to the general $U\in M(\unicode[STIX]{x1D709})$ . Put the natural determinantal scheme structure on the locus

$$\begin{eqnarray}D_{V}=\overline{\{U\in M^{s}(\unicode[STIX]{x1D709}):h^{1}(U\otimes V)\neq 0\}}.\end{eqnarray}$$

1. (1) $D_{V}$ is an effective divisor.

2. (2) If $\unicode[STIX]{x1D707}_{1,1}(U\otimes V)>-4$ , then ${\mathcal{O}}_{M(\unicode[STIX]{x1D709})}(D_{V})\cong \unicode[STIX]{x1D706}_{M}(\unicode[STIX]{x1D701})$ .

3. (3) If $\unicode[STIX]{x1D707}_{1,1}(U\otimes V)<0$ , then ${\mathcal{O}}_{M(\unicode[STIX]{x1D709})}(D_{V})\cong \unicode[STIX]{x1D706}_{M}(\unicode[STIX]{x1D701})^{\ast }\cong \unicode[STIX]{x1D706}_{M}(-\unicode[STIX]{x1D701})$ .

Given a Brill–Noether divisor, a natural question to ask is whether it lies in the primary or the secondary half of $\operatorname{NS}(M(\unicode[STIX]{x1D709}))$ . The answer is immediate from the computation of the class of the Brill–Noether divisors.

Note that since $\unicode[STIX]{x1D707}_{1,1}(U\otimes V)$ cannot be between $-4$ and $0$ there is no contradiction in these results.

2.10 The interpolation problem

As we previously mentioned, in order to construct a Brill–Noether divisor, it is necessary to find a sheaf $V$ for which $h^{1}(U\otimes V)=0$ for the general $U\in M(\unicode[STIX]{x1D709})$ and the easiest way to do so is to find a sheaf that is cohomologically orthogonal to $U$ . Cohomological orthogonality implies that $\unicode[STIX]{x1D712}(U\otimes V)=0$ . The vanishing of that Euler characteristic is a strictly weaker condition. For example, it might be the case that $h^{0}=h^{1}=1$ and $h^{2}=0$ . We would like an added condition which would make them equivalent. A bundle $V$ is nonspecial with respect to $U$ if $\unicode[STIX]{x1D712}(U\otimes V)$ determines the cohomology groups (i.e., they are lowest ranks allowed by the Euler characteristic). We can rephrase cohomological orthogonality as $V$ having $\unicode[STIX]{x1D712}(U\otimes V)=0$ and being nonspecial with respect to $U$ . In general, there are many such sheaves, but those that interest us are those of “minimum slope”. Finding a sheaf like this is a form of the interpolation problem.

Note, if we restrict our interest to line bundles on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ and $M(\unicode[STIX]{x1D709})=\left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)^{[m]}$ , this is the classical interpolation for points lying on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ : find the “lowest” bidegree $(a,b)$ such that $m$ points lie on a curve of type $(a,b)$ .

By construction of the Brill–Noether divisors, any solution of the interpolation problem gives an effective Brill–Noether divisor. Our goal is to give a method to construct Brill–Noether divisors which are sufficient to span the effective cone in many examples. We show that they span in examples by providing an alternate construction of the divisors. This description provides dual moving curves which prove that they are extremal divisors in the effective cone.

The alternate construction starts by using the cohomological vanishing guaranteed by these Brill–Noether divisors to resolve the general objects of $M(\unicode[STIX]{x1D709})$ . In general, we then use these resolutions to construct maps from $M(\unicode[STIX]{x1D709})$ to Picard rank one varieties that have positive dimensional fibers. The Brill–Noether divisors match up with the pull backs of an ample divisor on these Picard rank one varieties. We prove the extremality of each divisor using the dual moving curves that cover the fibers of the morphism.

2.11 Kronecker moduli spaces

The simpler varieties that we map to are moduli spaces of Kronecker modules.

The expected dimension of this space is

$$\begin{eqnarray}(\operatorname{edim})Kr_{N}(a,b)=1-\unicode[STIX]{x1D712}(r,r)=Nab-a^{2}-b^{2}+1.\end{eqnarray}$$

For Kronecker moduli spaces, we also know that they are nonempty and of the expected dimension if their expected dimension is nonnegative [Reference ReinekeRei]. Another fact about Kronecker moduli spaces is that they are Picard rank one [Reference DrézetDré1]. We use this fact later when we create effective divisors on our moduli spaces $M(\unicode[STIX]{x1D709})$ by constructing fibrations from them to these Kronecker moduli spaces and pulling back a generator of the ample cone of $Kr_{N}(a,b)$ . In order to do this, we need to get Kronecker modules from maps between exceptional bundles.

3.1 Bundles with vanishing Euler characteristic

First, we find bundles $V$ with $\unicode[STIX]{x1D712}(U\otimes V)=0$ . As we are trying to compute the effective cone, scaling Chern characters is relatively unimportant. In particular, we can scale a Chern character so that $\operatorname{ch}_{0}=1$ (unless it was 0 to start).

Definition 3.1. When a Chern character $\unicode[STIX]{x1D709}$ has positive rank, the orthogonal surface to $\unicode[STIX]{x1D709}$ is

$$\begin{eqnarray}Q_{\unicode[STIX]{x1D709}}=\{(\unicode[STIX]{x1D707},\unicode[STIX]{x1D6E5}):(1,\unicode[STIX]{x1D707},\unicode[STIX]{x1D6E5})\text{ lies in }\unicode[STIX]{x1D709}^{\bot }\}\subset \mathbb{R}^{3}.\end{eqnarray}$$

Using the orthogonal surface rather than the full $\unicode[STIX]{x1D709}^{\bot }$ has the advantage of working in the three dimensional $\left(\unicode[STIX]{x1D707},\unicode[STIX]{x1D6E5}\right)$ -space instead of in the full four dimensional $K(\mathbb{P}^{1}\times \mathbb{P}^{1})$ . We define the reference surface, $Q_{\unicode[STIX]{x1D709}_{0}}$ , to be the orthogonal surface to $\unicode[STIX]{x1D709}_{0}=\operatorname{ch}({\mathcal{O}})=(1,(0,0),0)$ .

Using Hirzebruch–Riemann–Roch to compute the equation of the orthogonal surface to $\unicode[STIX]{x1D709}$ gives the formula

$$\begin{eqnarray}Q_{\unicode[STIX]{x1D709}}:P(\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})+\unicode[STIX]{x1D707})-\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D6E5}\end{eqnarray}$$

where $P(x,y)=(x+1)(y+1)$ . This equation defines a saddle surface that is a shift of the reference surface. It has unique saddle point at the point $\left(-1-x_{0},-1-y_{0}\right)$ where $(x_{0},y_{0})=\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709})$ . Consequently, any two such surfaces intersect in a parabola that lies over a line in the slope plane.

Using this language, the condition that $\unicode[STIX]{x1D712}(U\otimes V)=0$ can be rephrased as saying that $V$ must lie on $Q_{\unicode[STIX]{x1D709}}$ . In other words, we can restrict our search for solutions to the interpolation problem to bundles that lie on the orthogonal surface (to $\unicode[STIX]{x1D709}$ ).

As a side note, we can now give an alternative description of each fractal part of the $\unicode[STIX]{x1D6FF}$ surface using orthogonal surfaces. Given an exceptional bundle $E_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FF}_{E}$ can equivalently be written as

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{E}(\unicode[STIX]{x1D707})=\left\{\begin{array}{@{}ll@{}}Q_{E^{\ast }}(\unicode[STIX]{x1D707}) & \text{if }\unicode[STIX]{x1D707}_{1,1}(E)-4<\unicode[STIX]{x1D707}_{1,1}(\unicode[STIX]{x1D707})\text{ and }\overline{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D707})<\overline{\unicode[STIX]{x1D6FE}}(E),\\ Q_{E^{\ast }(K)}(\unicode[STIX]{x1D707}) & \text{if }\unicode[STIX]{x1D707}_{1,1}(\unicode[STIX]{x1D707})<\unicode[STIX]{x1D707}_{1,1}(E)+4\text{ and }\overline{\unicode[STIX]{x1D6FE}}(E)<\overline{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D707}).\end{array}\right.\end{eqnarray}$$

Thus, every saddle subsurface of the surface $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D707})=\unicode[STIX]{x1D6E5}$ can be seen to be a portion of some surface $Q_{E}$ .

3.2 Cohomologically nonspecial bundles

Now that we have found the bundles with $\unicode[STIX]{x1D712}(U\otimes V)=0$ , we would like to impose the second condition of cohomological orthogonality, being nonspecial. In some cases, we can find numerical conditions defining being nonspecialty as well. Those conditions are in terms of certain exceptional bundles that we pick out. Picking them out begins with studying $Q_{\unicode[STIX]{x1D709}}$ again.

Note that $Q_{\unicode[STIX]{x1D709}}$ intersects the plane $\unicode[STIX]{x1D6E5}=\frac{1}{2}$ . The exceptional bundles that we want to pick out are those exceptional bundles that control this intersection.

As promised above, each controlling exceptional bundle provides a necessary condition for a stable Chern character to be nonspecial with respect to the general element of $M(\unicode[STIX]{x1D709})$ . The condition that a controlling exceptional bundles imposes on a Chern character is that the Chern character must be on or above a surface corresponding to that exceptional bundle.

Definition 3.3. The corresponding surface to an exceptional bundle $E_{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D709}$ is defined as

$$\begin{eqnarray}Q_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D708})=\left\{\begin{array}{@{}ll@{}}Q_{E_{\unicode[STIX]{x1D6FC}}^{\ast }}(\unicode[STIX]{x1D708}) & \text{if }\unicode[STIX]{x1D712}\left(E_{\unicode[STIX]{x1D6FC}}^{\ast },U\right)>0,\\ Q_{E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)}(\unicode[STIX]{x1D708}) & \text{if }\unicode[STIX]{x1D712}\left(E_{\unicode[STIX]{x1D6FC}}^{\ast },U\right)<0.\end{array}\right.\end{eqnarray}$$

Then in some, if not all, cases, the Chern characters $\unicode[STIX]{x1D708}$ such that $Q_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D708})>0$ for all controlling bundles $E_{\unicode[STIX]{x1D6FC}}$ of $\unicode[STIX]{x1D709}$ are precisely the nonspecial Chern characters with respect to $\unicode[STIX]{x1D709}$ .

3.3 Potential extremal rays

The solutions to interpolation are the intersection of the orthogonal surface and the maximum of the corresponding surfaces. Each part of this intersection is where the orthogonal surface intersects a corresponding surface, that is, where $Q_{\unicode[STIX]{x1D709}}=Q_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D709}}$ for a controlling exceptional $E_{\unicode[STIX]{x1D6FC}}$ of $\unicode[STIX]{x1D709}$ . The corners of the intersection are where the orthogonal surface intersects two different corresponding surfaces, that is, where $Q_{\unicode[STIX]{x1D709}}=Q_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D709}}=Q_{\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D709}}$ for controlling exceptionals $E_{\unicode[STIX]{x1D6FC}}$ and $E_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D709}$ .

As $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FD}$ , and $\unicode[STIX]{x1D709}$ are three linearly independent rational Chern characters, $Q_{\unicode[STIX]{x1D709}}\cap Q_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D709}}\cap Q_{\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D709}}$ is a single point that corresponds to the intersection of the 3 planes $\unicode[STIX]{x1D6FC}^{\bot }\cap \unicode[STIX]{x1D6FD}^{\bot }\cap \unicode[STIX]{x1D6FE}^{\bot }$ , $(\unicode[STIX]{x1D6FC}+K)^{\bot }\cap \unicode[STIX]{x1D6FD}^{\bot }\cap \unicode[STIX]{x1D6FE}^{\bot }$ , $\unicode[STIX]{x1D6FC}^{\bot }\cap (\unicode[STIX]{x1D6FD}+K)^{\bot }\cap \unicode[STIX]{x1D6FE}^{\bot }$ , or $(\unicode[STIX]{x1D6FC}+K)^{\bot }\cap (\unicode[STIX]{x1D6FD}+K)^{\bot }\cap \unicode[STIX]{x1D6FE}^{\bot }$ . We determine which intersection it is by which cases of $Q_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D709}}$ and $Q_{\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D709}}$ we are using. We want to find the corners made in this way in order to get effective divisors. To find those divisors, we first define all of the possible triple intersections that might work.

(2) and (3) can occur simultaneously, but you can treat them individually so we say “the orthogonal point”.

Some of the corresponding orthogonal points are not actually what we want, as they can both fail to satisfy interpolation (by being below one of the other corresponding surfaces) or by not being extremal (as they have a slope that is not linearly independent of other solutions). We want to pick out the only the solutions that satisfy interpolation and are extremal among those solutions.

We now want to turn these points back into Chern characters for convenience.

We call them potential because we see in the next section that we need a few additional conditions to make sure that they actually span an extremal ray of the effective cone. We give an approach to showing that some of the potential primary orthogonal Chern characters span the solutions of the interpolation problem and that the effective cone of $M(\unicode[STIX]{x1D709})$ is spanned by the Brill–Noether divisors $D_{V}$ for many examples of $M(\unicode[STIX]{x1D709})$ where $V$ are bundles whose Chern character is an orthogonal Chern character $\unicode[STIX]{x1D709}^{+}$ .

The behavior of these extremal rays depends on the sign of $\unicode[STIX]{x1D712}(E_{\unicode[STIX]{x1D6FC}}^{\ast },U)$ and $\unicode[STIX]{x1D712}(E_{\unicode[STIX]{x1D6FD}}^{\ast },U)$ . Keeping the identification of $\operatorname{NS}\left(M(\unicode[STIX]{x1D709})\right)\cong \unicode[STIX]{x1D709}^{\bot }$ in mind, recall that the primary half of the space corresponds to characters of positive rank.

1. (1) If $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D709})>0$ and $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D709})>0$ , the ray is spanned by a positive rank Chern character in $\unicode[STIX]{x1D709}^{\bot }\cap \unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FC}}^{\bot }\cap \unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FD}}^{\bot }$ .

2. (2) If $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D709})<0$ and $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D709})>0$ , the ray is spanned by a positive rank Chern character in $\unicode[STIX]{x1D709}^{\bot }\cap \left(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FC}+K}\right)^{\bot }\cap \unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FD}}^{\bot }$ .

3. (3) If $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D709})<0$ and $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D709})<0$ , the ray is spanned by a positive rank Chern character in $\unicode[STIX]{x1D709}^{\bot }\cap \left(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FC}+K}\right)^{\bot }\cap \left(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FD}+K}\right)^{\bot }$ .

4. (4) If $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D709})=0$ or $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D709}_{-\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D709})=0$ , the ray is spanned by $\unicode[STIX]{x1D6FC}$ or $\unicode[STIX]{x1D6FD}$ , respectively.

4.1 The “mixed type” spectral sequence

Assume $\unicode[STIX]{x1D712}\left(E_{-\unicode[STIX]{x1D6FC}},U\right)\leqslant 0$ and $\unicode[STIX]{x1D712}\left(E_{-\unicode[STIX]{x1D6FD}},U\right)\geqslant 0$ . Let the right mutated coil of $\left(E_{-\unicode[STIX]{x1D6FC}}(K),F_{0}^{\ast },F_{-1}^{\ast },E_{-\unicode[STIX]{x1D6FD}}\right)$ be $\left(E_{-\unicode[STIX]{x1D6FD}},E_{-1}^{\ast },E_{-2}^{\ast },E_{-\unicode[STIX]{x1D6FC}}\right)$ . Let $\unicode[STIX]{x1D6E5}_{i}$ be as in the spectral sequence with input these two coils.

Given an extremal pair, we can resolve the general object of $M(\unicode[STIX]{x1D709})$ .

Theorem 4.4. The general $U\in M(\unicode[STIX]{x1D709})$ admits a resolution of the following form

$$\begin{eqnarray}0\rightarrow E_{-\unicode[STIX]{x1D6FC}}(K)^{m_{3}}\bigoplus \left(F_{0}^{\ast }\right)^{m_{2}}\stackrel{\unicode[STIX]{x1D719}}{\rightarrow }\left(F_{-1}^{\ast }\right)^{m_{1}}\bigoplus E_{-\unicode[STIX]{x1D6FD}}^{m_{0}}\rightarrow U\rightarrow 0.\end{eqnarray}$$

Proof. Consider a bundle $U$ defined by the sequence

$$\begin{eqnarray}0\rightarrow E_{-\unicode[STIX]{x1D6FC}}(K)^{m_{3}}\bigoplus \left(F_{0}^{\ast }\right)^{m_{2}}\stackrel{\unicode[STIX]{x1D719}}{\rightarrow }\left(F_{-1}^{\ast }\right)^{m_{1}}\bigoplus E_{-\unicode[STIX]{x1D6FD}}^{m_{0}}\rightarrow U\rightarrow 0\end{eqnarray}$$

where the map $\unicode[STIX]{x1D719}\in \operatorname{Hom}(E_{-\unicode[STIX]{x1D6FC}}(K)^{m_{3}}\bigoplus \left(F_{0}^{\ast }\right)^{m_{2}}\!,\left(F_{-1}^{\ast }\right)^{m_{1}}\bigoplus E_{-\unicode[STIX]{x1D6FD}}^{m_{0}})$ is general.

The proof proceeds in 4 steps: calculate that $\operatorname{ch}(U)=\unicode[STIX]{x1D709}$ , show $\unicode[STIX]{x1D719}$ is injective, confirm the expected vanishings in the spectral sequence, and prove that $U$ is stable.

Step 1. Calculate $\operatorname{ch}(U)=\unicode[STIX]{x1D709}$ . We do not know if $\unicode[STIX]{x1D719}$ is injective yet, but we can compute the Chern character of the mapping cone of $\unicode[STIX]{x1D719}$ in the derived category. Assuming $\unicode[STIX]{x1D719}$ is injective, this computes the Chern character of $U$ .

This computation follows from the generalized Beilinson spectral sequence’s convergence to $U$ . Specifically, we have a spectral sequence with $E_{1}$ -page

$$\begin{eqnarray}\displaystyle E_{-\unicode[STIX]{x1D6FC}}(K)\otimes \mathbb{C}^{m_{32}} & \rightarrow & \displaystyle F_{0}^{\ast }\otimes \mathbb{C}^{m_{22}}\rightarrow F_{-1}^{\ast }\otimes \mathbb{C}^{m_{12}}\rightarrow 0\nonumber\\ \displaystyle E_{-\unicode[STIX]{x1D6FC}}(K)\otimes \mathbb{C}^{m_{31}} & \rightarrow & \displaystyle F_{0}^{\ast }\otimes \mathbb{C}^{m_{21}}\rightarrow F_{-1}^{\ast }\otimes \mathbb{C}^{m_{11}}\rightarrow E_{-\unicode[STIX]{x1D6FD}}\otimes \mathbb{C}^{m_{01}}\nonumber\\ \displaystyle 0 & \rightarrow & \displaystyle F_{0}^{\ast }\otimes \mathbb{C}^{m_{20}}\rightarrow F_{-1}^{\ast }\otimes \mathbb{C}^{m_{10}}\rightarrow E_{-\unicode[STIX]{x1D6FD}}\otimes \mathbb{C}^{m_{00}}\nonumber\end{eqnarray}$$

that converges in degree 0 to the sheaf $U$ and to 0 in all other degrees. We omitted all nonzero rows and columns. In particular, row 3 is zero by the vanishing of $h^{2}$ for all the sheaves (follows from Definition 4.3). Also note that either the top or bottom element of each of the middle two rows vanishes (depending on the value of $\unicode[STIX]{x1D6E5}_{1}$ and $\unicode[STIX]{x1D6E5}_{2}$ ). An easy computation shows that

$$\begin{eqnarray}\displaystyle \operatorname{ch}(U) & = & \displaystyle -(m_{32}-m_{31})\operatorname{ch}(E_{-\unicode[STIX]{x1D6FC}}(K))+(m_{22}-m_{21}+m_{20})\operatorname{ch}(F_{0}^{\ast })\nonumber\\ \displaystyle & & \displaystyle -\,(m_{12}-m_{11}+m_{10})\operatorname{ch}(F_{-1}^{\ast })+(m_{00}-m_{01})\operatorname{ch}(E_{-\unicode[STIX]{x1D6FD}}).\nonumber\end{eqnarray}$$

In our situation, we see that this gives

$$\begin{eqnarray}\unicode[STIX]{x1D709}=-m_{-3}\operatorname{ch}(E_{-\unicode[STIX]{x1D6FC}}(K))+m_{-2}\operatorname{ch}(F_{0}^{\ast })-m_{-1}\operatorname{ch}(F_{-1}^{\ast })+m_{0}\operatorname{ch}(E_{-\unicode[STIX]{x1D6FD}})\end{eqnarray}$$

where the $m_{i}$ are defined in the obvious way.

Step 2. Show $\unicode[STIX]{x1D719}$ is injective. The sheaves $\mathscr{H}\!om(E_{-\unicode[STIX]{x1D6FC}}(K),F_{-1}^{\ast })$ , $\mathscr{H}\!om(E_{-\unicode[STIX]{x1D6FC}}(K),E_{-\unicode[STIX]{x1D6FD}})$ , $\mathscr{H}\!om(F_{0}^{\ast },F_{-1}^{\ast })$ , and $\mathscr{H}\!om(F_{0}^{\ast },E_{-\unicode[STIX]{x1D6FD}})$ are all globally generated by the controlling pair being an extremal pair. Those bundles being globally generated immediately implies that

$$\begin{eqnarray}\mathscr{H}\!om\left(E_{-\unicode[STIX]{x1D6FC}}(K)^{m_{3}}\bigoplus \left(F_{0}^{\ast }\right)^{m_{2}}\!,\left(F_{-1}^{\ast }\right)^{m_{1}}\bigoplus E_{-\unicode[STIX]{x1D6FD}}^{m_{0}}\right)\end{eqnarray}$$

is globally generated as well. Using a Bertini-type theorem [Reference HuizengaHui2, Prop. 2.6] and the fact that the virtually computed rank of $U$ is positive, we see that $\unicode[STIX]{x1D719}$ is injective.

Step 3. Verify $U$ ’s spectral sequence has the correct vanishings. We know that $\unicode[STIX]{x1D712}(E_{-\unicode[STIX]{x1D6FC}},U)\leqslant 0$ and $\unicode[STIX]{x1D712}(E_{-\unicode[STIX]{x1D6FD}},U)\geqslant 0$ . We also know that ( $\unicode[STIX]{x1D712}(E_{-2}^{\ast },U)\leqslant 0$ and $\unicode[STIX]{x1D6E5}_{2}=0$ ) or ( $\unicode[STIX]{x1D712}(E_{-2}^{\ast },U)\geqslant 0$ and $\unicode[STIX]{x1D6E5}_{2}=1$ ). Analogously, we know that ( $\unicode[STIX]{x1D712}(E_{-1}^{\ast },U)\leqslant 0$ and $\unicode[STIX]{x1D6E5}_{1}=0$ ) or ( $\unicode[STIX]{x1D712}(E_{-1}^{\ast },U)\geqslant 0$ and $\unicode[STIX]{x1D6E5}_{1}=1$ ).

Since we know that all of the groups other than possibly $\operatorname{ext}^{1}$ and $\hom$ vanish, it is enough to check that

$$\begin{eqnarray}\hom (E_{-\unicode[STIX]{x1D6FC}},U)=\operatorname{ext}^{\unicode[STIX]{x1D6E5}_{1}}(E_{-1}^{\ast },U)=\operatorname{ext}^{\unicode[STIX]{x1D6E5}_{2}}(E_{-2}^{\ast },U)=\operatorname{ext}^{1}(E_{-\unicode[STIX]{x1D6FD}},U)=0.\end{eqnarray}$$

These vanishings for the specific $U$ we have resolved follow from the orthogonality properties of exceptional bundles and the relevant long exact sequences. Once they vanish for a specific $U$ , they vanish for a general $U$ as needed. We show these four vanishings in order.

First, $\hom (E_{-\unicode[STIX]{x1D6FC}},F_{-1}^{\ast })$ , $\hom (E_{-\unicode[STIX]{x1D6FC}},E_{-\unicode[STIX]{x1D6FD}})$ , and $\operatorname{ext}^{1}(E_{-\unicode[STIX]{x1D6FC}},F_{0}^{\ast })$ all vanish since $\left(F_{0}^{\ast },F_{-1}^{\ast },E_{-\unicode[STIX]{x1D6FD}},E_{-\unicode[STIX]{x1D6FC}}\right)$ is a coil and $\operatorname{ext}^{1}(E_{-\unicode[STIX]{x1D6FC}},E_{-\unicode[STIX]{x1D6FC}}(K))$ is equal to $\operatorname{ext}^{1}(E_{-\unicode[STIX]{x1D6FC}}(K),E_{-\unicode[STIX]{x1D6FC}}(K))$ which vanishes since exceptional bundles are rigid. This gives $\hom (E_{-\unicode[STIX]{x1D6FC}},U)=0$ as desired.

For the next vanishing, we have two cases: $\unicode[STIX]{x1D6E5}_{1}=0$ and $\unicode[STIX]{x1D6E5}_{1}=1$ . We assume that $\unicode[STIX]{x1D6E5}_{1}=0$ , and the other case is similar. Assuming $\unicode[STIX]{x1D6E5}_{1}=0$ , we need to show that $\hom (E_{-1}^{\ast },U)=0$ . Next, $\operatorname{ext}^{1}(E_{-1}^{\ast },E_{-\unicode[STIX]{x1D6FC}}(K))$ , $\hom (E_{-1}^{\ast },E_{-\unicode[STIX]{x1D6FD}})$ , and $\operatorname{ext}^{1}(E_{-1}^{\ast },F_{0}^{\ast })$ all vanish since $\left(E_{-\unicode[STIX]{x1D6FC}}(K),F_{0}^{\ast },E_{-\unicode[STIX]{x1D6FD}},E_{-1}^{\ast }\right)$ is a coil. Then the only remaining vanishing we need to prove is $\hom (E_{-1}^{\ast },F_{-1}^{\ast })$ . By assumption on $\unicode[STIX]{x1D6E5}_{1}$ , we can write $0\rightarrow F_{-1}^{\ast }\rightarrow E_{-\unicode[STIX]{x1D6FD}}^{a}\rightarrow E_{-1}^{\ast }\rightarrow 0$ where $a=\unicode[STIX]{x1D712}(E_{-\unicode[STIX]{x1D6FD}},E_{-1})$ . From this resolution, we see $\hom (E_{-1}^{\ast },F_{-1}^{\ast }){\hookrightarrow}\hom (E_{-1}^{\ast },E_{-\unicode[STIX]{x1D6FD}})=0$ , and the vanishing follows.

For the next vanishing, we have two cases: $\unicode[STIX]{x1D6E5}_{1}=0$ and $\unicode[STIX]{x1D6E5}_{1}=1$ . We assume that $\unicode[STIX]{x1D6E5}_{1}=0$ , and the other case is similar. Assuming $\unicode[STIX]{x1D6E5}_{2}=0$ , we need to show that $\hom (E_{-2}^{\ast },U)=0$ . Then, $\operatorname{ext}^{1}(E_{-2}^{\ast },E_{-\unicode[STIX]{x1D6FC}}(K))$ , $\hom (E_{-2}^{\ast },E_{-\unicode[STIX]{x1D6FD}})$ , and $\hom (E_{-2}^{\ast },F_{-1}^{\ast })$ all vanish since $\left(E_{-\unicode[STIX]{x1D6FC}}(K),F_{-1}^{\ast },E_{-\unicode[STIX]{x1D6FD}},E_{-2}^{\ast }\right)$ is a coil. Then the only remaining vanishing we need to prove is $\operatorname{ext}^{1}(E_{-2}^{\ast },F_{0}^{\ast })$ . By assumption on $\unicode[STIX]{x1D6E5}_{2}$ , we can write $0\rightarrow F_{0}^{\ast }\rightarrow E_{-\unicode[STIX]{x1D6FD}}^{a}\rightarrow L_{E_{-1}^{\ast }}E_{-2}^{\ast }\rightarrow 0$ where $a=\unicode[STIX]{x1D712}(E_{-\unicode[STIX]{x1D6FD}},L_{E_{-1}^{\ast }}E_{-2}^{\ast })$ . From this resolution we see that $\operatorname{ext}^{1}(E_{-2}^{\ast },F_{0}^{\ast })\cong \hom (E_{-2}^{\ast },L_{E_{-1}^{\ast }}E_{-2}^{\ast })$ since $\hom (E_{-2}^{\ast },E_{-\unicode[STIX]{x1D6FD}})=\operatorname{ext}^{1}(E_{-2}^{\ast },E_{-\unicode[STIX]{x1D6FD}})=0$ . Thus, we reduce to showing that $\hom (E_{-2}^{\ast },L_{E_{-1}^{\ast }}E_{-2}^{\ast })=0$ . Again, by assumption on $\unicode[STIX]{x1D6E5}_{2}$ , we know that we can write $0\rightarrow L_{E_{-1}^{\ast }}E_{-2}^{\ast }\rightarrow (E_{-1}^{\ast })^{a}\rightarrow E_{-2}^{\ast }\rightarrow 0$ from which we can easily see $\hom (E_{-2}^{\ast },L_{E_{-1}^{\ast }}E_{-2}^{\ast }){\hookrightarrow}\hom (E_{-2}^{\ast },E_{-1}^{\ast })=0$ , and the vanishing follows.

Finally, we show $\operatorname{ext}^{1}(E_{-\unicode[STIX]{x1D6FD}},U)=0$ . Then $\operatorname{ext}^{1}(E_{-\unicode[STIX]{x1D6FD}},F_{-1}^{\ast })$ , $\operatorname{ext}^{2}(E_{-\unicode[STIX]{x1D6FD}},E_{-\unicode[STIX]{x1D6FC}}(K))$ , and $\operatorname{ext}^{2}(E_{-\unicode[STIX]{x1D6FD}},F_{0}^{\ast })$ all vanish since $(E_{-\unicode[STIX]{x1D6FC}}(K),F_{0}^{\ast },F_{-1}^{\ast },E_{-\unicode[STIX]{x1D6FD}})$ is a coil while $\operatorname{ext}^{1}(E_{-\unicode[STIX]{x1D6FD}},E_{-\unicode[STIX]{x1D6FD}})=0$ since exceptional bundles are rigid. This gives the vanishing of $\operatorname{ext}^{1}(E_{-\unicode[STIX]{x1D6FD}},U)$ .

Step 4. Prove that $U$ is stable. Let

$$\begin{eqnarray}S\subset \operatorname{Hom}\left(E_{-\unicode[STIX]{x1D6FC}}(K)^{m_{3}}\bigoplus \left(F_{0}^{\ast }\right)^{m_{2}}\!,\left(F_{-1}^{\ast }\right)^{m_{1}}\bigoplus E_{-\unicode[STIX]{x1D6FD}}^{m_{0}}\right)\end{eqnarray}$$

be the open subset of sheaf maps that are injective and have torsion-free cokernels.

By the argument of 5.3 of [Reference Coskun, Huizenga and WoolfCHW], it follows that $S$ is nonempty.

Consider the family $^{{\mathcal{U}}}\!/_{S}$ of quotients parameterized by $S$ . We need to show that ${\mathcal{U}}$ is a complete family of prioritary sheaves. Recall that a prioritary sheaf is a torsion-free sheaf $U$ such that $\operatorname{Ext}^{2}(U,U(0,-1))=0$ or $\operatorname{Ext}^{2}(U,U(-1,0))=0$ or, equivalently in our case, that $\operatorname{Hom}(U,U(-1,-2))=0$ or $\operatorname{Hom}(U,U(-2,-1))=0$ . By Definition 4.3, the elements of ${\mathcal{U}}$ are prioritary. Again by the general argument of [Reference Coskun, Huizenga and WoolfCHW, 5.3], the family is a complete family. By [Reference WalterWal, Thm. 1], the Artin stack of prioritary sheaves with Chern character $\unicode[STIX]{x1D709}$ is an irreducible stack that contains the stack of semistable sheaves with Chern character $\unicode[STIX]{x1D709}$ as a dense open subset. It is then clear that $S$ parameterizes the general sheaves in $M(\unicode[STIX]{x1D709})$ .

4.2 The “negative type” spectral sequence

Assume $\unicode[STIX]{x1D712}\left(E_{-\unicode[STIX]{x1D6FC}},U\right)\leqslant 0$ and $\unicode[STIX]{x1D712}\left(E_{-\unicode[STIX]{x1D6FD}},U\right)\leqslant 0$ . Let the right mutated coil of $\left(E_{-\unicode[STIX]{x1D6FD}}(K),E_{-\unicode[STIX]{x1D6FC}}(K),F_{0}^{\ast },F_{-1}^{\ast }\right)$ be $\left(F_{-1}^{\ast },F_{-2}^{\ast },E_{-\unicode[STIX]{x1D6FE}},E_{-\unicode[STIX]{x1D6FD}}\right)$ . Let $\unicode[STIX]{x1D6E5}_{i}$ be as in the spectral sequence with input these two coils.

Given an extremal pair, we can resolve the general object of $M(\unicode[STIX]{x1D709})$ .

Theorem 4.6. The general $U\in M(\unicode[STIX]{x1D709})$ admits a resolution of the following forms

$$\begin{eqnarray}0\rightarrow E_{-\unicode[STIX]{x1D6FD}}(K)^{m_{3}}\bigoplus E_{-\unicode[STIX]{x1D6FC}}(K)^{m_{2}}\bigoplus \left(F_{0}^{\ast }\right)^{m_{1}}\rightarrow \left(F_{-1}^{\ast }\right)^{m_{0}}\rightarrow U\rightarrow 0.\end{eqnarray}$$

As the proof is similar to the proof of the previous theorem, we omit its proof.

4.3 The “positive type” spectral sequence

Assume $\unicode[STIX]{x1D712}\left(E_{-\unicode[STIX]{x1D6FC}},U\right)\geqslant 0$ and $\unicode[STIX]{x1D712}\left(E_{-\unicode[STIX]{x1D6FD}},U\right)\geqslant 0$ . Let the right mutated coil of $\left(F_{0}^{\ast },F_{-1}^{\ast },E_{-\unicode[STIX]{x1D6FD}},E_{-\unicode[STIX]{x1D6FC}}\right)$ be $\left(E_{-\unicode[STIX]{x1D6FC}},E_{-\unicode[STIX]{x1D6FE}},F_{1}^{\ast }(-K),F_{0}^{\ast }(-K)\right)$ . Let $\unicode[STIX]{x1D6E5}_{i}$ be as in the spectral sequence with input these two coils.

Given an extremal pair, we can resolve the general object of $M(\unicode[STIX]{x1D709})$ .

Theorem 4.8. The general $U\in M(\unicode[STIX]{x1D709})$ admits a resolution of the following forms

$$\begin{eqnarray}0\rightarrow \left(F_{0}^{\ast }\right)^{m_{3}}\rightarrow \left(F_{-1}^{\ast }\right)^{m_{2}}\bigoplus E_{-\unicode[STIX]{x1D6FD}}^{m_{1}}\bigoplus E_{-\unicode[STIX]{x1D6FC}}^{m_{0}}\rightarrow U\rightarrow 0.\end{eqnarray}$$

As the proof is similar to the proof of the second previous theorem, we omit its proof.

5.1 The case when two powers vanish

First note that the cases $m_{3}=m_{2}=0$ and $m_{1}=m_{0}=0$ cannot occur due to the form of the spectral sequence. This leaves four cases where two exponents vanish to deal with. The proof of the proposition is identical in each case after you replace the two bundles with nonzero exponents so we only explicitly prove the proposition in the first case.

5.1.1 When the second and third powers vanish

For this subsubsection, assume that $m_{2}=0$ and $m_{1}=0$ in the resolution

$$\begin{eqnarray}0\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\bigoplus F_{0}^{\ast m_{2}}\stackrel{\unicode[STIX]{x1D719}}{\rightarrow }F_{-1}^{\ast m_{1}}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U\rightarrow 0.\end{eqnarray}$$

We see that $U$ determines a complex of the form

$$\begin{eqnarray}W:E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\rightarrow E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\end{eqnarray}$$

which in turn determines a Kronecker $\operatorname{Hom}(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K),E_{\unicode[STIX]{x1D6FD}}^{\ast })$ -module.

Conversely, given a general such module and its determined complex, $W^{\prime }$ , there exists an element $U^{\prime }\in M(\unicode[STIX]{x1D709})$ such that $W^{\prime }$ is the associated complex of $U^{\prime }$ by Theorem 4.4. Assuming that the Kronecker module associated to a general $W$ is semistable, this constructs a rational map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:M(\unicode[STIX]{x1D709}){\dashrightarrow}Kr_{N}(m_{3},m_{0})\end{eqnarray}$$

where $N=\hom (E_{\unicode[STIX]{x1D6FC}}^{\ast }(K),E_{\unicode[STIX]{x1D6FD}}^{\ast })$ . In order to show that the Kronecker module associated to a general $W$ is semistable, it suffices by Section 2.11 to show that $Kr_{N}(m_{3},m_{0})$ is positive dimensional.

Proposition 5.2. With the above notation, $Kr_{N}(m_{3},m_{0})$ is positive dimensional, and the dominant rational map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:M(\unicode[STIX]{x1D709}){\dashrightarrow}Kr_{N}(m_{3},m_{0})\end{eqnarray}$$

is a birational map.

Proof. By construction, the primary orthogonal Chern characters to $\unicode[STIX]{x1D709}$ are all semistable so $\unicode[STIX]{x1D709}^{+}$ is semistable. By assumption, the general $U\in M(\unicode[STIX]{x1D709})$ has the resolution

$$\begin{eqnarray}0\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\rightarrow E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U\rightarrow 0.\end{eqnarray}$$

A general map in a complex of the form

$$\begin{eqnarray}W:E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\rightarrow E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\end{eqnarray}$$

is injective and has a semistable cokernel with Chern character $\unicode[STIX]{x1D709}$ by Theorem 4.4. We also know that any isomorphism of two general elements of $M(\unicode[STIX]{x1D709})$ is induced by an isomorphism of their resolutions.

Recall from earlier in the section that $W$ corresponds to a Kronecker $\hom (E_{\unicode[STIX]{x1D6FC}}^{\ast }(K),E_{\unicode[STIX]{x1D6FD}}^{\ast })$ -module $e$ with dimension vector $(m_{3},m_{0})$ . Then we compute that

$$\begin{eqnarray}\dim (M(\unicode[STIX]{x1D709}))=1-\unicode[STIX]{x1D712}(U,U)=1-\unicode[STIX]{x1D712}(e,e)=(\operatorname{edim})Kr_{N}(m_{3},m_{0}).\end{eqnarray}$$

As $\dim (M(\unicode[STIX]{x1D709}))>0$ , we have that $(\operatorname{edim})Kr_{N}(m_{3},m_{0})>0$ . By the properties of Kronecker moduli spaces, $Kr_{N}(m_{3},m_{0})$ is positive dimensional. Thus, the general such module is stable. As isomorphism of complexes corresponds exactly with isomorphism of Kronecker modules, we obtain a birational map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:M(\unicode[STIX]{x1D709}){\dashrightarrow}Kr_{N}(m_{3},m_{0}).\end{eqnarray}$$

5.2 When all of the powers are nonzero

For this subsection, assume that $m_{i}\neq 0$ for all $i$ in the resolution

$$\begin{eqnarray}0\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\bigoplus F_{0}^{\ast m_{2}}\stackrel{\unicode[STIX]{x1D719}}{\rightarrow }F_{-1}^{\ast m_{1}}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U\rightarrow 0.\end{eqnarray}$$

Forgetting most of the information of the resolution, $U$ determines a complex of the form

$$\begin{eqnarray}W:F_{0}^{\ast m_{2}}\rightarrow F_{-1}^{\ast m_{1}}\end{eqnarray}$$

which in turn determines a Kronecker $\operatorname{Hom}(F_{0}^{\ast },F_{-1}^{\ast })$ -module.

Conversely, given a general such module and its determined complex, $W^{\prime }$ , there exists an element $U^{\prime }\in M(\unicode[STIX]{x1D709})$ such that $W^{\prime }$ is the associated complex of $U^{\prime }$ by Theorem 4.4. Assuming that there is a semistable Kronecker module associated to a general $W$ , this constructs a rational map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:M(\unicode[STIX]{x1D709}){\dashrightarrow}Kr_{N}(m_{2},m_{1})\end{eqnarray}$$

where $N=\hom (F_{0}^{\ast },F_{-1}^{\ast })$ . In order to verify that we get Kronecker modules, we have to prove a result about the map $E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U$ and $U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast m_{3}}(K)[1]$ being canonical. To show that the general associated Kronecker module is semistable, we again have to show that the Kronecker moduli space is nonempty. We now show that the map $E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U$ is canonical.

Proposition 5.3. With the notation of this subsection, let $U\in M(\unicode[STIX]{x1D709})$ be general. Let $W^{\prime }\in D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{1})$ be the mapping cone of the canonical evaluation map

$$\begin{eqnarray}E_{\unicode[STIX]{x1D6FD}}^{\ast }\otimes \operatorname{Hom}(E_{\unicode[STIX]{x1D6FD}}^{\ast },U)\rightarrow U,\end{eqnarray}$$

so that there is a distinguished triangle

$$\begin{eqnarray}E_{\unicode[STIX]{x1D6FD}}^{\ast }\otimes \operatorname{Hom}(E_{\unicode[STIX]{x1D6FD}}^{\ast },U)\rightarrow U\rightarrow W^{\prime }\rightarrow \cdot .\end{eqnarray}$$

Then $W^{\prime }$ is isomorphic to a complex of the form

$$\begin{eqnarray}\left(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\otimes \mathbb{C}^{m_{3}}\right)\bigoplus \left(F_{0}^{\ast }\otimes \mathbb{C}^{m_{2}}\right)\rightarrow \left(F_{-1}^{\ast }\otimes \mathbb{C}^{m_{1}}\right)\end{eqnarray}$$

sitting in degrees $-1$ and $0$ .

Furthermore, $W^{\prime }$ is also isomorphic to the complex

$$\begin{eqnarray}\left(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\otimes \mathbb{C}^{m_{3}}\right)\bigoplus \left(F_{0}^{\ast }\otimes \mathbb{C}^{m_{2}}\right)\rightarrow \left(F_{-1}^{\ast }\otimes \mathbb{C}^{m_{1}}\right)\end{eqnarray}$$

appearing in the Beilinson spectral sequence for $U$ .

Proof. It is easy to show that if

$$\begin{eqnarray}0\rightarrow A\rightarrow B\oplus C\rightarrow D\rightarrow 0\end{eqnarray}$$

is an exact sequence of sheaves, then the mapping cone of $C\rightarrow D$ is isomorphic to the complex $A\rightarrow B$ sitting in degrees $-1$ and $0$ .

Suppose we have a resolution

$$\begin{eqnarray}0\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\bigoplus F_{0}^{\ast m_{2}}\stackrel{\unicode[STIX]{x1D719}}{\rightarrow }F_{-1}^{\ast m_{1}}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U\rightarrow 0\end{eqnarray}$$

of a general $U$ as in Theorem 4.4. We have $m_{0}=\hom (E_{\unicode[STIX]{x1D6FD}}^{\ast },U)$ . Then since $U$ is semistable, the map $E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U$ can be identified with the canonical evaluation $E_{\unicode[STIX]{x1D6FD}}^{\ast }\otimes \operatorname{Hom}(E_{\unicode[STIX]{x1D6FD}}^{\ast },U)\rightarrow U$ . Thus, the mapping cone of this evaluation is the complex given by the first component of $\unicode[STIX]{x1D719}$ . By Lemma 5.1, any two complexes of the form $E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\bigoplus F_{0}^{\ast m_{2}}\rightarrow F_{-1}^{m_{1}}$ which are isomorphic in the derived category are in the same orbit of the $\operatorname{GL}(m_{3})\times \operatorname{GL}(m_{2})\times M_{m_{2},m_{3}}(\operatorname{Hom}(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K),F_{0}^{\ast }))\times \operatorname{GL}(m_{1})$ action.

Finally, we show that $W^{\prime }$ is isomorphic to the complex which appears in the Beilinson spectral sequence for $U$ . For simplicity, we assume we are in the case $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6E5}_{2}=1$ . We recall how to compute the $E_{1}^{p,q}$ -page of the spectral sequence. Let $p_{i}:\left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)\times \left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)\rightarrow \mathbb{P}^{1}\times \mathbb{P}^{1}$ be the projections, and let $\unicode[STIX]{x1D6E5}\subset \left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)\times \left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)$ be the diagonal. There is a resolution of the diagonal

$$\begin{eqnarray}0\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\boxtimes E_{\unicode[STIX]{x1D6FC}}\bigoplus F_{0}^{\ast }\boxtimes E_{0}\rightarrow F_{-1}^{\ast }\boxtimes E_{1}\rightarrow E_{\unicode[STIX]{x1D6FD}}^{\ast }\boxtimes E_{\unicode[STIX]{x1D6FD}}\rightarrow {\mathcal{O}}_{\unicode[STIX]{x1D6E5}}\rightarrow 0.\end{eqnarray}$$

We split the resolution of the diagonal into two short exact sequences

$$\begin{eqnarray}\displaystyle 0 & \rightarrow & \displaystyle E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\boxtimes E_{\unicode[STIX]{x1D6FC}}\bigoplus F_{0}^{\ast }\boxtimes E_{0}\rightarrow F_{-1}^{\ast }\boxtimes E_{1}\rightarrow M\rightarrow 0\qquad \text{and}\nonumber\\ \displaystyle 0 & \rightarrow & \displaystyle M\rightarrow E_{\unicode[STIX]{x1D6FD}}^{\ast }\boxtimes E_{\unicode[STIX]{x1D6FD}}\rightarrow {\mathcal{O}}_{\unicode[STIX]{x1D6E5}}\rightarrow 0.\nonumber\end{eqnarray}$$

Tensoring with $p_{2}^{\ast }(U)$ and applying $Rp_{1\ast }$ , we get triangles

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\boxtimes E_{\unicode[STIX]{x1D6FC}}}(U)\bigoplus \unicode[STIX]{x1D6F7}_{F_{0}^{\ast }\boxtimes E_{0}}(U) & \rightarrow & \displaystyle \unicode[STIX]{x1D6F7}_{F_{-1}^{\ast }\boxtimes E_{1}}(U)\rightarrow \unicode[STIX]{x1D6F7}_{M}(U)\rightarrow \cdot \qquad \text{and}\nonumber\\ \displaystyle \unicode[STIX]{x1D6F7}_{M}(U) & \rightarrow & \displaystyle \unicode[STIX]{x1D6F7}_{E_{\unicode[STIX]{x1D6FD}}^{\ast }\boxtimes E_{\unicode[STIX]{x1D6FD}}}(U)\rightarrow \unicode[STIX]{x1D6F7}_{{\mathcal{O}}_{\unicode[STIX]{x1D6E5}}}(U)\rightarrow \cdot ,\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}_{F}:D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{1})\rightarrow D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{1})$ is the Fourier–Mukai transform with kernel $F$ . Computing these transforms using Proposition 4.4, we obtain two different complexes

$$\begin{eqnarray}\displaystyle & & \displaystyle E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\otimes H^{1}(E_{\unicode[STIX]{x1D6FC}}\otimes U)\bigoplus F_{0}^{\ast }\otimes H^{0}(E_{0}\otimes U)\nonumber\\ \displaystyle & & \displaystyle \quad \rightarrow F_{-1}^{\ast }\otimes H^{0}(E_{1}\otimes U)\rightarrow \unicode[STIX]{x1D6F7}_{M}(U)[1]\rightarrow \cdot \nonumber\\ \displaystyle & & \displaystyle \text{and}\quad \unicode[STIX]{x1D6F7}_{M}(U)\rightarrow E_{\unicode[STIX]{x1D6FD}}^{\ast }\otimes \operatorname{Hom}(E_{\unicode[STIX]{x1D6FD}}^{\ast },U)\rightarrow U\rightarrow \cdot \nonumber\end{eqnarray}$$

involving $\unicode[STIX]{x1D6F7}_{M}(U)$ ; notice that the map $E_{\unicode[STIX]{x1D6FD}}^{\ast }\otimes \operatorname{Hom}(E_{\unicode[STIX]{x1D6FD}}^{\ast },U)\rightarrow U$ is the canonical one since the map

$$\begin{eqnarray}E_{\unicode[STIX]{x1D6FD}}^{\ast }\boxtimes E_{\unicode[STIX]{x1D6FD}}\rightarrow {\mathcal{O}}_{\unicode[STIX]{x1D6E5}}\end{eqnarray}$$

is the trace map. Therefore, $\unicode[STIX]{x1D6F7}_{M}(U)[1]$ is isomorphic to $W$ by the second triangle. On the other hand, $\unicode[STIX]{x1D6F7}_{M}(U)[1]$ is also isomorphic to the complex in the Beilinson spectral sequence by the first triangle.

We now turn to showing that the map $U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{0}}[1]$ is canonical in order to show that the general $W$ is associated to a Kronecker module.

Proposition 5.4. With the notation of this subsection, let $U\in M(\unicode[STIX]{x1D709})$ be general. Let $W^{\prime }\in D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{1})$ be the mapping cone of the canonical evaluation map

$$\begin{eqnarray}U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1]\otimes \operatorname{Hom}(U,E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1]),\end{eqnarray}$$

so that there is a distinguished triangle

$$\begin{eqnarray}U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1]\otimes \operatorname{Hom}(U,E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1])\rightarrow W^{\prime }\rightarrow \cdot .\end{eqnarray}$$

Then $W^{\prime }$ is isomorphic to a complex of the form

$$\begin{eqnarray}\left(F_{0}^{\ast }\otimes \mathbb{C}^{m_{2}}\right)\rightarrow \left(F_{-1}^{\ast }\otimes \mathbb{C}^{m_{1}}\right)\bigoplus \left(E_{\unicode[STIX]{x1D6FD}}^{\ast }\otimes \mathbb{C}^{m_{0}}\right)\end{eqnarray}$$

sitting in degrees $-2$ and $-1$ .

Furthermore, $W^{\prime }$ is also isomorphic to the complex $\left(F_{0}^{\ast }\otimes \mathbb{C}^{m_{2}}\right)\rightarrow \left(F_{-1}^{\ast }\otimes \mathbb{C}^{m_{1}}\right)\bigoplus \left(E_{\unicode[STIX]{x1D6FD}}^{\ast }\otimes \mathbb{C}^{m_{0}}\right)$ appearing in the Beilinson spectral sequence for $U$ .

Proof. It is easy to show that if

$$\begin{eqnarray}0\rightarrow A\oplus B\rightarrow C\rightarrow D\rightarrow 0\end{eqnarray}$$

is an exact sequence of sheaves, then the mapping cone of $D\rightarrow A[1]$ is isomorphic to the complex $B\rightarrow C$ sitting in degrees $-2$ and $-1$ . Suppose we have a resolution

$$\begin{eqnarray}0\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\bigoplus F_{0}^{\ast m_{2}}\stackrel{\unicode[STIX]{x1D719}}{\rightarrow }F_{-1}^{\ast m_{1}}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U\rightarrow 0\end{eqnarray}$$

of a general $U$ as in Theorem 4.4. We have $m_{3}=\hom (U,E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1])$ . Then since $U$ is semistable, the map $U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}[1]$ can be identified with the canonical co-evaluation $U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1]\otimes \operatorname{Hom}(U,E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1])$ . Thus, the mapping cone of this co-evaluation is the complex given by the second component of $\unicode[STIX]{x1D719}$ . By Lemma 5.1, any two complexes of the form $F_{0}^{\ast m_{2}}\rightarrow F_{-1}^{m_{1}}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}$ which are isomorphic in the derived category are in the same orbit of the $\operatorname{GL}(m_{2})\times \operatorname{GL}(m_{1})\times \operatorname{GL}(m_{0})\times M_{m_{0},m_{1}}(\operatorname{Hom}(F_{-1}^{\ast },E_{\unicode[STIX]{x1D6FD}}^{\ast }))$ action.

Finally, we show that $W^{\prime }$ is isomorphic to the complex which appears in the Beilinson spectral sequence for $U$ . For simplicity, we assume we are in the case $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6E5}_{2}=0$ . Recall how to compute the $E_{1}^{p,q}$ -page of the spectral sequence. Let $p_{i}:\left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)\times \left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)\rightarrow \mathbb{P}^{1}\times \mathbb{P}^{1}$ be the projections, and let $\unicode[STIX]{x1D6E5}\subset \left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)\times \left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)$ be the diagonal. There is a resolution of the diagonal

$$\begin{eqnarray}0\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\boxtimes E_{\unicode[STIX]{x1D6FC}}\rightarrow F_{0}^{\ast }\boxtimes E_{0}\rightarrow F_{-1}^{\ast }\boxtimes E_{1}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast }\boxtimes E_{\unicode[STIX]{x1D6FD}}\rightarrow {\mathcal{O}}_{\unicode[STIX]{x1D6E5}}\rightarrow 0.\end{eqnarray}$$

We split the resolution of the diagonal into two short exact sequences

$$\begin{eqnarray}\displaystyle 0 & \rightarrow & \displaystyle F_{0}^{\ast }\boxtimes E_{0}\rightarrow F_{-1}^{\ast }\boxtimes E_{1}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast }\boxtimes E_{\unicode[STIX]{x1D6FD}}\rightarrow M\rightarrow 0\qquad \text{and}\nonumber\\ \displaystyle 0 & \rightarrow & \displaystyle M\rightarrow {\mathcal{O}}_{\unicode[STIX]{x1D6E5}}\rightarrow \left(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\boxtimes E_{\unicode[STIX]{x1D6FC}}\right)[1].\nonumber\end{eqnarray}$$

Tensoring with $p_{2}^{\ast }(U)$ and applying $Rp_{1\ast }$ , we get triangles

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{F_{0}^{\ast }\boxtimes E_{0}}(U) & \rightarrow & \displaystyle \unicode[STIX]{x1D6F7}_{F_{-1}^{\ast }\boxtimes E_{1}}(U)\bigoplus \unicode[STIX]{x1D6F7}_{E_{\unicode[STIX]{x1D6FD}}^{\ast }\boxtimes E_{\unicode[STIX]{x1D6FD}}}(U)\rightarrow \unicode[STIX]{x1D6F7}_{M}(U)\rightarrow \cdot \qquad \text{and}\nonumber\\ \displaystyle \unicode[STIX]{x1D6F7}_{M}(U) & \rightarrow & \displaystyle \unicode[STIX]{x1D6F7}_{{\mathcal{O}}_{\unicode[STIX]{x1D6E5}}}(U)\rightarrow \unicode[STIX]{x1D6F7}_{(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\boxtimes E_{\unicode[STIX]{x1D6FC}})[1]}(U)\rightarrow \cdot ,\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}_{F}:D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{1})\rightarrow D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{1})$ is the Fourier–Mukai transform with kernel $F$ . Computing these transforms using Theorem 4.4, we obtain two different complexes

$$\begin{eqnarray}\displaystyle F_{0}^{\ast }\otimes H^{1}(E_{0}\otimes U) & \rightarrow & \displaystyle F_{-1}^{\ast }\otimes H^{1}(E_{1}\otimes U)\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast }\otimes \operatorname{Hom}(E_{\unicode[STIX]{x1D6FD}}^{\ast },U)\nonumber\\ \displaystyle & \rightarrow & \displaystyle \unicode[STIX]{x1D6F7}_{M}(U)\rightarrow \cdot \nonumber\\ \displaystyle \text{and}\qquad \unicode[STIX]{x1D6F7}_{M}(U) & \rightarrow & \displaystyle U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\otimes \operatorname{Hom}(U,E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1])\rightarrow \cdot \nonumber\end{eqnarray}$$

involving $\unicode[STIX]{x1D6F7}_{M}(U)$ ; notice that the map $U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1]\otimes \operatorname{Hom}(U,E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)[1])$ is the canonical one since the map

$$\begin{eqnarray}{\mathcal{O}}_{\unicode[STIX]{x1D6E5}}\rightarrow (E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)\boxtimes E_{\unicode[STIX]{x1D6FC}})[1]\end{eqnarray}$$

is the cotrace map. Therefore, $\unicode[STIX]{x1D6F7}_{M}(U)$ is isomorphic to $W^{\prime }$ by the second triangle. On the other hand, $\unicode[STIX]{x1D6F7}_{M}(U)$ is also isomorphic to the complex in the Beilinson spectral sequence by the first triangle.

We now use that these maps are canonical to establish that each resolution is associated to a Kronecker module.

Proposition 5.5. Let $U\in M(\unicode[STIX]{x1D709})$ be a general object with the resolution

$$\begin{eqnarray}0\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\bigoplus F_{0}^{\ast m_{2}}\rightarrow F_{-1}^{\ast m_{1}}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U\rightarrow 0\end{eqnarray}$$

with subcomplex

$$\begin{eqnarray}W:F_{0}^{\ast m_{2}}\rightarrow F_{-1}^{\ast m_{1}}.\end{eqnarray}$$

Then $W$ is the complex appearing in the spectral sequence.

Proof. Let $W^{\prime }:F_{0}^{\ast m_{2}}\rightarrow F_{-1}^{\ast m_{1}}$ be the complex appearing in $U$ ’ spectral sequence. By a Bertini-like statement [Reference HuizengaHui2], $W^{\prime }$ is either surjective or injective.

Assume that it is injective. This means that the $E_{2}$ page of the spectral sequence is

$$\begin{eqnarray}\begin{array}{@{}cccc@{}}E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}} & 0 & 0 & 0\\ 0 & 0 & K & 0\\ 0 & 0 & 0 & E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\\ \end{array}\end{eqnarray}$$

where $K=\operatorname{coker}(W^{\prime })$ . In turn this gives that, in the resolution coming from the spectral sequence, the map from $F_{0}^{\ast m_{2}}$ to $E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}$ is zero. Then, by Proposition 5.4, we have two short exact sequences

$$\begin{eqnarray}\displaystyle 0 & \rightarrow & \displaystyle F_{0}^{\ast m_{2}}\stackrel{\unicode[STIX]{x1D719}}{\rightarrow }F_{-1}^{\ast m_{1}}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow L\rightarrow 0\qquad \text{and}\nonumber\\ \displaystyle 0 & \rightarrow & \displaystyle F_{0}^{\ast m_{2}}\stackrel{\unicode[STIX]{x1D713}}{\rightarrow }F_{-1}^{\ast m_{1}}\bigoplus E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow L\rightarrow 0\nonumber\end{eqnarray}$$

where $W$ and $W^{\prime }$ are subcomplexes of the respective sequences and the second component of $\unicode[STIX]{x1D713}$ is zero. The identity map on $L$ induces an isomorphism of its resolutions, which implies that the first component of $\unicode[STIX]{x1D719}$ is a scalar multiple of the first component of $\unicode[STIX]{x1D713}$ . This then gives an isomorphism between $W$ and $W^{\prime }$ given by dividing by that scalar multiple in the degree $-1$ and the identity in degree $0$ . Thus, $W$ is the complex in the spectral sequence converging to $U$ .

The case of surjectivity is similar but uses Proposition 5.3.

We now finish the construction of the map $M(\unicode[STIX]{x1D709}){\dashrightarrow}Kr_{N}(m_{2},m_{1})$ .

Proposition 5.6. With the above notation, $Kr_{N}(m_{2},m_{1})$ is nonempty and there is a dominant rational map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:M(\unicode[STIX]{x1D709}){\dashrightarrow}Kr_{N}(m_{2},m_{1}).\end{eqnarray}$$

Proof. By construction, we know that the primary orthogonal Chern characters to $\unicode[STIX]{x1D709}$ are all semistable so $\unicode[STIX]{x1D709}^{+}$ is semistable. Let $V\in M(\unicode[STIX]{x1D709}^{+})$ be general. Then, by Theorem 4.4, $V$ has the resolution

$$\begin{eqnarray}0\rightarrow E_{1}^{n_{1}}\rightarrow V\rightarrow E_{0}^{n_{2}}\rightarrow 0\end{eqnarray}$$

or one of the equivalent resolutions which have the same Kronecker module structure. Similarly, the general $U\in M(\unicode[STIX]{x1D709})$ has the resolution

$$\begin{eqnarray}E_{\unicode[STIX]{x1D6FD}}^{\ast m_{0}}\rightarrow U\rightarrow \left(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\bigoplus W\right)[1]\end{eqnarray}$$

where $W$ is the complex

$$\begin{eqnarray}F_{0}^{\ast m_{2}}\rightarrow F_{-1}^{\ast m_{1}}.\end{eqnarray}$$

As the point $(\unicode[STIX]{x1D707}^{+},\unicode[STIX]{x1D6E5}^{+})$ lies on the surface $Q_{\unicode[STIX]{x1D709}}$ , we have that $\unicode[STIX]{x1D712}(V^{\ast },U)=0$ . By design, the resolution of $V$ immediately forces $\unicode[STIX]{x1D712}(V^{\ast },E_{\unicode[STIX]{x1D6FD}}^{\ast })=0$ and $\unicode[STIX]{x1D712}(V^{\ast },E_{\unicode[STIX]{x1D6FC}}^{\ast }(K))=0$ because of the orthogonality properties of the coil $\{E_{\unicode[STIX]{x1D6FC}}^{\ast }(K),E_{\unicode[STIX]{x1D6FD}}^{\ast },E_{1}^{\ast },E_{0}^{\ast }\}$ . Then vanishings of $\unicode[STIX]{x1D712}(V^{\ast },U)$ , $\unicode[STIX]{x1D712}(V^{\ast },E_{\unicode[STIX]{x1D6FD}}^{\ast })$ , and $\unicode[STIX]{x1D712}(V^{\ast },E_{\unicode[STIX]{x1D6FC}}^{\ast }(K))$ force $\unicode[STIX]{x1D712}(V^{\ast },W)=0$ . Since $\unicode[STIX]{x1D712}(V^{\ast },W)=0$ and $\unicode[STIX]{x1D712}(E_{\unicode[STIX]{x1D6FD}}^{\ast },W)=0$ , we have that $\unicode[STIX]{x1D712}(L_{E_{\unicode[STIX]{x1D6FD}}^{\ast }}V^{\ast },W)=0$ . Shifting only shifts the indices so we have $\unicode[STIX]{x1D712}(L_{E_{\unicode[STIX]{x1D6FD}}^{\ast }}V^{\ast }[1],W)=0$ as well. In the derived category $L_{E_{\unicode[STIX]{x1D6FD}}^{\ast }}V^{\ast }[1]$ is isomorphic to the complex

$$\begin{eqnarray}F_{0}^{\ast n_{1}}\rightarrow F_{-1}^{\ast n_{2}}\end{eqnarray}$$

sitting in degrees $-1$ and 0. Thus, $L_{E_{\unicode[STIX]{x1D6FD}}^{\ast }}V^{\ast }[1]$ and $W$ both correspond to Kronecker $\operatorname{Hom}(F_{0}^{\ast },F_{1}^{\ast })$ -modules. Call them $e$ and $f$ respectively.

Then $\unicode[STIX]{x1D712}(L_{E_{\unicode[STIX]{x1D6FD}}^{\ast }}V^{\ast }[1],W)=0$ tells us that $\unicode[STIX]{x1D712}(e,f)=0$ which implies that $\text{}\underline{\dim }\,f$ if a right-orthogonal dimension vector to $\text{}\underline{\dim }\,e$ . Since $M(V)$ is nonempty, Proposition 5.2 shows that $Kr_{N}(n_{1},n_{2})$ is nonempty. If $Kr_{N}(n_{1},n_{2})$ is positive or $0$ dimensional, the discussion at the end of [Reference Coskun, Huizenga and WoolfCHW, Section 6.1] shows that $Kr_{N}(m_{2},m_{1})$ is as well. Thus, $Kr_{N}(m_{2},m_{1})$ is nonempty as promised.

6.1 The zero dimensional Kronecker moduli space case

Theorem 6.1. Let $\unicode[STIX]{x1D709}^{+}$ be a primary orthogonal Chern character to $\{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\}$ for the Chern character $\unicode[STIX]{x1D709}$ with $\dim (K)=0$ and let $V\in M(\unicode[STIX]{x1D709}^{+})$ be the element. Then the Brill–Noether divisor

$$\begin{eqnarray}D_{V}=\{U^{\prime }\in M(\unicode[STIX]{x1D709}):h^{1}(U^{\prime }\otimes V)\neq 0\}\end{eqnarray}$$

is on an edge of the effective cone of $M(\unicode[STIX]{x1D709})$ . Using the isomorphism $\operatorname{NS}\left(M(\unicode[STIX]{x1D709})\right)\cong \unicode[STIX]{x1D709}^{\bot }$ , $D_{V}$ corresponds to $\unicode[STIX]{x1D709}^{+}$ .

Proof. The general element $U\in M(\unicode[STIX]{x1D709})$ fits into the short exact sequence

$$\begin{eqnarray}0\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\bigoplus F_{0}^{\ast m_{2}}\rightarrow F_{-1}^{\ast m_{1}}\bigoplus \left(E_{\unicode[STIX]{x1D6FD}}^{\ast }\right)^{m_{0}}\rightarrow U\rightarrow 0.\end{eqnarray}$$

In order to show that $D_{V}$ is an effective Brill–Noether divisor, we have to show that $V$ is cohomologically orthogonal to $U$ . This means showing that $U\otimes V=\mathscr{H}\!om(U^{\ast },V)$ has no cohomology. How we show this orthogonality depends upon which if any of the $m_{i}$ vanish. Note that $\dim (K)=0$ implies that at most one of the $m_{i}$ is zero because if two are zero then $\dim (K)=\dim (M(\unicode[STIX]{x1D709}))$ .

Assume that none of the $m_{i}$ are zero. Then the general element, $U\in M(\unicode[STIX]{x1D709})$ , fits into the triangle

$$\begin{eqnarray}\left(E_{\unicode[STIX]{x1D6FD}}^{\ast }\right)^{m_{0}}\rightarrow U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}[1]\bigoplus W\end{eqnarray}$$

where $W$ is the complex $\left(F_{0}^{\ast }\right)^{m_{2}}\rightarrow \left(F_{-1}^{\ast }\right)^{m_{1}}$ sitting in degrees -1 and 0. Similarly, the general element, $V\in M(\unicode[STIX]{x1D709}^{+})$ , fits into the triangle

$$\begin{eqnarray}E_{1}^{n_{2}}\rightarrow V\rightarrow E_{0}^{n_{1}}.\end{eqnarray}$$

By choice of resolving exceptional bundles, $\mathscr{H}\!om(E_{\unicode[STIX]{x1D6FD}}^{\ast },\!V)$ and $\mathscr{H}\!om(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K),\!V)$ have no cohomology. Thus, to construct the divisor it suffices to show that $\mathscr{H}\!om(W^{\ast },V)$ has no cohomology. As $\mathscr{H}\!om(W^{\ast },E_{\unicode[STIX]{x1D6FD}})$ has no cohomology, this is equivalent to $\mathscr{H}\!om(W^{\ast },R_{\unicode[STIX]{x1D6FD}}V)$ having no cohomology. We reduce further to showing that $\mathscr{H}\!om(W^{\ast },R_{\unicode[STIX]{x1D6FD}}V[1])$ has no cohomology as shifting merely shifts the cohomology. Then if $f$ and $e$ are the Kronecker modules corresponding to $W^{\ast }$ and $R_{\unicode[STIX]{x1D6FD}}V[1]$ , respectively, the vanishing of these cohomologies is equivalent to the vanishing of the $\operatorname{Hom}(f,e)$ , but that vanishing follows directly from [Reference Coskun, Huizenga and WoolfCHW, Thm. 6.1]. Thus, we have the orthogonality that we needed.

If one of the $m_{i}$ is zero (in which case we have not constructed a Kronecker fibration explicitly), then $V$ is one of the exceptional bundles $E_{\unicode[STIX]{x1D6FD}}$ , $E_{1}$ , $E_{0}$ , or $E_{\unicode[STIX]{x1D6FC}}$ . Then $V$ is cohomologically orthogonal to all three bundles that appear in the resolution of $U$ so it is automatically cohomologically orthogonal.

Thus, we have shown the cohomological orthogonality in either case. The class of $D_{V}$ and the fact that it is effective is computed using Proposition 2.16.

To show it lies on an edge, we construct a moving curve by varying a map in the resolution.

If $m_{3}\neq 0$ and $m_{0}\neq 0$ , fix every map except $E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\rightarrow \left(E_{\unicode[STIX]{x1D6FD}}^{\ast }\right)^{m_{0}}$ , let

$$\begin{eqnarray}S=\mathbb{P}\operatorname{Hom}\left(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}},\left(E_{\unicode[STIX]{x1D6FD}}^{\ast }\right)^{m_{0}}\right),\end{eqnarray}$$

and let ${\mathcal{U}}/S$ be the universal cokernel sheaf (of the fixed map plus the varying part).

If $m_{3}=0$ , fix every map except $F_{0}^{\ast m_{2}}\rightarrow \left(E_{\unicode[STIX]{x1D6FD}}^{\ast }\right)^{m_{0}}$ , let

$$\begin{eqnarray}S=\mathbb{P}\operatorname{Hom}\left(F_{0}^{\ast m_{2}},\left(E_{\unicode[STIX]{x1D6FD}}^{\ast }\right)^{m_{0}}\right),\end{eqnarray}$$

and let ${\mathcal{U}}/S$ be the universal cokernel sheaf (of the fixed map plus the varying part).

If $m_{0}=0$ , fix every map except $E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}\rightarrow \left(F_{0}^{\ast }\right)^{m_{1}}$ , let

$$\begin{eqnarray}S=\mathbb{P}\operatorname{Hom}\left(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}},\left(F_{0}^{\ast }\right)^{m_{1}}\right),\end{eqnarray}$$

and let ${\mathcal{U}}/S$ be the universal cokernel sheaf (of the fixed map plus the varying part).

In any case, we have our ${\mathcal{U}}$ and our $S$ . Because $M(\unicode[STIX]{x1D709})$ is positive dimensional and the general sheaf in it has a resolution of this form, $S$ is nonempty. Then ${\mathcal{U}}$ is a complete family of prioritary sheaves whose fixed Chern character lies above the $\unicode[STIX]{x1D6FF}$ surface. Define the open set $S^{\prime }\subset S$ by

$$\begin{eqnarray}S^{\prime }:=\{s\in S:{\mathcal{U}}_{s}\text{ is stable}\}.\end{eqnarray}$$

Thus, by assumption, the complement of $S^{\prime }$ has codimension at least 2 which allows us to find a complete curve in $S^{\prime }$ containing the point corresponding to $U$ for the general element $U\in M(\unicode[STIX]{x1D709})$ . Notice that this is a moving curve by the codimension statement. Any curve in $S^{\prime }$ is disjoint from $D_{V}$ which makes the curve dual to it.

This curve makes $D_{V}$ be on an edge. As the resolution only provides one moving curve, this resolution only shows that it lies on an edge of the cone, not that it is an extremal ray.

6.2 The positive dimensional Kronecker moduli space case

Theorem 6.2. Let $\unicode[STIX]{x1D709}^{+}$ be a primary orthogonal Chern character to $\{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\}$ for the Chern character $\unicode[STIX]{x1D709}$ with $\dim (K)>0$ and $V\in M(\unicode[STIX]{x1D709}^{+})$ be a general element. Then the Brill–Noether divisor

$$\begin{eqnarray}D_{V}=\{U^{\prime }\in M(\unicode[STIX]{x1D709}):h^{1}(U^{\prime }\otimes V)\neq 0\}\end{eqnarray}$$

lies on the edge of the effective cone of $M(\unicode[STIX]{x1D709})$ . Using the isomorphism $\operatorname{NS}\left(M(\unicode[STIX]{x1D709})\right)\cong \unicode[STIX]{x1D709}^{\bot }$ , $D_{V}$ corresponds to $\unicode[STIX]{x1D709}^{+}$ .

Proof. Recall that we have a dominant rational map $\unicode[STIX]{x1D70B}:M(\unicode[STIX]{x1D709}){\dashrightarrow}K$ . There are two possibilities; either $\unicode[STIX]{x1D70B}$ is a birational map or $\unicode[STIX]{x1D70B}$ has positive dimensional fibers.

Birational case In this case, either zero or two of the $m_{i}$ can vanish. If zero vanish, we show that $V$ is cohomologically orthogonal to $U$ by the same arguments as the previous theorem. If two vanish, then $V$ is again one of the exceptional bundles so orthogonality is immediate. The class of $D_{V}$ and the fact that it is effective is computed using Proposition 2.16, and $D_{V}$ is the exceptional locus of $\unicode[STIX]{x1D70B}$ . Using that fact, we get a dual moving curve to $D_{V}$ by varying the Kronecker module. Formally, because $K$ is Picard rank one, there is a moving curve $C$ . Then $[\unicode[STIX]{x1D70B}^{\ast }(C)]$ is a moving curve which is dual to the exceptional locus of $\unicode[STIX]{x1D70B}$ (i.e., dual to $D_{V}$ ). Thus, $D_{V}$ is on the edge of the effective cone.

Positive dimensional fiber case In this case, none of the $m_{i}$ is zero.

We first show cohomological orthogonality. This means showing that $U\otimes V=\mathscr{H}\!om(U^{\ast },V)$ has no cohomology. In this case, the general element, $U\in M(\unicode[STIX]{x1D709})$ , fits into the triangle

$$\begin{eqnarray}\left(E_{\unicode[STIX]{x1D6FD}}^{\ast }\right)^{m_{0}}\rightarrow U\rightarrow E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{3}}[1]\bigoplus W\end{eqnarray}$$

where $W$ is the complex $\left(F_{0}^{\ast }\right)^{m_{2}}\rightarrow \left(F_{-1}^{\ast }\right)^{m_{1}}$ sitting in degrees -1 and 0. Similarly, the general element, $V\in M(\unicode[STIX]{x1D709}^{+})$ , fits into the triangle

$$\begin{eqnarray}E_{1}^{n_{2}}\rightarrow V\rightarrow E_{0}^{n_{1}}.\end{eqnarray}$$

By choice of resolving exceptional bundles, $\mathscr{H}\!om(E_{\unicode[STIX]{x1D6FD}}^{\ast },\!V)$ and $\mathscr{H}\!om(E_{\unicode[STIX]{x1D6FC}}^{\ast }(K),\!V)$ have no cohomology. Thus, to construct the divisor it suffices to show that $\mathscr{H}\!om(W^{\ast },V)$ has no cohomology. As $\mathscr{H}\!om(W^{\ast },E_{\unicode[STIX]{x1D6FD}})$ has no cohomology, we have that this is equivalent to $\mathscr{H}\!om(W^{\ast },R_{\unicode[STIX]{x1D6FD}}V)$ having no cohomology. We reduce further to showing that $\mathscr{H}\!om(W^{\ast },R_{\unicode[STIX]{x1D6FD}}V[1])$ has no cohomology as shifting merely shifts the cohomology. Then if $f$ and $e$ are the Kronecker modules corresponding to $W^{\ast }$ and $R_{\unicode[STIX]{x1D6FD}}V[1]$ , respectively, the vanishing of these cohomologies is equivalent to the vanishing of the $\operatorname{Hom}(f,e)$ , but that vanishing follows directly from [Reference Coskun, Huizenga and WoolfCHW, Thm. 6.1]

We have now established the cohomological orthogonality. The class of $D_{V}$ and the fact that it gives an effective divisor are computed using Proposition 2.16.

As the general $U$ has the given resolution, the fibers of map to $K$ are covered by varying the other maps of the resolution as we did in the last proof. As these moving curves sit inside fibers, they are dual to $D_{V}$ since $V$ is dual to the Kronecker modules in $K$ .

If we can vary two different maps in the resolution other than the Kronecker module independently, than $D_{V}$ has the same class as the pullback of an ample divisor of $K$ . This immediately implies that $D_{V}$ is in the moving cone. We can vary two maps independently if the no $m_{i}$ is zero and no subcomplex of the resolution has enough dimensions to account for all of the dimensions of our moduli space.

These theorems together give an effective divisor on $M(\unicode[STIX]{x1D709})$ . These conjecturally might give a spanning set of effective divisors for the effective cone of $M(\unicode[STIX]{x1D709})$ , but even if this method does not do that, it gives a way to construct effective divisors on many of these moduli spaces. In addition, this same method works to give secondary extremal rays when the rank of the moduli space is at least three (for rank less than three all secondary rays have special meaning).

In the case that no power in the resolution vanishes, this effective divisor immediately spans an extremal ray if $\hom (E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{0}},K)-m_{0}^{2}<\dim (M(\unicode[STIX]{x1D709}))$ and $\hom (E_{\unicode[STIX]{x1D6FC}}^{\ast }(K)^{m_{0}},E_{\unicode[STIX]{x1D6FD}}^{\ast m_{3}})-m_{0}^{2}-m_{3}^{2}<\dim (M(\unicode[STIX]{x1D709}))$ . Otherwise, this divisor is known to span an extremal ray as long as both controlling exceptional bundles are part of another extremal pair.

7.1 The effective cones of Hilbert schemes of at most sixteen points

The most classical example of a moduli space of sheaves on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ is Hilbert scheme of $n$ points on it. For these Hilbert schemes, the Picard group has a classical basis, $\{B,H_{1},H_{2}\}$ . Each element of this basis has an extremely geometric interpretation. $B$ is the locus of nonreduced schemes or equivalently the schemes supported on $n-1$ or fewer closed points. $H_{1}$ is the schemes whose support intersects a fixed line of type $(1,0)$ . Similarly, $H_{2}$ is the schemes whose support intersects a fixed line of type $(0,1)$ .

Using this basis, every ray in the Néron–Severi space is spanned by a ray of the form $B$ , $aH_{1}+bH_{2}+B$ , $aH_{1}+bH_{2}$ , or $iH_{1}+jH_{2}-B/2$ . Then we fix the notation for the last two types of ray as

$$\begin{eqnarray}\displaystyle Y_{a,b} & = & \displaystyle aH_{1}+bH_{2}\qquad \text{and}\nonumber\\ \displaystyle X_{i,j} & = & \displaystyle iH_{1}+jH_{2}-{\textstyle \frac{1}{2}}B.\nonumber\end{eqnarray}$$

Using this notation, we list the extremal rays of the effective cones of $\left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)^{\left[n\right]}$ for $n\leqslant 16$ , explicitly work out the case of $n=7$ , prove that some sequences of rays are extremal for varying $n$ , and then finally explicitly work out each remaining extremal ray for $n\leqslant 16$ . These are all new results except for the cases of $n\leqslant 5$ .

7.2 The effective cone of the Hilbert scheme of $7$ points

It is worth showing how the theorem is applied in one of these cases to compute the effective cone. Recall that the general strategy to compute an effective cone has two steps. First, provide effective divisors. Second, provide moving curves which are dual to the effective divisors.

We use our main theorem to do this for the primary extremal rays of the effective cone; we have to deal with the secondary extremal rays separately. There is a single secondary extremal ray which is spanned by $B$ . $B$ is clearly an effective divisor as it is the locus of nonreduced schemes. In order to show that $B$ spans an extremal ray, we just have to construct two distinct dual moving curves.

We now construct these moving curves, $C_{1}$ and $C_{2}$ . We construct $C_{1}$ by fixing $6$ general points and then varying a seventh point along a curve of type $(1,0)$ . Similarly, we construct $C_{2}$ by fixing $6$ general points and then varying a seventh point along a curve of type $(0,1)$ . Any set of $7$ distinct points lies on at least one curve of type $C_{1}$ , so it is a moving curve. Similarly, $C_{2}$ is a moving curve.

We now show that $C_{1}$ and $C_{2}$ are dual to $B$ . Starting with six general points, we can find a line $l$ of type $(1,0)$ that does not contain any of the points. We get a curve $C^{\prime }$ of type $C_{1}$ in the Hilbert scheme by varying the seventh point along $l$ . As $l$ does not contain any of the six general points, every point in $C^{\prime }$ corresponds to seven distinct points, so $C^{\prime }$ does not intersect $B$ . Thus, we get

$$\begin{eqnarray}C_{1}\cdot B=C^{\prime }\cdot B=0.\end{eqnarray}$$

Similarly, we get that

$$\begin{eqnarray}C_{2}\cdot B=0.\end{eqnarray}$$

The only thing left to do in order to show that $B$ spans an extremal ray is to show that $C_{1}$ and $C_{2}$ have distinct classes. Starting with six general points, we find lines $l$ and $l^{\prime }$ of type $(1,0)$ that does not contain any of the points. Again, we get a curve $C^{\prime }$ of type $C_{1}$ in the Hilbert scheme by varying the seventh point along $l$ . Analogously, we get a divisor $H^{\prime }$ of type $H_{1}$ as the locus of schemes whose support intersects $l^{\prime }$ . As $l^{\prime }$ does not contain any of the general fixed points and does not intersect $l$ , we get that $H_{1}$ and $C^{\prime }$ are disjoint. Thus,

$$\begin{eqnarray}C_{1}\cdot H_{1}=C^{\prime }\cdot H^{\prime }=0.\end{eqnarray}$$

Using the same six general points, we find a line $l_{0}$ of type $(0,1)$ that does not contain any of the points. We get a curve $C_{0}$ of type $C_{2}$ by varying the seventh point along $l_{0}$ . Then $l_{0}$ does not contain any of the six general points by construction but does intersect $l^{\prime }$ in exactly one point. Thus,

$$\begin{eqnarray}C_{2}\cdot H_{1}=C_{0}\cdot H^{\prime }=1.\end{eqnarray}$$

As $C_{1}\cdot H_{1}\neq C_{2}\cdot H_{1}$ , we know that $C_{1}$ and $C_{2}$ are distinct classes. This observation completes the proof that $B$ spans an extremal ray.

While constructing the primary extremal rays, we construct two moving curves dual to $B$ . These curves show that $B$ is the only secondary extremal ray. Also as the slope of the ideal sheaf is $(0,0)$ , the effective cone is symmetric in the coordinates of $H_{1}$ and $H_{2}$ so we only deal with the primary rays spanned by $X_{i,j}$ where $i\geqslant j$ . Keeping that in mind, we move to computing the primary extremal rays using our theorem.

One way to think about the main results of this paper are that they give an algorithm to compute the primary extremal rays of the effective cone of  $M(\unicode[STIX]{x1D709})$ . That algorithm breaks down roughly into four steps: find the extremal pairs, use the extremal pairs to resolve the general object of $M(\unicode[STIX]{x1D709})$ , use those resolutions to construct maps to moduli spaces of Kronecker modules, and analyze these maps to find divisors spanning extremal rays. Let us follow those steps in this specific case.

Step 1. As we noted above, the first step is to find all of the extremal pairs. Proceed by finding all controlling exceptional bundles, finding the controlling pairs, and then finding which are extremal pairs.

Controlling exceptional bundles are those controlling the $\unicode[STIX]{x1D6FF}$ -surface over the locus

$$\begin{eqnarray}\left\{X\in \left(1,\unicode[STIX]{x1D707},{\textstyle \frac{1}{2}}\right)\subset K(\mathbb{P}^{1}\times \mathbb{P}^{1}):\unicode[STIX]{x1D712}(X\otimes {\mathcal{I}}_{z})=0\quad \text{for }{\mathcal{I}}_{z}\in \left(\mathbb{P}^{1}\times \mathbb{P}^{1}\right)^{[n]}\right\}.\end{eqnarray}$$

Using Mathematica, we find that these controlling exceptional bundles are

$$\begin{eqnarray}\displaystyle & \displaystyle \{\!\ldots ,\{0,6,1\},\{0,7,1\},\{0,8,1\},\{0,9,1\},\{0,10,1\},\{0,11,1\},\{0,12,1\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \{0,13,1\},\{0,14,1\},\{1,27,5\},\{1,13,3\},\{2,11,3\},\{1,2,1\},\{1,3,1\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \{1,4,1\},\{6,12,5\},\{2,1,1\},\{2,2,1\},\ldots \!\} & \displaystyle \nonumber\end{eqnarray}$$

where we record an exceptional bundle with Chern character $\left(r,(\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}),\unicode[STIX]{x1D6E5}\right)$ as $(\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2},r)$ . We truncated the list before and after all bundles which are useful in computing the effective cone. We can see that the ones we have truncated do not matter as our first resolution is dual to $B$ .

There are many, many controlling exceptional pairs, but we do not need to see all of them.

Finally, we check to see which of these are extremal pairs. They are whittled down by eliminating each pair that does not have each of the properties of an extremal pair. The only four controlling pairs that are extremal pairs are

$$\begin{eqnarray}\displaystyle & \displaystyle \{{\mathcal{O}}(6,0),{\mathcal{O}}(7,0)\},\{{\mathcal{O}}(3,1),{\mathcal{O}}(6,0)\},\{{\mathcal{O}}(2,1),{\mathcal{O}}(3,1)\},\qquad \text{and} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \{{\mathcal{O}}(2,1),{\mathcal{O}}(2,2)\}. & \displaystyle \nonumber\end{eqnarray}$$

Each extremal pair controls an extremal ray of the effective cone. Recall that given an extremal pair $\{A,B\}$ , the extremal ray it corresponds to is spanned by the primary orthogonal Chern character of the pair: $\operatorname{ch}(A)$ , $\operatorname{ch}(B)$ , or

$$\begin{eqnarray}p=\{X\in K(\mathbb{P}^{1}\times \mathbb{P}^{1}):Q_{\unicode[STIX]{x1D709},A}(X)=Q_{\unicode[STIX]{x1D709},B}(X)=\unicode[STIX]{x1D712}({\mathcal{I}}_{z}\otimes X)=0\}.\end{eqnarray}$$

Then the primary orthogonal Chern character for our exceptional pairs are $\left(1,(6,0),0\right)$ , $\left(1,(6,0),0\right)$ , $\left(5,(12,6),12\right)$ , and $\left(2,(4,3),5\right)$ , respectively. These Chern characters correspond to the extremal rays $X_{6,0}$ , $X_{6,0}$ , $X_{\frac{12}{5},\frac{6}{5}}$ , and $X_{2,\frac{3}{2}}$ , respectively. Notice that one of the rays is repeated twice. This repetition is because we need all of these extremal pairs to share each of their elements with another extremal pair in order to link neighboring extremal rays with moving curves.