2.1 Curves

We refer to [Reference Arbarello, Cornalba, Griffiths and HarrisACG⁺85, Chapters IV--V] for a general treatment of Brill–Noether theory for curves.

Let C be a smooth proper curve of genus g. For $1 \le d \le 2g-3$ and $r \ge 0$ , the Brill–Noether locus ${\mathrm{W}}_{d}^r(C) \subset \mathrm{Pic}^d(C)$ is defined as

\begin{equation*}{\mathrm{W}}_d^r(C) \mathrel{:=} \{ {\mathcal{L}} \in \mathrm{Pic}^d(C) \mid h^0({\mathcal{L}}) \ge r + 1,\ h^1({\mathcal{L}}) \ge 1 \}.\end{equation*}

If a line bundle ${\mathcal{L}}$ corresponds to a point of ${\mathrm{W}}_d^r(C) \setminus {\mathrm{W}}_d^{r+1}(C)$ the cotangent space to ${\mathrm{W}}_d^r(C)$ at this point is equal to the cokernel of the Petri map

\begin{equation*}{\mathrm{H}}^0(C,{\mathcal{L}}) \otimes {\mathrm{H}}^0(C, {\mathcal{L}}^{-1}(K_C)) \to {\mathrm{H}}^0(C, {\mathcal{O}}_C(K_C)).\end{equation*}

It is well known (see, for example, [Reference Arbarello, Cornalba, Griffiths and HarrisACG⁺85, Theorem V.1.1]) that for $r \ge d - g$ one has

\begin{equation*}\dim {\mathrm{W}}_d^r(C) \ge g - (r + 1)(g - d + r),\end{equation*}

and that if the right-hand side (the expected dimension of ${\mathrm{W}}_d^r(C)$ ) is non-negative, ${\mathrm{W}}_d^r(C) \ne \varnothing$ .

Definition 2.1. A smooth proper curve C is Brill–Noether general (BN-general) if the locus ${\mathrm{W}}_d^r(C)$ is non-empty if and only if $(r + 1)(g - d + r) \le g$ ; in other words,

\begin{equation*}h^0({\mathcal{L}}) \cdot h^1({\mathcal{L}}) \le g\end{equation*}

for any line bundle ${\mathcal{L}}$ on C. Furthermore, we will say that C is Brill–Noether–Petri general (BNP-general) if it is BN-general and the Petri map is injective for any line bundle ${\mathcal{L}}$ on C.

A general curve of genus g is known to be BNP-general (see [Reference GiesekerGie82] and [Reference LazarsfeldLaz86]). For BNP-general curves the locus ${\mathrm{W}}_d^r(C) \setminus {\mathrm{W}}_d^{r+1}(C)$ is smooth and non-empty of dimension $g - (r + 1)(g - d + r)$ whenever this number is non-negative ([Reference Arbarello, Cornalba, Griffiths and HarrisACG⁺85, Proposition IV.4.2 and Theorem V.1.7]).

The following definition axiomatizes the situation studied in [Reference MukaiMuk93, 3].

Definition 2.2. A pair of line bundles $(\xi,\eta)$ on a curve C of genus $g = r \cdot s$ is a Mukai pair of type (r,s) if $\xi \otimes \eta \cong {\mathcal{O}}_C(K_C)$ ,

(8) \begin{equation}\deg(\xi) = (r - 1)(s + 1),\quad h^0(\xi) = r,\quad \text{hence}\quad h^1(\xi) = s,\end{equation}

and both $\xi$ and $\eta$ are globally generated.

If $(\xi,\eta)$ is a Mukai pair of type (r,s) then Serre duality implies

\begin{equation*}\deg(\eta) = (r + 1)(s - 1),\quad h^0(\eta) = s,\quad \text{and}\quad h^1(\eta) = r,\end{equation*}

and $(\eta,\xi)$ is a Mukai pair of type (s,r). Thus, the definition is symmetric.

The following lemma deduces the existence of Mukai pairs from Brill–Noether theory.

If C is BNP-general and $(\xi,\eta)$ is a Mukai pair, then the Petri maps for $\xi$ and $\eta$ are isomorphisms.

Proof. If $\deg(\xi) = (r - 1)(s + 1)$ then $\chi(\xi) = (r - 1)(s + 1) + 1 - rs = r - s$ by Riemann–Roch. So, if $h^0(\xi) \gt r$ then $h^1(\xi) \gt s$ , which contradicts the BN-generality of C, hence $h^0(\xi) = r$ and $h^1(\xi) = s$ . Similarly, if $\xi$ is not globally generated at a point P then $\xi(-P)$ violates the BN property. Therefore, $\xi$ is globally generated, and by the symmetry of the definition the same holds for $\eta \mathrel{:=} \xi^{-1}(K_C)$ . Thus, $(\xi,\eta)$ is a Mukai pair of type (r,s).

Since $r \cdot s = g$ , the expected dimension of ${\mathrm{W}}_{(r-1)(s+1)}^{r-1}(C)$ is zero, hence ${\mathrm{W}}_{(r-1)(s+1)}^{r-1}(C) \ne \varnothing$ by [Reference Arbarello, Cornalba, Griffiths and HarrisACG⁺85, Theorem V.1.1]. This means that there is a line bundle $\xi$ of degree $(r-1)(s+1)$ with $h^0(\xi) \ge r$ , hence $(\xi,\xi^{-1}(K_C))$ is a Mukai pair of type (r,s).

Finally, if C is BNP-general, the Petri map for $\xi$ is injective, and as the dimensions of its source and target are the same, it is an isomorphism; again, the same property for $\eta$ follows.

Remark 2.4. If C is BNP-general, then the Brill–Noether locus ${\mathrm{W}}^{r-1}_{(r-1)(s+1)}(C)$ is reduced and zero-dimensional of length equal to

\begin{equation*}{\mathrm{N}}(r,s) \mathrel{:=}(rs)! \, \frac{\prod\nolimits_{i=1}^{r-1} i! \prod\nolimits_{i=1}^{s-1} i!}{ \prod\nolimits_{i=1}^{r+s-1} i!},\end{equation*}

the degree of the Grassmannian $\mathrm{Gr}(r,r+s)$ (see [Reference Eisenbud and HarrisEH87, Theorem 1]). Therefore, this degree is exactly the number of Mukai pairs of type (r,s) on a BNP-general curve.

The following proposition will be important for the study of Lazarsfeld bundles. Part (a), for any r, is attributed to Castelnuovo in [Reference CilibertoCil83, Theorem 1.11].

1. (a) If $(\xi, \eta)$ is a Mukai pair on C of type (r,s) then the natural morphism

   \begin{equation*}{\mathrm{H}}^0(C,\xi) \otimes {\mathrm{H}}^0(C, {\mathcal{O}}_C(K_C)) \to {\mathrm{H}}^0(C, \xi(K_C))\end{equation*}
   is surjective.

2. (b) If $(\xi_1, \eta_1)$ and $(\xi_2, \eta_2)$ are Mukai pairs of type (r,s) with $\xi_1 \not\cong \eta_2$ then the natural morphism

   \begin{equation*}{\mathrm{H}}^0(C, \xi_1) \otimes {\mathrm{H}}^0(C, \eta_2(K_C)) \to {\mathrm{H}}^0(C, \xi_1 \otimes \eta_2(K_C))\end{equation*}
   is surjective.

Proof. (a) First, assume $r = 2$ . As $\xi$ is globally generated by definition, we have an exact sequence

\begin{equation*}0 \to \xi^{-1} \to {\mathrm{H}}^0(C, \xi) \otimes {\mathcal{O}}_C \to \xi \to 0.\end{equation*}

Twisting this by $K_C$ , we obtain

\begin{equation*}0 \to \eta \to {\mathrm{H}}^0(C, \xi) \otimes {\mathcal{O}}_C(K_C) \to \xi(K_C) \to 0.\end{equation*}

As $h^1(\eta) = h^0(\xi) = h^1({\mathrm{H}}^0(C, \xi) \otimes {\mathcal{O}}_C(K_C))$ by Serre duality and $h^1(\xi(K_C)) = 0$ because the degree of $\xi$ is positive, this exact sequence induces a surjection of global sections, which is our claim.

Now assume $r = 3$ , and let $V = {\mathrm{H}}^0(C, \xi)$ . Since $\xi$ is globally generated, it induces a morphism $C \to {\mathbb{P}}^2 = {\mathbb{P}}(V^{\vee})$ . The pullback of the Koszul complex on ${\mathbb{P}}^2$ is an exact sequence

\begin{equation*}0 \to \wedge^3 V \otimes \xi^{-3} \to \wedge^2 V \otimes \xi^{-2} \to V \otimes \xi^{-1} \to {\mathcal{O}}_C \to 0.\end{equation*}

Twisting this by $\xi(K_C)$ , we obtain an exact sequence

\begin{equation*}0 \to \wedge^3 V \otimes \xi^{-2}(K_C) \to \wedge^2 V \otimes \xi^{-1}(K_C) \to V \otimes {\mathcal{O}}_C(K_C) \to \xi(K_C) \to 0.\end{equation*}

The ${\mathbf{E}}_1$ page of its hypercohomology spectral sequence has the form

Since it converges to zero, surjectivity of the last arrow in the lower row is equivalent to the dashed arrow in the ${\mathbf{E}}_2$ page being zero. To show this, it is enough to check that the upper row is exact in the second term. Tensoring it with $\wedge^3 V^\vee$ , using the natural identifications $V \otimes \wedge^3 V^\vee = \wedge^2 V^\vee$ and $\wedge^2 V \otimes \wedge^3 V^\vee = V^\vee$ , and dualizing it afterwards, we obtain the sequence

\begin{equation*}\wedge^2 V \to V \otimes V \to {\mathrm{H}}^0(C,\xi^2),\end{equation*}

where the first arrow is the natural embedding. Therefore, it is enough to check that the natural map $\mathrm{Sym}^2V = \mathrm{Sym}^2{\mathrm{H}}^0(C,\xi) \to {\mathrm{H}}^0(C,\xi^2)$ is injective. But if it is not injective, the image of the map $C \to {\mathbb{P}}^2$ given by $\xi$ is contained in a conic in ${\mathbb{P}}^2$ , hence this map factors through a map $C \to {\mathbb{P}}^1$ of degree $1/2\deg(\xi) = 1/2(r - 1)(s + 1) = s + 1$ , hence ${\mathrm{W}}_{s + 1}^1(C) \ne \varnothing$ , which contradicts the BN-generality of the curve C as $g - 2 (g - s) = 2s - g < 0$ .

(b) As before, consider the morphism $C \to {\mathbb{P}}({\mathrm{H}}^0(C, \xi_1)^\vee)$ and the pullback of the Koszul complex

\begin{equation*}\dots \to \wedge^2{\mathrm{H}}^0(C, \xi_1) \otimes \xi_1^{-2} \to {\mathrm{H}}^0(C, \xi_1) \otimes \xi_1^{-1} \to {\mathcal{O}}_C \to 0.\end{equation*}

Tensoring this by $\xi_1 \otimes \eta_2(K_C)$ , we obtain an exact sequence

\begin{equation*}\dots \to \wedge^2{\mathrm{H}}^0(C, \xi_1) \otimes \xi_1^{-1} \otimes \eta_2(K_C) \to {\mathrm{H}}^0(C, \xi_1) \otimes \eta_2(K_C) \to \xi_1 \otimes \eta_2(K_C) \to 0.\end{equation*}

Now note that $\deg(\xi_1) = r \le s = \deg(\eta_2)$ , and if this is an equality, $\xi_1^{-1} \otimes \eta_2$ is a non-trivial (by the assumption $\xi_1 \not\cong \eta_2$ ) line bundle of degree zero, hence ${\mathrm{H}}^1(C, \xi_1^{-1} \otimes \eta_2(K_C)) = 0$ . Therefore, the hypercohomology spectral sequence proves the surjectivity of the required map.

2.2 Quasipolarized K3 surfaces

In § 5 we will have to deal with polarized K3 surfaces with du Val singularities; these will appear as special members of generic pencils of hyperplane sections of terminal Fano threefolds. A minimal resolution of such a surface is a smooth quasipolarized K3 surface. In this subsection we explain this relation and prove a few useful results. For readers only interested in the case where X is smooth, it is enough to consider the case of K3 surfaces that have at most one ordinary double point, since in this case a generic pencil is a Lefschetz pencil.

So, let $(\bar{S},\bar{H})$ be a K3 surface with du Val singularities and an ample class $\bar{H} \in \mathrm{Pic}(\bar{S})$ . Let

\begin{equation*}\sigma \colon S \to \bar{S},\quad H \mathrel{:=} \sigma^*(\bar{H})\end{equation*}

be the minimal resolution of singularities and the pullback of the ample class. Then S is a smooth K3 surface and $H \in \mathrm{Pic}(S)$ is a quasipolarization, that is, a big and nef divisor class. The exceptional locus of $\sigma$ is formed by a finite number of smooth rational curves $R_i \subset S$ that are characterized by the condition $H \cdot R_i = 0$ and form an ADE configuration that we denote by ${\mathfrak{R}}(S,H)$ .

Conversely, if (S,H) is a quasipolarized K3 surface, all irreducible curves orthogonal to H are smooth and rational and they form an ADE configuration ${\mathfrak{R}}(S,H)$ (see [Reference Saint-DonatSD74, (4.2)]); moreover, a multiple of H defines a proper birational morphism $\sigma \colon S \to \bar{S}$ onto a K3 surface $\bar{S}$ with du Val singularities contracting the configuration ${\mathfrak{R}}(S,H)$ so that H is the pullback of an ample class $\bar{H} \in \mathrm{Pic}(\bar{S})$ . The morphism $\sigma$ will be referred to as the contraction of ${\mathfrak{R}}(S,H)$ .

The two above constructions are mutually inverse, so the language of polarized du Val K3 surfaces is equivalent to the language of quasipolarized smooth K3 surfaces. Below we use the language that is more convenient, depending on the situation.

If H is a quasipolarization of a smooth (or a polarization of a du Val) K3 surface S then

(9) \begin{equation}H^2 = 2g - 2,\quad h^0({\mathcal{O}}_S(H)) = g + 1,\quad \text{and}\quad h^1({\mathcal{O}}_S(H)) = h^2({\mathcal{O}}_S(H)) = 0,\end{equation}

where $g \ge 2$ is an integer called the genus of (S,H). The line bundle ${\mathcal{O}}_S(H)$ is globally generated if and only if the linear system $|H|$ contains a smooth connected curve $C \subset S$ of genus g; one direction follows from Bertini’s theorem and [Reference Saint-DonatSD74, Proposition 2.6], the other is [Reference Saint-DonatSD74, Theorem 3.1].

If (S,H) is a quasipolarized smooth K3 surface, the irreducible curves $R_i$ that form the ADE configuration ${\mathfrak{R}}(S,H)$ are called the simple roots; they satisfy $R_i^2 = -2$ and generate a root system in $H^\perp \subset \mathrm{Pic}(S) \otimes {\mathbb{R}}$ . Its positive roots are non-negative linear combinations $R = \sum a_i R_i$ such that $R^2 = -2$ , and the reflections in $R_i$ generate an action of the corresponding Weyl group.

A basic fact about Weyl groups (see, for example, Corollary 2 to Proposition IV.17 in [Reference BourbakiBou02]) tells us that for any non-simple positive root R there is a simple root $R_j$ such that $R \cdot R_j = -1$ . Then $R' = R - R_j$ is also a positive root and $R' \cdot R_j = 1$ . Therefore, considering R and R’ as Cartier divisors on S, we obtain exact sequences

(10) \begin{equation}0 \to {\mathcal{O}}_{R_j}(-1) \to {\mathcal{O}}_R \to {\mathcal{O}}_{R'} \to 0\end{equation}

and

(11) \begin{equation}0 \to {\mathcal{O}}_{R'}(-R_j) \to {\mathcal{O}}_R \to {\mathcal{O}}_{R_j} \to 0.\end{equation}

These sequences are useful for inductive arguments about positive roots.

Lemma 2.6. If $D \in \mathrm{Pic}(S)$ and $R = \sum a_iR_i$ is a positive root in ${\mathfrak{R}}(S,H)$ then

\begin{equation*}\chi({\mathcal{O}}_R(D)) = 1 + D \cdot R.\end{equation*}

Moreover, if $D \cdot R_i \le 0$ for all $R_i$ with $a_i \gt 0$ and $D \cdot R < 0$ then $h^0({\mathcal{O}}_R(D)) = 0$ .

The second statement is clear if R is a simple root. Otherwise, we consider exact sequences (10) and (11), where $R_j$ is a simple root such that $R \cdot R_j = -1$ and $R' = R- R_j$ , so that $R' \cdot R_j = 1$ . Since R’ is a positive root, we have $a_j \gt 0$ , hence $D \cdot R_j \le 0$ by the hypothesis of the lemma.

If $D \cdot R' < 0$ , twisting (10) by D and noting that $h^0({\mathcal{O}}_{R_j}(D \cdot R_j - 1)) = 0$ because $D \cdot R_j \le 0$ , we see that $h^0({\mathcal{O}}_R(D)) \le h^0({\mathcal{O}}_{R'}(D))$ , which is zero by induction, hence $h^0({\mathcal{O}}_R(D)) = 0$ .

On the other hand, the hypotheses of the lemma imply that $D \cdot R' = \sum_i (a_i - \delta_{i,j}) D \cdot R_i \le 0$ , so if $D \cdot R' \ge 0$ , it follows that $D \cdot R' = 0$ and $D \cdot R_j < 0$ , hence $h^0({\mathcal{O}}_{R_j}(D)) = 0$ . It also follows that $a_j = 1$ ; therefore, twisting (11) by D, we obtain $h^0({\mathcal{O}}_R(D)) = h^0({\mathcal{O}}_{R'}(D - R_j))$ , and since $(D - R_j) \cdot R' = -R' \cdot R_j = -1$ , this is zero, again by induction.

Using further the terminology of Weyl groups, we will say that a divisor class D on a quasipolarized K3 surface (S,H) is minuscule if

\begin{equation*}D \cdot R \in \{-1,0,1\}\end{equation*}

for any positive root R in the ADE configuration ${\mathfrak{R}}(S,H)$ .

If D is minuscule, by [Reference BourbakiBou02, Exercises 23, 24 to Chapter VI.2] there is a sequence of minuscule divisor classes $D_0 = D, D_1, \dots, D_n$ such that for each $1 \le k \le n$ the difference $R_{i_k} \mathrel{:=} D_{k-1} - D_{k}$ is a simple root satisfying $D_k \cdot R_{i_k} = - D_{k-1} \cdot R_{i_k} = 1$ , so that there are exact sequences

(12) \begin{equation}0 \to {\mathcal{O}}_S(D_k) \to {\mathcal{O}}_S(D_{k-1}) \to {\mathcal{O}}_{R_{i_k}}(-1) \to 0,\end{equation}

and $D_n \cdot R \in \{0,1\}$ for all positive roots R in ${\mathfrak{R}}(S,H)$ ; in other words, $D_n$ is nef over the contraction $\bar{S}$ of ${\mathfrak{R}}(S,H)$ . The sequence $\{D_k\}$ as above is not unique, but the last element $D_n$ is unique; we will call it the dominant replacement for D and denote it by $D_+$ . Note that

(13) \begin{equation}D_+^2 = D^2,\quad H \cdot D_+ = H \cdot D,\quad \text{and}\quad {\mathrm{H}}^\bullet(S, {\mathcal{O}}_S(D_+)) \cong {\mathrm{H}}^\bullet(S, {\mathcal{O}}_S(D)).\end{equation}

Moreover, if $\sigma \colon S \to \bar{S}$ is the contraction of ${\mathfrak{R}}(S,H)$ then

(14) \begin{equation}{\mathrm{R}}\sigma_*{\mathcal{O}}_S(D) \cong {\mathrm{R}}\sigma_*{\mathcal{O}}_S(D_+).\end{equation}

Indeed, both (13) and (14) follow from the definition and sequences (12) by induction.

Recall that a coherent sheaf ${\mathcal{F}}$ on a scheme X is maximal Cohen–Macaulay if $\mathcal{E}\!\mathit{xt}^i({\mathcal{F}}, {\mathcal{O}}_X) = 0$ for all $i \ge 1$ (see [Reference BuchweitzBuc21, Definition 4.2.1]); in other words, if the derived dual of ${\mathcal{F}}$ is a pure sheaf. We refer to [Reference BuchweitzBuc21] for the basic properties of maximal Cohen–Macaulay sheaves.

Lemma 2.7. Let (S,H) be a quasipolarized K3 surface of genus $g = r \cdot s$ . Let $\sigma \colon S \to \bar{S}$ be the contraction of the ADE configuration ${\mathfrak{R}}(S,H)$ . If $D \in \mathrm{Pic}(S)$ is a minuscule divisor class on S then ${\mathrm{R}}^1\sigma_*{\mathcal{O}}_S(D) = 0$ and

(15) \begin{equation}(\sigma_*{\mathcal{O}}_S(D))^\vee \cong \sigma_*{\mathcal{O}}_S(-D).\end{equation}

In particular, $\sigma_*{\mathcal{O}}_S(D)$ is a maximal Cohen–Macaulay sheaf on $\bar{S}$ . If, moreover, the dominant replacement $D_+$ is globally generated, then so is the sheaf $\sigma_*{\mathcal{O}}_S(D)$ .

Proof. The dominant replacement $D_+$ for D is nef over $\bar{S}$ ; hence, we have ${\mathrm{R}}^1\sigma_*{\mathcal{O}}_S(D_+) = 0$ by the Kawamata–Viehweg vanishing theorem. Applying (14), we obtain ${\mathrm{R}}^1\sigma_*{\mathcal{O}}_S(D) = 0$ . Furthermore,

\begin{equation*}\mathrm{R}\mathcal{H}\mathit{om}(\sigma_*{\mathcal{O}}_{S}(D), {\mathcal{O}}_{\bar{S}}) \cong {\mathrm{R}}\sigma_*\mathrm{R}\mathcal{H}\mathit{om}({\mathcal{O}}_{S}(D), \sigma^!{\mathcal{O}}_{\bar{S}}) \cong {\mathrm{R}}\sigma_*\mathrm{R}\mathcal{H}\mathit{om}({\mathcal{O}}_{S}(D), {\mathcal{O}}_{S}) \cong {\mathrm{R}}\sigma_*{\mathcal{O}}_{S}(-D)\end{equation*}

(where the first isomorphism is the Grothendieck duality and the second is the crepancy of $\sigma$ ). Clearly, the divisor $-D$ is also minuscule, hence the right-hand side is a pure sheaf, hence $\sigma_*{\mathcal{O}}_{S}(D)$ is a maximal Cohen–Macaulay sheaf and (15) holds.

If D is minuscule and $D_+$ is globally generated then the zero locus of a general pair of global sections of ${\mathcal{O}}_S(D_+)$ is a zero-dimensional subscheme of S, disjoint from the union $\cup R_i$ of all simple roots in ${\mathfrak{R}}(S,H)$ . Therefore, the corresponding Koszul complex

\begin{equation*}0 \to {\mathcal{O}}_S(-D_+) \to {\mathcal{O}}_S \oplus {\mathcal{O}}_S \to {\mathcal{O}}_S(D_+) \to 0\end{equation*}

is exact in a neighbourhood of $\cup R_i$ . Pushing it forward to $\bar{S}$ and taking into account that $-D_+$ is also minuscule, and hence ${\mathrm{R}}^1\sigma_*{\mathcal{O}}_S(-D_+) = 0$ , we see that $\sigma_*{\mathcal{O}}_S(D_+)$ is globally generated in a neighbourhood of $\mathrm{Sing}(\bar{S})$ . On the other hand, over the complement of $\mathrm{Sing}(\bar{S})$ the morphism $\sigma$ is an isomorphism, hence $\sigma_*{\mathcal{O}}_S(D_+)$ is globally generated away from $\mathrm{Sing}(\bar{S})$ , hence everywhere. Since $\sigma_*{\mathcal{O}}_S(D) \cong \sigma_*{\mathcal{O}}_S(D_+)$ by (14), it is also globally generated.

2.3 Brill–Noether theory for K3 surfaces

The following definition was introduced in [Reference MukaiMuk02, Definition 3.8] for polarized K3 surfaces; we generalize it to the case of quasipolarizations.

Definition 2.8. A quasipolarized K3 surface (S,H) is called Brill–Noether general (BN-general) if

\begin{equation*}h^0({\mathcal{O}}_S(D)) \cdot h^0({\mathcal{O}}_S(H - D)) < h^0({\mathcal{O}}_S(H)) = g + 1\end{equation*}

for all $D \not\in \{0,H\}$ , where g is the genus of (S,H) (see (9)).

Remark 2.9. In general, a quasipolarization H may have base points, but if (S,H) is BN-general, it is base-point-free. Indeed, if H is not base-point-free, then by [Reference Saint-DonatSD74, Corollary 3.2] the linear system $|H|$ has a fixed component D. In particular, $h^0({\mathcal{O}}_S(H-D)) = h^0({\mathcal{O}}_S(H)) = g+1$ and

\begin{equation*}h^0({\mathcal{O}}_S(D)) \cdot h^0({\mathcal{O}}_S(H-D)) = 1 \cdot (g+1) = g+1,\end{equation*}

contradicting the assumption that (S, H) is BN-general.

The following theorem relates Brill–Noether properties of curves and K3 surfaces.

1. (a) If $|H|$ contains a smooth connected BN-general curve $C \subset S$ then (S,H) is BN-general.

2. (b) If every curve in $|H|$ is reduced and irreducible then a general curve in $|H|$ is BNP-general.

Proof. Part (b) is easy. Assume $h^0({\mathcal{O}}_S(D)) \cdot h^0({\mathcal{O}}_S(C-D)) \ge g + 1$ . Using the exact sequence

\begin{equation*}0 \to {\mathcal{O}}_S(D - C) \to {\mathcal{O}}_S(D) \to {\mathcal{O}}_C(D\vert_C) \to 0\end{equation*}

and effectivity and non-triviality of $C - D$ , we obtain $h^0({\mathcal{O}}_C(D\vert_C)) \ge h^0({\mathcal{O}}_S(D))$ . Using the similar inequality for $C - D$ , we obtain

\begin{equation*}h^0({\mathcal{O}}_C(D\vert_C)) \cdot h^0({\mathcal{O}}_C((C-D)\vert_C)) \ge g + 1\end{equation*}

which contradicts the BN-generality of C.

Part (b) is much harder; it is proved in [Reference LazarsfeldLaz86].

The following definition is analogous to Definition 2.2 of a Mukai pair on a curve; it plays an important role in the rest of the paper.

Definition 2.12. Let $r, s \ge 2$ . A special Mukai class of type (r,s) on a quasipolarized smooth K3 surface (S,H) of genus $g = r\cdot s$ is a globally generated class $\Xi \in \mathrm{Pic}(S)$ such that

(16) \begin{equation}\Xi \cdot H = (r - 1)(s + 1),\quad h^0({\mathcal{O}}_S(\Xi)) = r,\quad \text{and}\quad h^1({\mathcal{O}}_S(\Xi)) = h^2({\mathcal{O}}_S(\Xi)) = 0.\end{equation}

Recall from § 2.2 the definition of minuscule divisor classes.

Lemma 2.13. If $\Xi$ is a special Mukai class of type (r,s) on a BN-general quasipolarized K3 surface (S,H) of genus $g = r \cdot s$ then

\begin{equation*}(H - \Xi) \cdot H = (r + 1)(s - 1),\quad h^0({\mathcal{O}}_S(H - \Xi)) = s,\!\quad \text{and}\quad \!h^1({\mathcal{O}}_S(H - \Xi)) = h^2({\mathcal{O}}_S(H - \Xi)) = 0.\end{equation*}

Moreover, $\Xi$ and $H - \Xi$ are minuscule.

Proof. We have $H \cdot (H - \Xi) = 2rs - 2 - (r - 1)(s + 1) = (r + 1)(s - 1) \gt 0$ , hence $h^0({\mathcal{O}}_S(\Xi - H)) = 0$ and the natural exact sequence

\begin{equation*}0 \to {\mathcal{O}}_S(\Xi - H) \to {\mathcal{O}}_S(\Xi) \to {\mathcal{O}}_C(\Xi\vert_C) \to 0\end{equation*}

shows that $h^0({\mathcal{O}}_C(\Xi\vert_C)) \ge h^0({\mathcal{O}}_S(\Xi)) = r$ . Since, on the other hand, $\deg({\mathcal{O}}_S(\Xi)\vert_C) = (r - 1)$ $(s + 1)$ , it follows that $h^1({\mathcal{O}}_C(\Xi\vert_C)) \ge s$ , and then the above exact sequence implies that

\begin{equation*}h^0({\mathcal{O}}_S(H - \Xi)) = h^2({\mathcal{O}}_S(\Xi - H)) = h^1({\mathcal{O}}_C(\Xi\vert_C)) \ge s.\end{equation*}

By BN-generality of S, this must be an equality, and then, using Riemann–Roch and taking the vanishing $h^2({\mathcal{O}}_S(H - \Xi)) = h^0({\mathcal{O}}_S(\Xi - H))$ into account, we deduce the first part of the lemma.

For the second part, since $(H - \Xi) \cdot R = - \Xi \cdot R$ , it is enough to show that $\Xi$ is minuscule. So, let R be a positive root. We have $\Xi \cdot R \ge 0$ because $\Xi$ is globally generated, hence nef, and R is effective, and it remains to show that $\Xi \cdot R \le 1$ . So, assume $\Xi \cdot R \ge 2$ . Then $(\Xi + R) \cdot R \ge 0$ , hence $\chi({\mathcal{O}}_R(\Xi + R)) \gt 0$ by Lemma 2.6, hence $h^0({\mathcal{O}}_R(\Xi + R)) \gt 0$ , and the exact sequence

\begin{equation*}0 \to {\mathcal{O}}_S(\Xi) \to {\mathcal{O}}_S(\Xi + R) \to {\mathcal{O}}_R(\Xi + R) \to 0\end{equation*}

implies that $h^0({\mathcal{O}}_S(\Xi + R)) \gt h^0({\mathcal{O}}_S(\Xi)) = r$ .

On the other hand, $(H - \Xi) \cdot R_i = - \Xi \cdot R_i \le 0$ for any simple root $R_i$ , because $\Xi$ is nef, and $(H - \Xi) \cdot R = - \Xi \cdot R\le -2$ , hence we have $h^0({\mathcal{O}}_R(H - \Xi)) = 0$ , again by Lemma 2.6. Now the exact sequence

\begin{equation*}0 \to {\mathcal{O}}_S(H - \Xi - R) \to {\mathcal{O}}_S(H - \Xi) \to {\mathcal{O}}_R(H - \Xi) \to 0\end{equation*}

implies that $h^0({\mathcal{O}}_S(H - \Xi - R)) = h^0({\mathcal{O}}_S(H - \Xi)) = s$ . We conclude that

\begin{equation*}h^0({\mathcal{O}}_S(\Xi + R)) \cdot h^0({\mathcal{O}}_S(H - \Xi - R)) \gt rs = g,\end{equation*}

contradicting Brill–Noether generality of (S,H). Thus, $\Xi$ is minuscule.

The next result gives a precise relation between special Mukai classes and Mukai pairs. Recall from § 2.2 the definition of the dominant replacement $D_+$ of a minuscule divisor class D.

Assume to the contrary that $(H - \Xi)_+$ is not globally generated. We saw in the proof of Lemma 2.13 that $h^0({\mathcal{O}}_C(\Xi\vert_C)) \ge r$ . Since C is BN-general, $({\mathcal{O}}_S(\Xi)\vert_C, {\mathcal{O}}_S(H - \Xi)\vert_C)$ is a Mukai pair by Lemma 2.3, and in particular ${\mathcal{O}}_C((H - \Xi)_+\vert_C) \cong {\mathcal{O}}_C((H - \Xi)\vert_C)$ is globally generated. Since $h^1({\mathcal{O}}_S(-\Xi)) = 0$ , the restriction morphism ${\mathrm{H}}^0(S, {\mathcal{O}}_S((H - \Xi))) \to {\mathrm{H}}^0(C, {\mathcal{O}}_C((H - \Xi)\vert_C))$ is surjective, and then (13) implies that ${\mathrm{H}}^0(S, {\mathcal{O}}_S((H - \Xi)_{+})) \to {\mathrm{H}}^0(C, {\mathcal{O}}_C((H - \Xi)_{+}\vert_C))$ is surjective as well, hence $(H - \Xi)_+$ is globally generated in a neighbourhood of C. Since C intersects any curve on S except for those in ${\mathfrak{R}}(S,H)$ and since the base locus of a divisor on a smooth K3 surface is a Cartier divisor (see [Reference Saint-DonatSD74, Corollary 3.2]), we conclude that the base locus of $(H - \Xi)_+$ is contained in the ADE configuration ${\mathfrak{R}}(S,H)$ . Thus,

\begin{equation*}(H - \Xi)_+ \cdot \Gamma \ge 0\end{equation*}

for any irreducible curve not contained in ${\mathfrak{R}}(S,H)$ . But the same inequality also holds for any $\Gamma$ contained in ${\mathfrak{R}}(S,H)$ by the definition of dominant replacement. Therefore, $(H - \Xi)_+$ is nef. Since it is not globally generated by our assumption, [Reference ReidRei97, Theorem 3.8(b)] implies that $(H - \Xi)_+^2 \gt 0$ , then [Reference ReidRei97, Theorem 3.8(c)] shows that $(H - \Xi)_+$ is big, and [Reference ReidRei97, Theorem 3.8(d)] proves that

\begin{equation*}(H - \Xi)_+ = aE + R_i,\end{equation*}

where $R_i$ is a simple root, $|E|$ is an elliptic pencil, $E \cdot R_i = 1$ , and $a \ge 2$ ; moreover, computing the square of both sides, we find that $a = s - 1$ , in particular $s \ge 3$ . Now let $\zeta \mathrel{:=} {\mathcal{O}}_S(E)\vert_C$ . Then ${\mathcal{O}}_S((H - \Xi)_+\vert_C) \cong \zeta^{(s-1)}$ , and $\zeta$ is a line bundle with $h^0(\zeta) \ge 2$ of degree

\begin{equation*}\deg(\zeta) = \frac{1}{s-1} (H - \Xi)_+ \cdot H = \frac{1}{s-1}(r + 1)(s - 1) = r + 1.\end{equation*}

Then $\chi(\zeta) = r + 2 - rs$ , hence $h^1(\zeta) \ge r(s - 1)$ , hence $h^0(\zeta)h^1(\zeta) \ge 2r(s-1)$ ; since $s \ge 3$ , this is larger than the genus $g = rs$ of C. Thus, we obtain a contradiction with BN-generality of C.

Corollary 2.15. Let $\Xi$ be a special Mukai class of type (r,s) on a BN-general quasipolarized K3 surface (S,H) of genus $g = r \cdot s \ge 4$ . If $\sigma \colon S \to \bar{S}$ is the contraction of ${\mathfrak{R}}(S,H)$ then

\begin{equation*}(\sigma_*{\mathcal{O}}_S(\Xi))^\vee \cong \sigma_*{\mathcal{O}}_S(-\Xi).\end{equation*}

Moreover, $\sigma_*{\mathcal{O}}_S(\Xi)$ and $\sigma_*{\mathcal{O}}_S(H - \Xi)$ are globally generated maximal Cohen–Macaulay sheaves on $\bar{S}$ with $h^0(\sigma_*{\mathcal{O}}_S(\Xi)) = r$ , $h^0(\sigma_*{\mathcal{O}}_S(H - \Xi)) = s$ , and no higher cohomology.

3.1 The Lazarsfeld bundle

In this section, we explain a construction, due to Lazarsfeld, of a vector bundle on $\bar{S}$ with Mukai vector (17) and state a criterion for stability of this bundle. This will give us a construction of a Mukai bundle.

Note that if $(\bar{S},\bar{H})$ is a polarized du Val K3 surface, a smooth curve $C \subset \bar{S}$ in $|\bar{H}|$ is contained in the smooth locus of $\bar{S}$ (because it is a Cartier divisor). Similarly, if (S,H) is a quasipolarized smooth K3 surface, a smooth curve $C \subset S$ in $|H|$ does not intersect the simple roots $R_i$ . Thus, if $\sigma \colon S \to \bar{S}$ is the contraction of ${\mathfrak{R}}(S,H)$ and $H = \sigma^*(\bar{H})$ we have a bijection between smooth curves on $\bar{S}$ in $|\bar{H}|$ and smooth curves on S in $|H|$ . Given such a curve C, we will denote its embedding into both S and $\bar{S}$ by j, hoping that this will not cause any confusion.

Now assume that C is a smooth BNP-general curve. We fix a factorization $g = r \cdot s$ , where r and s are as in (18). By Lemma 2.3 the curve C has a Mukai pair $(\xi,\eta)$ of type (r,s). If $j \colon C \to S$ is an embedding into a smooth or du Val K3 surface, we define, following [Reference LazarsfeldLaz86] the Lazarsfeld bundle ${\mathbf{L}}_S(\xi)$ by the exact sequence

(19) \begin{equation}0 \to {\mathbf{L}}_{S}(\xi) \xrightarrow\quad {\mathrm{H}}^0(C,\xi) \otimes {\mathcal{O}}_{S} \xrightarrow{\ \mathrm{ev}\ } j_*\xi \to 0,\end{equation}

where ev is the evaluation morphism (note that $\xi$ is globally generated by Definition 2.2). Note that ${\mathbf{L}}_{S}(\xi) \cong \sigma^*{\mathbf{L}}_{\bar{S}}(\xi)$ .

The most important properties of ${\mathbf{L}}_{\bar{S}}(\xi)$ are summarized in the following.

Lemma 3.3. Let (S,H) be a smooth quasipolarized or du Val polarized K3 surface of genus $g = r \cdot s$ and let C be a smooth BNP-general curve in $|H|$ . If $(\xi,\eta)$ is a Mukai pair of type (r,s) on C then the sheaf ${\mathbf{L}}_{S}(\xi)$ defined by (19) is locally free, spherical, has Mukai vector $(r, -H, s)$ and satisfies

(20) \begin{equation}h^0({\mathbf{L}}_{S}(\xi)) = h^1({\mathbf{L}}_{S}(\xi)) = 0\quad \text{and}\quad h^2({\mathbf{L}}_{S}(\xi)) = r + s.\end{equation}

The dual Lazarsfeld bundle ${\mathbf{L}}_{S}(\xi)^\vee$ fits into an exact sequence

(21) \begin{equation}0 \to {\mathrm{H}}^0(C,\xi)^\vee \otimes {\mathcal{O}}_{S} \to {\mathbf{L}}_{S}(\xi)^\vee \to j_*\eta \to 0.\end{equation}

It is globally generated and satisfies

(22) \begin{equation}h^0({\mathbf{L}}_{S}(\xi)^\vee) = r + s\quad \text{and}\quad h^1({\mathbf{L}}_{S}(\xi)^\vee) = h^2({\mathbf{L}}_{S}(\xi)^\vee) = 0.\end{equation}

Equalities (20) follow from the cohomology exact sequence of (19) and (8). Applying Serre duality, we deduce (22). The sequence (21) follows from (19) by dualizing, and the global generation of ${\mathbf{L}}_{S}(\xi)^\vee$ follows from ${\mathrm{H}}^1(S, {\mathcal{O}}_S) = 0$ and the global generation of ${\mathcal{O}}_{S}$ and $\eta$ .

To compute $\mathrm{Ext}^\bullet({\mathbf{L}}_{S}(\xi), {\mathbf{L}}_{S}(\xi))$ and check that ${\mathbf{L}}_{S}(\xi)$ is spherical, we use the defining short exact sequence (19). Taking into account the Grothendieck duality isomorphism

\begin{equation*}\mathrm{Ext}^p(j_*\xi, {\mathcal{O}}_{S}) \cong \mathrm{Ext}^p(\xi, j^!{\mathcal{O}}_{S}) \cong \mathrm{Ext}^p(\xi, {\mathcal{O}}_C(K_C)[-1]) \cong \mathrm{Ext}^{p-1}(\xi, {\mathcal{O}}_C(K_C)),\end{equation*}

we obtain a self-dual spectral sequence with the ${\mathbf{E}}_1$ page

that converges to $\mathrm{Ext}^\bullet({\mathbf{L}}_{S}(\xi), {\mathbf{L}}_{S}(\xi))$ . Obviously, the arrow of the bottom row induces an isomorphism . Since $\mathrm{Hom}(j_*\xi, j_*\xi) \cong \mathrm{Hom}(\xi, \xi) \cong {\Bbbk}$ , to check that ${\mathbf{L}}_{S}(\xi)$ is simple, it is enough to show that the first arrow in the middle row is injective. For this, note that we have a natural self-dual exact sequence

\begin{equation*}0 \to \mathrm{Ext}^1(\xi, \xi) \to \mathrm{Ext}^1(j_*\xi, j_*\xi) \to \mathrm{Hom}(\xi, \xi(K_C)) \to 0\end{equation*}

and the composition

\begin{equation*}{\mathrm{H}}^0(\xi) \otimes \mathrm{Hom}(\xi, {\mathcal{O}}_C(K_C)) \to \mathrm{Ext}^1(j_*\xi, j_*\xi) \to \mathrm{Hom}(\xi, \xi(K_C))\end{equation*}

of the first arrow in the middle row of the spectral sequence with the second arrow in the exact sequence above is the Petri map of $\xi$ , hence it is an isomorphism by Lemma 2.3. Therefore, the first arrow in the middle row of the spectral sequence is injective, hence $\mathrm{Hom}({\mathbf{L}}_{S}(\xi), {\mathbf{L}}_{S}(\xi)) = {\Bbbk}$ . Finally, by (6) we have $\chi({\mathbf{L}}_{S}(\xi), {\mathbf{L}}_{S}(\xi)) = 2rs - H^2 = 2$ , hence ${\mathbf{L}}_{S}(\xi)$ is spherical.

Lemma 3.3 shows that ${\mathbf{L}}_{\bar{S}}(\xi)$ is a Mukai bundle if and only if it is $\bar{H}$ -Gieseker stable. The main result of this section is a criterion for stability of ${\mathbf{L}}_{\bar{S}}(\xi)$ in terms of special Mukai classes on the minimal resolution S of $\bar{S}$ defined in § 2.3 (see Definition 2.12).

Theorem 3.4. Let $(\bar{S},\bar{H})$ be a polarized du Val K3 surface of genus $g = r \cdot s$ with

\begin{equation*}r \in \{2,3\}\quad \text{and}\quad s \ge r.\end{equation*}

Let $\sigma \colon S \to \bar{S}$ be the minimal resolution and let $H \mathrel{:=} \sigma^*(\bar{H})$ . Assume $|\bar{H}|$ contains a (smooth) BNP-general curve $C \subset \bar{S}$ . Then the following conditions are equivalent.

1. (a) The Lazarsfeld bundle ${\mathbf{L}}_{\bar{S}}(\xi)$ associated with a Mukai pair $(\xi,\eta)$ of type (r,s) on C is not $\bar{H}$ -Gieseker stable.

2. (b) The surface (S,H) has a special Mukai class $\Xi$ of type (r,s) such that ${\mathcal{O}}_S(\Xi)\vert_C \cong \xi$ .

Moreover, if (S,H) does not have special Mukai classes of type (r,s) then ${\mathbf{L}}_{\bar{S}}(\xi)$ does not depend on the choice of the curve C and Mukai pair $(\xi,\eta)$ and it is a Mukai bundle of type (r,s) on $\bar{S}$ .

We will prove this theorem in § 3.4 after some preparation.

3.2 Multispherical filtration

In this and the next subsections we work on a smooth quasipolarized K3 surface S. We start with a few well-known results.

1. (a) If ${\mathcal{F}} \hookrightarrow {\mathcal{O}}_S^{\oplus n}$ is a monomorphism from a sheaf ${\mathcal{F}}$ with ${\mathrm{c}}_1({\mathcal{F}}) \ge 0$ then ${\mathcal{F}}^{\vee\vee} \cong {\mathcal{O}}_S^{\oplus m}$ .

2. (b) If ${\mathcal{O}}_S^{\oplus n} \twoheadrightarrow {\mathcal{F}}$ is an epimorphism onto a torsion-free sheaf ${\mathcal{F}}$ with ${\mathrm{c}}_1({\mathcal{F}}) \le 0$ then ${\mathcal{F}} \cong {\mathcal{O}}_S^{\oplus m}$ .

(b) Dualizing the epimorphism, we obtain a monomorphism ${\mathcal{F}}^\vee \hookrightarrow {\mathcal{O}}_S^{\oplus n}$ . Since ${\mathcal{F}}$ is torsion-free, we have ${\mathrm{c}}_1({\mathcal{F}}^\vee) = - {\mathrm{c}}_1({\mathcal{F}}) \ge 0$ . Therefore, (a) implies that ${\mathcal{F}}^\vee \cong {\mathcal{O}}_S^{\oplus m}$ , hence ${\mathcal{F}}^{\vee\vee} \cong {\mathcal{O}}_S^{\oplus m}$ . The composition

\begin{equation*}{\mathcal{O}}_S^{\oplus n} \twoheadrightarrow {\mathcal{F}} \to {\mathcal{F}}^{\vee\vee} \cong {\mathcal{O}}_S^{\oplus m}\end{equation*}

is generically surjective, hence it is surjective everywhere. Finally, the middle map is injective because ${\mathcal{F}}$ is torsion-free, hence it is an isomorphism, and therefore ${\mathcal{F}} \cong {\mathcal{O}}_S^{\oplus m}$ .

Assume $h^2(\det({\mathcal{F}}')) \ne 0$ . By Serre duality there is a generically injective morphism $\det({\mathcal{F}}') \to {\mathcal{O}}$ , hence ${\mathrm{c}}_1({\mathcal{F}}') \le 0$ . Applying Lemma 3.7(b), we conclude that ${\mathcal{F}}' \cong {\mathcal{O}}^{\oplus m}$ . Finally, since the composition ${\mathcal{O}}^{\oplus n} \twoheadrightarrow {\mathcal{F}} \twoheadrightarrow {\mathcal{F}}' \cong {\mathcal{O}}^{\oplus m}$ is surjective, there is a splitting ${\mathcal{O}}^{\oplus m} \hookrightarrow {\mathcal{O}}^{\oplus n}$ of the composition, which induces a splitting ${\mathcal{F}}' \hookrightarrow {\mathcal{F}}$ of the epimorphism ${\mathcal{F}} \twoheadrightarrow {\mathcal{F}}'$ .

We will also need the following three fundamental results of Mukai.

Let ${\mathcal{F}}$ be a rigid sheaf on a smooth K3 surface. If

\begin{equation*}0 \to {\mathcal{F}}_1 \to {\mathcal{F}} \to {\mathcal{F}}_2 \to 0\end{equation*}

is an exact sequence and $\mathrm{Hom}({\mathcal{F}}_1,{\mathcal{F}}_2) = 0$ then ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are both rigid.

Moreover, if $\varepsilon \in \mathrm{Ext}^1({\mathcal{F}}_2,{\mathcal{F}}_1)$ is the extension class of the sequence then

1. (a) if ${\mathcal{F}}_1$ is simple then the map $\mathrm{Hom}({\mathcal{F}}_2,{\mathcal{F}}_2) \xrightarrow{\ \varepsilon\ } \mathrm{Ext}({\mathcal{F}}_2,{\mathcal{F}}_1)$ is surjective; and

2. (b) if ${\mathcal{F}}_2$ is simple then the map $\mathrm{Hom}({\mathcal{F}}_1,{\mathcal{F}}_1) \xrightarrow{\ \varepsilon\ } \mathrm{Ext}({\mathcal{F}}_2,{\mathcal{F}}_1)$ is surjective.

The main goal of this subsection is to construct a filtration of any rigid sheaf on a smooth quasipolarized K3 surface (S,H) into stable spherical sheaves. Since Gieseker stability for a quasipolarization is not well defined, we use a notion of stability that is equivalent to Gieseker stability with respect to a small ample perturbation $H + \epsilon A$ of the quasipolarization H. By Riemann–Roch the reduced Hilbert polynomial ${\mathbf{p}}_{H + \epsilon A}({\mathcal{F}})(t)$ of a sheaf ${\mathcal{F}}$ with respect to polarization $H + \epsilon A$ can be expressed in terms of the slope and the reduced Euler characteristic of ${\mathcal{F}}$ defined in (7) as

\begin{equation*}{\mathbf{p}}_{H + \epsilon A}({\mathcal{F}})(t) = \tfrac{1}{2} (H + \epsilon A)^2 t^2 + \mu_H({\mathcal{F}}) t + \epsilon\,\mu_A({\mathcal{F}}) t + \delta({\mathcal{F}}).\end{equation*}

Consequently, $(H + \epsilon A)$ -stability for $0 < \epsilon \ll 1$ can be rephrased in the following terms.

Given two sheaves ${\mathcal{F}}_1, {\mathcal{F}}_2$ on S of positive rank, we write ${\mathbf{p}}_{H, A}({\mathcal{F}}_1) \prec {\mathbf{p}}_{H, A}({\mathcal{F}}_2)$ if

• – $\mu_H({\mathcal{F}}_1) < \mu_H({\mathcal{F}}_2)$ , or

• – $\mu_H({\mathcal{F}}_1) = \mu_H({\mathcal{F}}_2)$ and $\mu_A({\mathcal{F}}_1) < \mu_A({\mathcal{F}}_2)$ , or

• – $\mu_H({\mathcal{F}}_1) = \mu_H({\mathcal{F}}_2)$ and $\mu_A({\mathcal{F}}_1) = \mu_A({\mathcal{F}}_2)$ and $\delta({\mathcal{F}}_1) < \delta({\mathcal{F}}_2)$ .

Similarly, we write ${\mathbf{p}}_{H, A}({\mathcal{F}}_1) = {\mathbf{p}}_{H, A}({\mathcal{F}}_2)$ if

\begin{equation*}\mu_H({\mathcal{F}}_1) = \mu_H({\mathcal{F}}_2),\quad \mu_A({\mathcal{F}}_1) = \mu_A({\mathcal{F}}_2),\quad \text{and}\quad \delta({\mathcal{F}}_1) = \delta({\mathcal{F}}_2).\end{equation*}

We say that a torsion-free sheaf ${\mathcal{F}}$ is (H, A)-(semi)stable if ${\mathbf{p}}_{H, A}({\mathcal{F}}') \prec (\preceq)\ {\mathbf{p}}_{H, A}({\mathcal{F}})$ for every subsheaf ${\mathcal{F}}' \subset {\mathcal{F}}$ .

The following proposition is the main result of this section.

1. (a) for all i we have ${\mathcal{F}}_i \cong {\mathcal{G}}_i^{\oplus m_i}$ , where ${\mathcal{G}}_i$ is an (H, A)-stable spherical sheaf; and

2. (b) for all $ i \le j$ we have $\mu_H({\mathcal{F}}_i) \ge \mu_H({\mathcal{F}}_{j})$ and $\mathrm{Hom}({\mathcal{F}}_i, {\mathcal{F}}_{i+1}) = 0$ .

Moreover, $\sum m_i = 1$ if and only if ${\mathcal{F}}$ is (H, A)-stable.

We will call a filtration as in Proposition 3.14 a multispherical filtration of ${\mathcal{F}}$ . Note that we do not claim $\mathrm{Hom}({\mathcal{F}}_i, {\mathcal{F}}_j) = 0$ for $j \gt i+1$ ; indeed, a priori the same stable factor could appear multiple times in the filtration.

Now among all short exact sequences

\begin{equation*}0 \to {\mathcal{F}}' \to {\mathcal{F}} \to {\mathcal{F}}'' \to 0\end{equation*}

with the property that ${\mathcal{F}}'$ admits a filtration with all associated factors isomorphic to the sheaf ${\mathcal{G}}$ (hence ${\mathbf{p}}_{H,A}({\mathcal{F}}') = {\mathbf{p}}_{H,A}({\mathcal{G}}) = {\mathbf{p}}_{H,A}({\mathcal{F}})$ ), we consider one where ${\mathcal{F}}'$ has maximal possible rank.

Note that ${\mathcal{F}}''$ is torsion-free: otherwise, by Remark 3.13 the saturation ${\widetilde{\mathcal{F}}}'$ in ${\mathcal{F}}$ of the subsheaf ${\mathcal{F}}'$ has ${\mathbf{p}}_{H, A}({\widetilde{\mathcal{F}}}') \succ {\mathbf{p}}_{H, A}({\mathcal{F}}') = {\mathbf{p}}_{H, A}({\mathcal{F}})$ and destabilizes ${\mathcal{F}}$ . It is also easy to see that ${\mathcal{F}}''$ is (H, A)-semistable; otherwise the preimage in ${\mathcal{F}}$ of a destabilizing subsheaf of ${\mathcal{F}}''$ destabilizes ${\mathcal{F}}$ . We claim that $\mathrm{Hom}({\mathcal{G}}, {\mathcal{F}}'') = 0$ . Indeed, as ${\mathcal{G}}$ is (H, A)-stable with ${\mathbf{p}}_{H, A}({\mathcal{G}}) = {\mathbf{p}}_{H, A}({\mathcal{F}}'')$ any morphism ${\mathcal{G}} \to {\mathcal{F}}''$ would be injective, hence the preimage of ${\mathcal{G}}$ under the surjection $\mathcal{F} \twoheadrightarrow \mathcal{F}''$ would be an extension of ${\mathcal{G}}$ by ${\mathcal{F}}'$ of higher rank than ${\mathcal{F}}'$ , in contradiction to our assumption on ${\mathcal{F}}'$ .

It follows that $\mathrm{Hom}({\mathcal{F}}', {\mathcal{F}}'') = 0$ , and therefore by Lemma 3.11 both ${\mathcal{F}}'$ and ${\mathcal{F}}''$ are rigid. This implies that $\chi({\mathcal{F}}', {\mathcal{F}}') \gt 0$ . Since ${\mathcal{F}}'$ admits a filtration with factors isomorphic to ${\mathcal{G}}$ , its Mukai vector $\upsilon({\mathcal{F}}')$ is proportional to $\upsilon({\mathcal{G}})$ , hence we also have $\chi({\mathcal{G}}, {\mathcal{G}}) \gt 0$ . Since ${\mathcal{G}}$ is (H,A)-stable, hence simple, it must be spherical. In particular, $\mathrm{Ext}^1({\mathcal{G}}, {\mathcal{G}}) = 0$ , hence ${\mathcal{F}}' \cong {\mathcal{G}}^{\oplus m}$ .

By induction on the rank, this concludes the case where ${\mathcal{F}}$ is (H, A)-semistable.

Now assume that ${\mathcal{F}}$ is not (H, A)-semistable. The same arguments as for Gieseker stability (see, for example, [Reference Huybrechts and LehnHL10, Theorem 1.3.4]) prove the existence of a Harder–Narasimhan filtration for ${\mathcal{F}}$ with respect to (H, A)-stability. Applying Lemma 3.11 again, we see that every Harder–Narasimhan filtration factor of ${\mathcal{F}}$ is rigid. Combined with the filtrations of the semistable factors proven in the previous case, this immediately gives a filtration satisfying (a).

For (b), just note that our construction in fact implies ${\mathbf{p}}_{H,A}({\mathcal{F}}_i) \succeq {\mathbf{p}}_{H,A}({\mathcal{F}}_{i+1})$ , hence a fortiori $\mu_{H}({\mathcal{F}}_i) \ge \mu_{H}({\mathcal{F}}_{i+1})$ , and, moreover, $\mathrm{Hom}({\mathcal{F}}_i,{\mathcal{F}}_{i+1}) = 0$ .

3.3 Brill–Noether inequality

The following proposition is the key result for the proof of Theorem 3.4.

Proposition 3.15. Let (S,H) be a Brill–Noether general quasipolarized smooth K3 surface. Let ${\mathcal{F}}$ be a globally generated spherical bundle on S with $\upsilon({\mathcal{F}}) = (r, H, s)$ . If

\begin{equation*}0 \to {\mathcal{F}}_1 \to {\mathcal{F}} \to {\mathcal{F}}_2 \to 0\end{equation*}

is an exact sequence of non-trivial spherical bundles with $\mu_H({\mathcal{F}}_1) \ge \mu_H({\mathcal{F}}) \ge \mu_H({\mathcal{F}}_2)$ then

\begin{equation*}\mathrm{rk}({\mathcal{F}}_1) = {\mathrm{ch}}_2({\mathcal{F}}_2) + \mathrm{rk}({\mathcal{F}}_2) = 1.\end{equation*}

Moreover, the class $\Xi \mathrel{:=} {\mathrm{c}}_1({\mathcal{F}}_2)$ is a special Mukai class of type (r,s).

Proof. Let $\upsilon({\mathcal{F}}_i) = (r_i, D_i, s_i)$ , so that $\upsilon({\mathcal{F}}) = (r_1 + r_2, D_1 + D_2, s_1 + s_2)$ ; in particular, $D_1 + D_2 = H$ . Since ${\mathcal{F}}_1$ , ${\mathcal{F}}_2$ , and ${\mathcal{F}}$ are spherical, (6) implies that

(23) \begin{equation}D_1^2 = 2r_1s_1 - 2,\quad D_2^2 = 2r_2s_2 - 2,\quad (D_1 + D_2)^2 = 2(r_1 + r_2)(s_1 + s_2) - 2.\end{equation}

Subtracting the sum of the first two equalities from the last one and dividing by 2, we obtain

\begin{equation*}D_1 \cdot D_2 = r_1s_2 + r_2s_1 + 1.\end{equation*}

This implies $H \cdot D_1 = 2r_1s_1 + r_1s_2 + r_2s_1 - 1$ and $H \cdot D_2 = 2r_2s_2 + r_1s_2 + r_2s_1 - 1$ , hence

\begin{equation*}2s_1 + s_2 + \frac{r_2s_1 - 1}{r_1} =\frac1{r_1}H \cdot D_1 =\mu_H({\mathcal{F}}_1) \ge\mu_H({\mathcal{F}}_2) =\frac1{r_2}H \cdot D_2 =s_1 + 2s_2 + \frac{r_1s_2 - 1}{r_2}.\end{equation*}

Equivalently,

(24) \begin{equation}s_1 - s_2 \ge \frac{r_1s_2 - 1}{r_2} - \frac{r_2s_1 - 1}{r_1}.\end{equation}

This inequality will be used a few times below.

By Corollary 3.8 the line bundle $\det({\mathcal{F}}_2) \cong {\mathcal{O}}_S(D_2)$ is globally generated. Therefore, $D_2^2 \ge 0$ , hence $r_2s_2 \ge 1$ by (23), and since $r_2 \ge 1$ , we have $s_2 \ge 1$ .

Assume $s_1 \le 0$ . Since $r_1$ and $r_2$ are positive, we obtain $r_1s_2 - 1 \ge 0$ , $r_2s_1 - 1 < 0$ , hence the right-hand side of (24) is positive, while the left-hand side is negative, which is absurd. Therefore, $s_1 \ge 1$ .

Further, $h^2({\mathcal{O}}_S(D_2)) = 0$ by Corollary 3.8. Moreover, $h^2({\mathcal{O}}_S(D_1)) = 0$ ; indeed, otherwise Serre duality shows that $-D_1$ is effective, hence $\mu_H({\mathcal{F}}_1) \le 0$ , contradicting the assumption. Therefore,

\begin{equation*}h^0({\mathcal{O}}_S(D_i)) \ge \chi({\mathcal{O}}_S(D_i)) = \tfrac{1}{2}D_i^2 + 2 = r_is_i + 1.\end{equation*}

Since $r_i \ge 1$ and $s_i \ge 1$ , this is positive, and since $D_1 + D_2 = H$ and $g = (r_1 + r_2)(s_1 + s_2)$ by the last equality in (23), the Brill–Noether property of S implies that

\begin{equation*}(r_1s_1 + 1)(r_2s_2 + 1) \le h^0({\mathcal{O}}_S(D_1)) \cdot h^0({\mathcal{O}}_S(D_2)) \le (r_1 + r_2)(s_1 + s_2).\end{equation*}

Expanding the products, we obtain $r_1r_2s_1s_2 + r_1s_1 + r_2s_2 + 1 \le r_1s_1 + r_2s_2 + r_1s_2 + r_2s_1$ , hence

\begin{equation*}(r_1s_2 - 1)(r_2s_1 - 1) = r_1r_2s_1s_2 - r_1s_2 - r_2s_1 + 1 \le 0.\end{equation*}

But as we have shown above, both factors on the left-hand side are non-negative, therefore one of them is zero. If $r_1s_2 - 1 \ge 1$ , hence $r_2s_1 - 1 = 0$ , then $r_2 = s_1 = 1$ , the right-hand side of (24) is positive, hence $s_2 < s_1$ , in contradiction to $s_2 \ge 1$ . Therefore, $r_1s_2 - 1 = 0$ , hence $r_1 = s_2 = 1$ , that is,

\begin{equation*}\mathrm{rk}({\mathcal{F}}_1) = {\mathrm{ch}}_2({\mathcal{F}}_2) + \mathrm{rk}({\mathcal{F}}_2) = 1.\end{equation*}

Moreover, $h^0({\mathcal{O}}_S(D_1)) \ge r_1s_1 + 1 = s_1 + s_2 = s$ and $h^0({\mathcal{O}}_S(D_2)) \ge r_2s_2 + 1 = r_2 + r_1 = r$ , so the Brill–Noether property of S implies that these are equalities, hence $h^1({\mathcal{O}}_S(D_2)) = 0$ .

Finally, it follows that

\begin{equation*}D_2 \cdot H = 2r_2s_2 + r_1s_2 + r_2s_1 - 1 =2r_2 + 1 + r_2s_1 - 1 =r_2(s_1 + 2) =(r - 1)(s + 1),\end{equation*}

hence the divisor class $\Xi \mathrel{:=} D_2$ is a special Mukai class of type (r,s).

We will also need the following similar result.

Lemma 3.16. Let (S,H) be a Brill–Noether general quasipolarized smooth K3 surface. Let ${\mathcal{F}}$ be a globally generated spherical bundle on S with $\mathrm{rk}({\mathcal{F}}) = 3$ and ${\mathrm{c}}_1({\mathcal{F}}) = H$ . Then ${\mathcal{F}}$ cannot fit into an exact sequence

\begin{equation*}0 \to {\mathcal{F}}_1^{\oplus 2} \to {\mathcal{F}} \to {\mathcal{F}}_2 \to 0\quad \text{or}\quad 0 \to {\mathcal{F}}_1 \to {\mathcal{F}} \to {\mathcal{F}}_2^{\oplus 2} \to 0,\end{equation*}

where ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are line bundles and $\mu_H({\mathcal{F}}_1) \ge \mu_H({\mathcal{F}}_2)$ .

Assume we have the first sequence. Then $\upsilon({\mathcal{F}}) = (3, 2D_1 + D_2, 2s_1 + s_2)$ , and since ${\mathcal{F}}$ is spherical, we have $(2D_1 + D_2)^2 = 12s_1 + 6s_2 - 2$ , which implies

(25) \begin{equation}D_1 \cdot D_2 = s_1 + s_2 + 2.\end{equation}

Moreover, as in Proposition 3.15 the global generation of ${\mathcal{F}}_2$ implies $s_2 \ge 1$ , and since

\begin{equation*}2(2s_1 - 2) + (s_1 + s_2 + 2) = \mu_H({\mathcal{F}}_1) \ge \mu_H({\mathcal{F}}_2) = 2(s_1 + s_2 + 2) + (2s_2 - 2),\end{equation*}

we see that $3s_1 \ge 3s_2 + 4$ , hence $s_1 \ge 3$ . Finally, we have $h^2({\mathcal{O}}_S(D_1 + D_2)) = h^2({\mathcal{O}}_S(D_1)) = 0$ by the argument of Proposition 3.15, hence $h^0({\mathcal{O}}_S(D_1 + D_2)) \ge \chi({\mathcal{O}}_S(D_1 + D_2)) = 2s_1 + 2s_2 + 2$ and $h^0({\mathcal{O}}_S(D_1)) \ge \chi({\mathcal{O}}_S(D_1)) = s_1 + 1$ , and the Brill–Noether inequality gives

\begin{equation*}2(s_1 + s_2 + 1)(s_1 + 1) \le 6s_1 + 3s_2.\end{equation*}

This can be rewritten as $(2s_1 + 2s_2 - 1)(2s_1 - 1) + 3 \le 0$ , contradicting $s_1 \ge 3$ and $s_2 \ge 1$ .

Similarly, assuming the second sequence, we obtain $\upsilon({\mathcal{F}}) = (3, D_1 + 2D_2, s_1 + 2s_2)$ , which again implies (25). Arguing as before, we obtain $s_2 \ge 1$ and $3s_1 \ge 3s_2 - 4$ , hence $s_1 \ge 0$ , and this time the Brill–Noether inequality gives

\begin{equation*}2(s_1 + s_2 + 1)(s_2 + 1) \le 3s_1 + 6s_2\end{equation*}

which can be rewritten as $(2s_2 - 1)(2s_1 + 2s_2 - 1) + 3 \le 0$ , contradicting $s_1 \ge 0$ and $s_2 \ge 1$ .

3.4 Proof of the theorem

We can now finally start proving Theorem 3.4, so we return to the setup of § 3.1. We use the notation introduced therein. Also recall Definition 3.12 and notation (7).

First, we have $H = \sigma^*(\bar{H})$ and $\sigma_*({\mathrm{c}}_1({\mathcal{F}})) = {\mathrm{c}}_1(\sigma_*{\mathcal{F}})$ , hence the projection formula implies

\begin{equation*}\mu_H({\mathcal{F}}) =\frac{1}{\mathrm{rk}({\mathcal{F}})} {\mathrm{c}}_1({\mathcal{F}}) \cdot \sigma^*(\bar{H}) =\frac{1}{\mathrm{rk}(\sigma_*{\mathcal{F}})} {\mathrm{c}}_1(\sigma_*{\mathcal{F}}) \cdot \bar{H} =\mu_{\bar{H}}(\sigma_*{\mathcal{F}}) \ge\mu_{\bar{H}}({\mathbf{L}}_{\bar{S}}(\xi)) =\mu_H({\mathbf{L}}_S(\xi)).\end{equation*}

Moreover, if this is an equality then we must have $\delta(\sigma_*{\mathcal{F}}) \ge \delta({\mathbf{L}}_{\bar{S}}(\xi)) = \delta({\mathbf{L}}_{S}(\xi))$ .

Next, for any curve $R_i$ in ${\mathfrak{R}}(S,H)$ we have $H \cdot R_i = 0$ , hence ${\mathrm{c}}_1({\mathbf{L}}_S(\xi)) \cdot A = 0$ . Note also that ${\mathbf{L}}_S(\xi)$ is trivial on $R_i$ , because it is a pullback along $\sigma$ . Thus, ${\mathcal{F}}\vert_{R_i}$ is a subsheaf in a trivial vector bundle, hence we have ${\mathrm{c}}_1({\mathcal{F}}) \cdot R_i \le 0$ for all i. Therefore, we have ${\mathrm{c}}_1({\mathcal{F}}) \cdot A \ge 0$ , hence

(26) \begin{equation}\mu_{A}({\mathcal{F}}) =\frac{1}{\mathrm{rk}({\mathcal{F}})}{{\mathrm{c}}_1({\mathcal{F}})\cdot A} \ge0 =\frac{1}{\mathrm{rk}({\mathbf{L}}_S(\xi))}{{\mathrm{c}}_1({\mathbf{L}}_S(\xi))\cdot A} =\mu_{A}({\mathbf{L}}_S(\xi)).\end{equation}

Finally, if (26) is an equality, we have ${\mathrm{c}}_1({\mathcal{F}}) \cdot R_i = 0$ for each $R_i$ , hence ${\mathcal{F}}$ is trivial on each of these curves, hence ${\mathcal{F}} \cong \sigma^*(\sigma_*{\mathcal{F}})$ , hence $\chi({\mathcal{F}}) = \chi(\sigma_*{\mathcal{F}})$ , hence $\delta({\mathcal{F}}) = \delta(\sigma_*{\mathcal{F}})$ , and we conclude that ${\mathbf{p}}_{H,A}({\mathcal{F}}) \succeq {\mathbf{p}}_{H,A}({\mathbf{L}}_S(\xi))$ , so that ${\mathbf{L}}_S(\xi)$ is not (H,A)-stable.

Lemma 3.18. If $r \in \{2,3\}$ and the Lazarsfeld bundle ${\mathbf{L}}_{\bar{S}}(\xi)$ is not $\bar{H}$ -Gieseker stable, then its pullback ${\mathbf{L}}_S(\xi) \cong \sigma^*{\mathbf{L}}_{\bar{S}}(\xi)$ fits into an exact sequence

\begin{equation*}0 \to {\mathcal{G}}_1^{\oplus m_1} \to {\mathbf{L}}_S(\xi) \to {\mathcal{G}}_2^{\oplus m_2} \to 0,\end{equation*}

where ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ are spherical bundles with $\mu_H({\mathcal{G}}_1) \ge \mu_H({\mathcal{G}}_2)$ and $(m_1,m_2) \in \{(1,1), (1,2), (2,1)\}$ .

Now assume $r = 3$ . The only case where the conclusion is not immediately obvious is where the multispherical filtration of ${\mathbf{L}}_S(\xi)$ has three spherical factors ${\mathcal{G}}_1 \not\cong {\mathcal{G}}_2 \not\cong {\mathcal{G}}_3$ , By Proposition 3.10 the sheaves ${\mathcal{G}}_i$ are locally free, hence they are line bundles. Note also that

\begin{equation*}\mathrm{Hom}({\mathcal{G}}_1,{\mathcal{G}}_2) = \mathrm{Hom}({\mathcal{G}}_2,{\mathcal{G}}_3) = \mathrm{Hom}({\mathcal{G}}_1,{\mathcal{G}}_3) = 0.\end{equation*}

Indeed, the first two spaces are zero by Proposition 3.14. Moreover, if $\mu_H({\mathcal{G}}_1) \gt \mu_H({\mathcal{G}}_3)$ , the third space is zero by (H,A)-stability of ${\mathcal{G}}_1$ and ${\mathcal{G}}_3$ , and otherwise we would have $\mu_H({\mathcal{G}}_i) = \mu_H({\mathbf{L}}_S(\xi))$ for all i, hence $\mu_H({\mathbf{L}}_S(\xi))$ would be integral, while in fact .

Now consider the sheaves ${\mathcal{G}}_{1,2} \mathrel{:=} \mathrm{Ker}({\mathbf{L}}_S(\xi) \to {\mathcal{G}}_3)$ and ${\mathcal{G}}_{2,3} \mathrel{:=} {\mathbf{L}}_S(\xi)/{\mathcal{G}}_1$ . It follows that

\begin{equation*}\mathrm{Hom}({\mathcal{G}}_{1,2}, {\mathcal{G}}_3) = \mathrm{Hom}({\mathcal{G}}_1, {\mathcal{G}}_{2,3}) = 0,\end{equation*}

hence ${\mathcal{G}}_{1,2}$ and ${\mathcal{G}}_{2,3}$ are rigid by Lemma 3.11. Furthermore, Lemma 3.11 implies that the maps

\begin{equation*}{\Bbbk} = \mathrm{Hom}({\mathcal{G}}_i,{\mathcal{G}}_i) \to \mathrm{Ext}^1({\mathcal{G}}_i,{\mathcal{G}}_{i-1})\quad \text{and}\quad {\Bbbk} = \mathrm{Hom}({\mathcal{G}}_i,{\mathcal{G}}_i) \to \mathrm{Ext}^1({\mathcal{G}}_{i+1},{\mathcal{G}}_i)\end{equation*}

are surjective; in particular, the spaces $\mathrm{Ext}^1({\mathcal{G}}_2,{\mathcal{G}}_1)$ and $\mathrm{Ext}^1({\mathcal{G}}_3,{\mathcal{G}}_2)$ are at most one-dimensional.

If $\mathrm{Hom}({\mathcal{G}}_2,{\mathcal{G}}_1) \ne 0$ and $\mathrm{Hom}({\mathcal{G}}_3,{\mathcal{G}}_2) \ne 0$ , the composition of non-trivial morphisms ${\mathcal{G}}_3 \to {\mathcal{G}}_2$ and ${\mathcal{G}}_2 \to {\mathcal{G}}_1$ is non-trivial, hence $\mathrm{Hom}({\mathcal{G}}_3,{\mathcal{G}}_1) \ne 0$ , which is impossible by Lemma 3.6 because ${\mathbf{L}}_S(\xi)$ is spherical. Thus, one of the above spaces must be zero.

Assume $\mathrm{Hom}({\mathcal{G}}_2,{\mathcal{G}}_1) = 0$ and $\mathrm{Hom}({\mathcal{G}}_3,{\mathcal{G}}_2) \ne 0$ . If $\mathrm{Ext}^1({\mathcal{G}}_2,{\mathcal{G}}_1) = 0$ then ${\mathcal{G}}_{1,2} \cong {\mathcal{G}}_1 \oplus {\mathcal{G}}_2$ , hence we have $\mathrm{Hom}({\mathcal{G}}_3, {\mathcal{G}}_{1,2}) \ne 0$ , which is impossible by Lemma 3.6. Therefore, $\mathrm{Ext}^1({\mathcal{G}}_2,{\mathcal{G}}_1) = {\Bbbk}$ , hence we have $\chi({\mathcal{G}}_{1,2},{\mathcal{G}}_{1,2}) = 2$ , and since the sheaf ${\mathcal{G}}_{1,2}$ is rigid, it is spherical. Therefore, the filtration $0 \to {\mathcal{G}}_{1,2} \to {\mathbf{L}}_S(\xi) \to {\mathcal{G}}_3 \to 0$ has the required properties.

The case where $\mathrm{Hom}({\mathcal{G}}_2,{\mathcal{G}}_1) \ne 0$ and $\mathrm{Hom}({\mathcal{G}}_3,{\mathcal{G}}_2) = 0$ is considered analogously; in this case the filtration $0 \to {\mathcal{G}}_1 \to {\mathbf{L}}_S(\xi) \to {\mathcal{G}}_{2,3}\to 0$ has the required properties.

Finally, assume $\mathrm{Hom}({\mathcal{G}}_2,{\mathcal{G}}_1) = \mathrm{Hom}({\mathcal{G}}_3,{\mathcal{G}}_2) = 0$ . If also $\mathrm{Ext}^1({\mathcal{G}}_2,{\mathcal{G}}_1) = \mathrm{Ext}^1({\mathcal{G}}_3,{\mathcal{G}}_2) = 0$ then it is easy to see that ${\mathcal{G}}_2$ is a direct summand of ${\mathbf{L}}_S(\xi)$ , which is impossible because ${\mathbf{L}}_S(\xi)$ is spherical. Therefore, $\mathrm{Ext}^1({\mathcal{G}}_i,{\mathcal{G}}_{i-1}) = {\Bbbk}$ for $i = 2$ or $i = 3$ , hence ${\mathcal{G}}_{i-1,i}$ is spherical and as in one of the two previous cases we obtain a filtration of ${\mathbf{L}}_S(\xi)$ with the required properties.

Proof of Theorem 3.4. Since the surface S contains a BNP-general curve, it is BN-general (see Theorem 2.10(a)). Assume ${\mathbf{L}}_{\bar{S}}(\xi)$ is not $\bar{H}$ -Gieseker stable. Dualizing the sequence produced by Lemma 3.18, we obtain an exact sequence

\begin{equation*}0 \to {\mathcal{F}}_1 \to {\mathbf{L}}_S(\xi)^\vee \to {\mathcal{F}}_2 \to 0\end{equation*}

of multispherical sheaves, and we deduce from Lemma 3.16 that the cases where $(m_1,m_2) = (1,2)$ or $(m_1,m_2) = (2,1)$ are impossible, hence both ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ must be spherical. Therefore, Proposition 3.15 proves that $\Xi \mathrel{:=} {\mathrm{c}}_1({\mathcal{F}}_2)$ is a special Mukai class of type (r,s) on S and ${\mathcal{F}}_1$ is a line bundle, hence ${\mathcal{F}}_1 \cong {\mathcal{O}}_S(H - \Xi)$ . Consider the composition

\begin{equation*}{\mathcal{O}}_S(H - \Xi) \cong {\mathcal{F}}_1 \hookrightarrow {\mathbf{L}}_S(\xi)^\vee \to j_*\eta,\end{equation*}

where the last arrow comes from exact sequence (21). If the composition vanishes, sequence (21) implies that the embedding ${\mathcal{O}}_S(H - \Xi) \hookrightarrow {\mathbf{L}}_S(\xi)^\vee$ factors through ${\mathrm{H}}^0(C,\xi)^\vee \otimes {\mathcal{O}}_{S}$ , which is absurd because $H \cdot (H-\Xi) = (r + 1)(s - 1) \gt 0$ . Therefore, the composition is non-zero, so it factors through a non-zero morphism

\begin{equation*}{\mathcal{O}}_S(H - \Xi)\vert_C \to \eta.\end{equation*}

The source and target are line bundles of the same degree $(r + 1)(s - 1)$ , hence the above morphism is an isomorphism, and we conclude that ${\mathcal{O}}_S(\Xi)\vert_C \cong \eta^{-1}(K_C) \cong \xi$ .

Conversely, assume $\Xi$ is a special Mukai class of type (r,s) on S. Let $C \subset S$ be a BNP-general curve in $|H|$ and let

\begin{equation*}\xi \mathrel{:=} {\mathcal{O}}_S(\Xi)\vert_C,\quad \eta \mathrel{:=} {\mathcal{O}}_S(H - \Xi)\vert_C.\end{equation*}

We know from Proposition 2.14 that $(\xi,\eta)$ is a Mukai pair of type (r,s) and the restriction induces an isomorphism ${\mathrm{H}}^0(S,{\mathcal{O}}_S(\Xi)) \cong {\mathrm{H}}^0(C,\xi)$ . Consider the commutative diagram with exact rows

The middle vertical arrow is surjective, because $\Xi$ is globally generated, and the kernel of the right vertical arrow is $\sigma^*{\mathbf{L}}_{S}(\xi)$ . It follows that there is an epimorphism ${\mathbf{L}}_{S}(\xi) \twoheadrightarrow {\mathcal{O}}_S(\Xi-H)$ and

\begin{equation*}\mathrm{Ker}\bigl({\mathbf{L}}_{S}(\xi) \twoheadrightarrow {\mathcal{O}}_S(\Xi-H)\bigr) \cong \mathrm{Ker}\bigl({\mathrm{H}}^0(S,{\mathcal{O}}_S(\Xi)) \otimes {\mathcal{O}}_S \twoheadrightarrow {\mathcal{O}}_S(\Xi)\bigr).\end{equation*}

Moreover, as $\sigma_*{\mathcal{O}}_S(\Xi)$ is globally generated by Corollary 2.15, the first direct image of the right-hand side vanishes, hence the same is true for the first direct image of the left-hand side, hence the induced morphism ${\mathbf{L}}_{\bar{S}}(\xi) \cong \sigma_*{\mathbf{L}}_{S}(\xi) \to \sigma_*{\mathcal{O}}_S(\Xi-H)$ is surjective, and since

\begin{equation*}\mu_{\bar{H}}({\mathbf{L}}_{\bar{S}}(\xi)) = \frac{2}{r} - 2s \ge -(r+1)(s-1) = \mu_{\bar{H}}(\sigma_*{\mathcal{O}}_S(\Xi-H)),\end{equation*}

such an epimorphism violates stability of ${\mathbf{L}}_{\bar{S}}(\xi)$ .

Now assume that (S,H) does not have special Mukai classes of type (r,s). Then for any BNP-general curve C and any Mukai pair $(\xi,\eta)$ on it the bundle ${\mathbf{L}}_{\bar{S}}(\xi)$ is stable, hence it is a Mukai bundle of type (r,s). Finally, since a Mukai bundle is unique by Lemma 3.2, ${\mathbf{L}}_{\bar{S}}(\xi)$ does not depend on C or $\xi$ .

4.1 Restriction of Lazarsfeld bundles

Here we establish some cohomology vanishings for the Lazarsfeld bundles and deduce some corollaries about their restrictions to curves.

The first result is elementary.

Lemma 4.1. Let (S,H) be a quasipolarized K3 surface of genus $g = r \cdot s \ge 4$ and let $C \subset S$ be a BNP-general curve in $|H|$ . Assume $s \ge r \in \{2,3\}$ . If $(\xi,\eta)$ is a Mukai pair of type (r,s) on C then

\begin{alignat*}{1}{\mathrm{H}}^2(S,{\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H)) &={\mathrm{H}}^1(S,{\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H)) = 0\\{\mathrm{H}}^1(S,{\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H)) &= {\mathrm{H}}^0(S,{\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H)) = 0.\end{alignat*}

Proof. Twisting (19) by ${\mathcal{O}}_S(H)$ , we obtain an exact sequence

\begin{equation*}0 \to {\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H) \xrightarrow\quad {\mathrm{H}}^0(C,\xi) \otimes {\mathcal{O}}_S(H) \xrightarrow{\ \mathrm{ev}\ } j_*(\xi(K_C)) \to 0.\end{equation*}

Since ${\mathrm{H}}^1(S, {\mathcal{O}}_S(H)) = {\mathrm{H}}^2(S, {\mathcal{O}}_S(H)) = 0$ by (9) and ${\mathrm{H}}^1(C, \xi(K_C)) = 0$ because $\deg(\xi) \gt 0$ , the vanishing of ${\mathrm{H}}^2(S,{\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H))$ is obvious, and to prove the vanishing of ${\mathrm{H}}^1(S,{\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H))$ it is enough to check that the morphism

\begin{equation*}{\mathrm{H}}^0(C,\xi) \otimes {\mathrm{H}}^0(C, {\mathcal{O}}_C(K_C)) \to {\mathrm{H}}^0(C, \xi(K_C))\end{equation*}

is surjective, which is proved in Proposition 2.5(a). The remaining two vanishings follow from the first two by Serre duality.

The second vanishing result is more complicated; in particular, it relies on Theorem 3.4.

Proposition 4.2. Let (S,H) be a quasipolarized K3 surface of genus $g = r \cdot s \ge 6$ and let $C \subset S$ be a BNP-general curve in $|H|$ . Assume $s \ge r \in \{2,3\}$ . If S does not have special Mukai classes of type (r,s), then for any Mukai pair $(\xi,\eta)$ of type (r,s) on C we have

\begin{equation*}{\mathrm{H}}^1(S, {\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(-H)) ={\mathrm{H}}^1(S, {\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H)) = 0.\end{equation*}

Proof. First, tensoring (21) by ${\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H)$ , we obtain an exact sequence

\begin{equation*}0 \to {\mathrm{H}}^0(C, \xi)^\vee \otimes {\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H) \to {\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H) \to j_*(j^*{\mathbf{L}}_S(\xi) \otimes \eta(K_C)) \to 0.\end{equation*}

Its first term has no first cohomology by Lemma 4.1, so to prove the vanishing of the first cohomology of the middle term, it is enough to check that ${\mathrm{H}}^1(C, j^*{\mathbf{L}}_S(\xi) \otimes \eta(K_C)) = 0$ .

We now note that the number of Mukai pairs on C of type (r,s) is equal to the degree ${\mathrm{N}}(r,s)$ of $\mathrm{Gr}(r,r + s)$ (see Remark 2.4). Since $r,s \ge 2$ and $(r,s) \ne (2,2)$ , this number is greater than 2; in particular, we can choose a Mukai pair $(\xi',\eta')$ of type (r,s) such that $\xi \not\cong \xi'$ , $\eta \not\cong \xi'$ . Applying Theorem 3.4, we obtain an isomorphism

\begin{equation*}{\mathbf{L}}_S(\xi) \cong {\mathbf{L}}_S(\xi'),\end{equation*}

hence $j^*{\mathbf{L}}_S(\xi) \cong j^*{\mathbf{L}}_S(\xi')$ , so it is enough to check that ${\mathrm{H}}^1(C, j^*{\mathbf{L}}_S(\xi') \otimes \eta(K_C)) = 0$ .

Now consider sequence (19) for the line bundle $\xi'$ and its restriction to C:

\begin{equation*}0 \to \xi'(-K_C) \to j^*{\mathbf{L}}_S(\xi') \to {\mathrm{H}}^0(C, \xi') \otimes {\mathcal{O}}_C \to \xi' \to 0.\end{equation*}

Tensoring this by $\eta(K_C)$ , we obtain the exact sequence

\begin{equation*}0 \to \xi' \otimes \eta \to j^*{\mathbf{L}}_S(\xi') \otimes \eta(K_C) \to {\mathrm{H}}^0(C, \xi') \otimes \eta(K_C) \to \xi' \otimes \eta(K_C) \to 0.\end{equation*}

We have ${\mathrm{H}}^1(C, \xi' \otimes \eta) \cong {\mathrm{H}}^1(C, \xi' \otimes \xi^{-1} \otimes {\mathcal{O}}_C(K_C))$ , and since $\xi' \otimes \xi^{-1}$ is a non-trivial line bundle of degree 0, this space vanishes. Moreover, ${\mathrm{H}}^1(C, \eta(K_C)) = 0$ because $\deg(\eta) \gt 0$ . Therefore,

\begin{equation*}{\mathrm{H}}^1(C, j^*{\mathbf{L}}_S(\xi') \otimes \eta(K_C)) \cong \mathrm{Coker}({\mathrm{H}}^0(C, \xi') \otimes {\mathrm{H}}^0(C, \eta(K_C)) \to {\mathrm{H}}^0(C, \xi' \otimes \eta(K_C))),\end{equation*}

and it remains to note that the morphism on the right-hand side is surjective by Proposition 2.5(b). Therefore, ${\mathrm{H}}^1(S, {\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(H)) = 0$ , and the other vanishing follows by Serre duality.

4.2 Lazarsfeld extension class

For any globally generated line bundle $\xi$ on a smooth curve C we consider the evaluation morphism $\mathrm{ev} \colon {\mathrm{H}}^0(C, \xi) \otimes {\mathcal{O}}_C \twoheadrightarrow \xi$ . Its dual gives an exact sequence

(27) \begin{equation}0 \to \xi^{-1} \xrightarrow{\ \mathrm{ev}^\vee\ } {\mathrm{H}}^0(C,\xi)^\vee \otimes {\mathcal{O}}_C \xrightarrow{\quad } {\mathbf{R}}_C(\xi^{-1}) \to 0,\end{equation}

defining a vector bundle ${\mathbf{R}}_C(\xi^{-1})$ on C.

If $(\xi,\eta)$ is a Mukai pair, the sheaf ${\mathbf{R}}_C(\xi^{-1})$ has stronger properties.

Lemma 4.4. If $(\xi,\eta)$ is a Mukai pair of type (r,s) with $s \ge r \in \{2,3\}$ on a BNP-general curve C of genus $g = r \cdot s$ , then

\begin{equation*}{\mathrm{H}}^0(C,{\mathbf{R}}_C(\xi^{-1})) \cong {\mathrm{H}}^0(C,\xi)^\vee\quad \text{and}\quad \mathrm{Hom}({\mathbf{R}}_C(\xi^{-1}), \eta) = 0.\end{equation*}

Moreover, the sheaf ${\mathbf{R}}_C(\xi^{-1})$ is simple.

Proof. Applying the functor $\mathrm{Hom}(-, \eta)$ to (27), we obtain a left-exact sequence

\begin{equation*}0 \to \mathrm{Hom}({\mathbf{R}}_C(\xi^{-1}), \eta) \to {\mathrm{H}}^0(C,\xi) \otimes {\mathrm{H}}^0(C, \eta) \to {\mathrm{H}}^0(C, \xi \otimes \eta).\end{equation*}

Its second arrow is induced by the evaluation morphism of $\xi$ , hence it coincides with the Petri map. Since the curve C is BNP-general and $(\xi,\eta)$ is a Mukai pair, this map is an isomorphism (see Lemma 2.3), hence $\mathrm{Hom}({\mathbf{R}}_C(\xi^{-1}), \eta) = 0$ .

Furthermore, since ${\mathrm{H}}^0(C,\xi^{-1}) = 0$ , the long exact sequence of cohomology of (27) looks like

\begin{equation*} 0 \to {\mathrm{H}}^0(C,\xi)^\vee \to {\mathrm{H}}^0(C,{\mathbf{R}}_C(\xi^{-1})) \to {\mathrm{H}}^1(C,\xi^{-1}) \to {\mathrm{H}}^0(C,\xi)^\vee \otimes {\mathrm{H}}^1(C,{\mathcal{O}}_C) \to {\mathrm{H}}^1(C,{\mathbf{R}}_C(\xi^{-1})) \to 0.\end{equation*}

We need to check that the connecting map ${\mathrm{H}}^0(C,{\mathbf{R}}_C(\xi^{-1})) \to {\mathrm{H}}^1(C,\xi^{-1})$ is zero, that is, that the map ${\mathrm{H}}^1(C,\xi^{-1}) \to {\mathrm{H}}^0(C,\xi)^\vee \otimes {\mathrm{H}}^1(C,{\mathcal{O}}_C)$ is injective, that is, that its dual map is surjective. But the dual map is nothing more than the map

\begin{equation*}{\mathrm{H}}^0(C,\xi) \otimes {\mathrm{H}}^0(C, {\mathcal{O}}_C(K_C)) \to {\mathrm{H}}^0(C, \xi(K_C))\end{equation*}

whose surjectivity was established in Proposition 2.5(a). Thus, ${\mathrm{H}}^0(C,{\mathbf{R}}_C(\xi^{-1})) \cong {\mathrm{H}}^0(C,\xi)^\vee$ .

Finally, since the second arrow in (27) induces an isomorphism of global sections, it is the evaluation morphism for ${\mathbf{R}}_C(\xi^{-1})$ , hence any endomorphism of ${\mathbf{R}}_C(\xi^{-1})$ extends to an endomorphism of the exact sequence (27). On the other hand, since $\mathrm{Hom}({\mathbf{R}}_C(\xi^{-1}), {\mathcal{O}}_C) = 0$ by Lemma 4.3, an endomorphism of the exact sequence (27) is determined uniquely by its endomorphism of $\xi^{-1}$ . As $\xi^{-1}$ is simple, the same holds for ${\mathbf{R}}_C(\xi^{-1})$ .

The sheaf ${\mathbf{R}}_C(\xi^{-1})$ is naturally related to the restriction $j^*{\mathbf{L}}_S(\xi)$ of the Lazarsfeld bundle.

Lemma 4.5. Let $(\xi,\eta)$ be a Mukai pair on a BNP-general curve C which lies on a smooth quasipolarized K3 surface S. If $\phi \colon {\mathbf{L}}_S(\xi)^\vee \to j_*\eta$ is the epimorphism from (21) there is a canonical exact sequence

(28) \begin{equation}0 \to {\mathbf{R}}_C(\xi^{-1}) \xrightarrow{\qquad} j^*{\mathbf{L}}_S(\xi)^\vee \xrightarrow{\ j^*(\phi)\ } \eta \to 0.\end{equation}

Proof. Pulling back (21) and using an isomorphism $\eta(-K_C) \cong \xi^{-1}$ , we obtain an exact sequence

\begin{equation*}0 \to \xi^{-1} \xrightarrow{\qquad}{\mathrm{H}}^0(C,\xi)^\vee \otimes {\mathcal{O}}_C \xrightarrow{\qquad}j^*{\mathbf{L}}_S(\xi)^\vee \xrightarrow{\ j^*(\phi)\ }\eta \to 0;\end{equation*}

in particular, $j^*(\phi)$ is surjective. Moreover, this sequence is dual to the exact sequence

\begin{equation*}0 \to \eta^{-1} \xrightarrow{\quad } j^*{\mathbf{L}}_S(\xi) \xrightarrow{\quad } {\mathrm{H}}^0(C,\xi) \otimes {\mathcal{O}}_C \xrightarrow{\ \mathrm{ev}\ } \xi \to 0\end{equation*}

obtained by pulling back (19) (because their middle maps are mutually dual), hence the first map in the first sequence is the dual evaluation morphism $\mathrm{ev}^\vee$ . Therefore, its cokernel is isomorphic to ${\mathbf{R}}_C(\xi^{-1})$ , hence $\mathrm{Ker}(j^*(\phi)) \cong {\mathbf{R}}_C(\xi^{-1})$ , and we obtain (28).

Definition 4.6. We call the extension class

\begin{equation*}\epsilon_{{\mathbf{L}}_S(\xi)} \in \mathrm{Ext}^1(\eta, {\mathbf{R}}_C(\xi^{-1}))\end{equation*}

of the exact sequence (28) the Lazarsfeld extension class.

The main result of this subsection is the following proposition.

Proposition 4.8. Let (S,H) be a smooth quasipolarized K3 surface of genus $g = r \cdot s \ge 6$ and let $C \subset S$ be a BNP-general curve in $|H|$ . Assume $s \ge r \in \{2,3\}$ . If S does not have special Mukai classes of type (r,s) then for any Mukai pair $(\xi,\eta)$ of type (r,s) on C the restriction $j^*{\mathbf{L}}_S(\xi)$ of the Lazarsfeld bundle to C is simple and the Lazarsfeld extension class $\epsilon_{{\mathbf{L}}_S(\xi)}$ is non-zero, but the connecting morphism

\begin{equation*}\epsilon_{{\mathbf{L}}_S(\xi)} \colon {\mathrm{H}}^0(C,\eta) \to {\mathrm{H}}^1(C, {\mathbf{R}}_C(\xi^{-1}))\end{equation*}

of the exact sequence (28) vanishes.

Proof. Consider the exact sequence

\begin{equation*}0 \to {\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(-H) \to {\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi) \to j_*j^*({\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi)) \to 0.\end{equation*}

We have $h^0({\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(-H)) = 0$ because ${\mathbf{L}}_S(\xi)$ is spherical, and in particular simple, by Lemma 3.3, and $h^1({\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi) \otimes {\mathcal{O}}_S(-H)) = 0$ by Proposition 4.2. Therefore,

\begin{equation*}\mathrm{End}(j^*{\mathbf{L}}_S(\xi)) ={\mathrm{H}}^0(C, j^*({\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi))) ={\mathrm{H}}^0(S, {\mathbf{L}}_S(\xi)^\vee \otimes {\mathbf{L}}_S(\xi)) =\mathrm{End}({\mathbf{L}}_S(\xi)).\end{equation*}

Since ${\mathbf{L}}_S(\xi)$ is simple, so is $j^*{\mathbf{L}}_S(\xi)$ , and a fortiori $j^*{\mathbf{L}}_S(\xi)$ is indecomposable. Thus, the exact sequence (28) does not split, hence $\epsilon_{{\mathbf{L}}_S(\xi)} \ne 0$ .

Furthermore, consider the exact sequence

\begin{equation*}0 \to {\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H) \to {\mathbf{L}}_S(\xi)^\vee \to j_*j^*({\mathbf{L}}_S(\xi)^\vee) \to 0.\end{equation*}

We have $h^0({\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H)) = h^1({\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H)) = 0$ by Lemma 4.1, hence

\begin{equation*}h^0(j^*{\mathbf{L}}_S(\xi)^\vee) = h^0({\mathbf{L}}_S(\xi)^\vee) = r + s,\end{equation*}

where the second equality uses (22). Finally, consider the cohomology exact sequence of (28):

\begin{equation*}0 \to {\mathrm{H}}^0(C,{\mathbf{R}}_C(\xi^{-1})) \to {\mathrm{H}}^0(C, j^*{\mathbf{L}}_S(\xi)^\vee) \to {\mathrm{H}}^0(C, \eta) \to {\mathrm{H}}^1(C,{\mathbf{R}}_C(\xi^{-1})).\end{equation*}

Its first three terms have dimension r (by Lemma 4.4), $r+s$ (proved above), and s, respectively, and therefore the last map vanishes.

4.3 Mukai extension class

In this subsection we show that the properties of the Lazarsfeld extension class established in Proposition 4.8 characterize it uniquely if $\mathrm{Pic}(S) = {\mathbb{Z}} \cdot H$ . We axiomatize these properties in the following definition.

Definition 4.9. Let $(\xi,\eta)$ be a Mukai pair on C. A Mukai extension is a non-split exact sequence

(29) \begin{equation}0 \to {\mathbf{R}}_C(\xi^{-1}) \to {\mathcal{G}} \to \eta \to 0\end{equation}

such that the connecting morphism ${\mathrm{H}}^0(C, \eta) \to {\mathrm{H}}^1(C, {\mathbf{R}}_C(\xi^{-1}))$ vanishes. The extension class

\begin{equation*}\epsilon \in \mathrm{Ext}^1(\eta,{\mathbf{R}}_C(\xi^{-1}))\end{equation*}

of a Mukai extension is called a Mukai extension class.

To study Mukai extension classes we use the following generalization of the Lazarsfeld construction. In § 3.1 we defined the Lazarsfeld bundle ${\mathbf{L}}_S(\xi)$ of a globally generated line bundle $\xi$ . The same construction can be applied to any globally generated vector bundle ${\mathcal{G}}$ on C; it defines a vector bundle ${\mathbf{L}}_S({\mathcal{G}})$ that fits into an exact sequence

\begin{equation*}0 \to {\mathbf{L}}_S({\mathcal{G}}) \xrightarrow{\quad } {\mathrm{H}}^0(C, {\mathcal{G}}) \otimes {\mathcal{O}}_S \xrightarrow{\ \mathrm{ev}\ } j_*{\mathcal{G}} \to 0.\end{equation*}

We will apply this construction to all bundles in (29).

Lemma 4.10. If $(\xi,\eta)$ is a Mukai pair of type (r,s) with $s \ge r \in \{2,3\}$ on a BNP-general curve $C \subset S$ in a linear system $|H|$ then

\begin{equation*}{\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1})) \cong {\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H).\end{equation*}

Proof. Recall that ${\mathbf{R}}_C(\xi^{-1})$ is globally generated by Lemma 4.3, hence ${\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1}))$ is well defined. Moreover, ${\mathrm{H}}^0(C,{\mathbf{R}}_C(\xi^{-1})) \cong {\mathrm{H}}^0(C,\xi)^\vee$ by Lemma 4.4. Therefore, twisting the defining exact sequence of ${\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1}))$ by ${\mathcal{O}}_S(H)$ , we obtain

\begin{equation*}0 \to {\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1})) \otimes {\mathcal{O}}_S(H) \to {\mathrm{H}}^0(C,\xi)^\vee \otimes {\mathcal{O}}_S(H) \to j_*({\mathbf{R}}_C(\xi^{-1}) \otimes {\mathcal{O}}_C(K_C)) \to 0.\end{equation*}

By (9) we have ${\mathrm{H}}^2(S, {\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1})) \otimes {\mathcal{O}}_S(H)) = {\mathrm{H}}^1(C, {\mathbf{R}}_C(\xi^{-1}) \otimes {\mathcal{O}}_C(K_C))$ , and by Lemma 4.3 this is zero. Applying Serre duality, we deduce ${\mathrm{H}}^0(S, {\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1}))^\vee \otimes {\mathcal{O}}_S(-H)) = 0$ .

Furthermore, consider the commutative diagram

where the bottom row is the pushforward of (27). The middle vertical arrow is induced by the natural morphism ${\mathcal{O}}_S \to j_*{\mathcal{O}}_C$ , hence it is surjective, therefore the exact sequence of kernels and cokernels takes the form

\begin{equation*}0 \to {\mathrm{H}}^0(C,\xi)^\vee \otimes {\mathcal{O}}_S(-H) \to {\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1})) \to j_*(\xi^{-1}) \to 0,\end{equation*}

and since $\mathrm{R}\mathcal{H}\mathit{om}(j_*(\xi^{-1}), {\mathcal{O}}_S) \cong j_*\xi \otimes {\mathcal{O}}_S(H)[-1]$ by Grothendieck duality, its dual sequence twisted by ${\mathcal{O}}_S(-H)$ takes the form

\begin{equation*}0 \to {\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1}))^\vee \otimes {\mathcal{O}}_S(-H) \to {\mathrm{H}}^0(C,\xi) \otimes {\mathcal{O}}_S \to j_*\xi \to 0.\end{equation*}

As we checked before, ${\mathrm{H}}^0(S, {\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1}))^\vee \otimes {\mathcal{O}}_S(-H)) = 0$ , hence the second arrow is the evaluation morphism, hence its kernel is isomorphic to the Lazarsfeld bundle ${\mathbf{L}}_S(\xi)$ and the required isomorphism ${\mathbf{L}}_S({\mathbf{R}}_C(\xi^{-1})) \cong {\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H)$ follows.

Proposition 4.11. Let (S,H) be a quasipolarized K3 surface of genus $g = r \cdot s \ge 6$ and let $C \subset S$ be a BNP-general curve in $|H|$ . Assume $s \ge r \in \{2,3\}$ . If S does not have special Mukai classes of type (r,s) then for any Mukai pair $(\xi,\eta)$ of type (r,s) on C and any Mukai extension (29) there is an exact sequence

(30) \begin{equation}0 \to {\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H) \to {\mathbf{L}}_S({\mathcal{G}}) \to {\mathbf{L}}_S(\eta) \to 0\end{equation}

of the corresponding Lazarsfeld bundles.

Proof. If (29) is a Mukai extension, we have an exact sequence of global sections

\begin{equation*}0 \to {\mathrm{H}}^0(C, \xi)^\vee \to {\mathrm{H}}^0(C, {\mathcal{G}}) \to {\mathrm{H}}^0(C, \eta) \to 0\end{equation*}

(where we use Lemma 4.4 to identify the first term and Definition 4.9 to prove exactness on the right). Moreover, since $\eta$ and ${\mathbf{R}}_C(\xi^{-1})$ are globally generated (by definition of a Mukai pair and Lemma 4.3, respectively), we conclude that ${\mathcal{G}}$ is also globally generated, hence we have a commutative diagram

(31)

with exact rows and columns, where the first term of the left column is identified in Lemma 4.10. Now the top row of the diagram gives (30).

From now on we work under the assumption that $\mathrm{Pic}(S) = {\mathbb{Z}} \cdot H$ .

Lemma 4.12. Assume $\mathrm{Pic}(S) = {\mathbb{Z}} \cdot H$ and $g = r \cdot s \ge 6$ with $s \ge r \in \{2,3\}$ . Let $(\xi,\eta)$ be a Mukai pair of type (r,s) on a curve C in $|H|$ and let

(32) \begin{equation}0 \to {\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H) \to {\mathcal{F}} \to {\mathbf{L}}_S(\eta) \to 0\end{equation}

be an exact sequence. If the sheaf ${\mathcal{F}}$ is not H-Gieseker stable then the sequence splits.

Proof. By Lemma 3.3 and Proposition 3.9, the subbundle ${\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H)$ and quotient bundle ${\mathbf{L}}_S(\eta)$ of ${\mathcal{F}}$ are stable with Mukai vectors

(33)

Note that $1/r - 1 \le - 1/2 < - 1/s$ because $r \ge 2$ and $s \ge 3$ .

Assume ${\mathcal{F}}$ is not H-Gieseker stable. Consider the Harder–Narasimhan filtration of a sheaf ${\mathcal{F}}$ and refine it (via a Jordan–Hölder filtration of every semistable factor) to a filtration with H-Gieseker stable factors ${\mathcal{F}}_1,\dots,{\mathcal{F}}_m$ . We will show that $m = 2$ , ${\mathcal{F}}_1 = {\mathbf{L}}_S(\eta)$ , and ${\mathcal{F}}_2 = {\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H)$ , so that this filtration is opposite to filtration (32).

First, (32) implies that the slopes of ${\mathcal{F}}_i$ are bounded by the slopes of ${\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H)$ and ${\mathbf{L}}_S(\eta)$ :

(34) \begin{equation}-\frac1s \ge \mu_H({\mathcal{F}}_1) \ge \dots \ge \mu_H({\mathcal{F}}_m) \ge \frac1r - 1.\end{equation}

Furthermore, since $\mathrm{Pic}(S) = {\mathbb{Z}} \cdot H$ , we can write $\upsilon({\mathcal{F}}_i) = (x_i, - y_iH, z_i)$ . Then $\mu_H({\mathcal{F}}_i) \le - 1/s < 0$ implies $y_i \gt 0$ and $\mu_H({\mathcal{F}}_1) \ge 1/r - 1 \gt -1$ implies $x_i \gt y_i$ . Moreover, a combination of (33) and (34) gives the relations

\begin{equation*}\sum x_i = r + s,\quad \sum y_i = r,\quad \sum z_i = (r-1)^2s + 2,\quad x_i \gt y_i{} \ge 1.\end{equation*}

The inequalities $x_i \gt y_i \ge 1$ , with the equality $\sum y_i = r$ , imply, in particular, that

(35) \begin{equation}m \le r\quad \text{and}\quad x_1, \dots, x_m \ge 2.\end{equation}

On the other hand, by Riemann–Roch we have $\chi({\mathcal{F}}_i,{\mathcal{F}}_i) = 2x_iz_i - (2rs - 2)y_i^2$ , and since the sheaves ${\mathcal{F}}_i$ are stable, they are simple, hence $\chi({\mathcal{F}}_i,{\mathcal{F}}_i) \le 2$ , hence

\begin{equation*}x_iz_i \le (rs - 1)y_i^2 + 1, 1 \le i \le m.\end{equation*}

Dividing the ith inequality by $x_i$ and summing them, we obtain

(36) \begin{equation}(r-1)^2s + 2 = \sum z_i \le \sum \frac{(rs - 1)y_i^2 + 1}{x_i}.\end{equation}

From now on we assume $r \in \{2,3\}$ . If $m = 3$ then (35) implies $r = 3$ and $y_i = 1$ for all i, hence the right-hand side of (36) equals $3s \sum 1/{x_i}$ , and we obtain

(37) \begin{equation}\sum \frac{1}{x_i} \ge\frac{4s + 2}{3s} =\frac43 + \frac{2}{3s} \gt \frac43.\end{equation}

Since $x_i \ge 2$ by (35), we must have $x_1 = x_2 = x_3 = 2$ , but then $s = 3$ , and (37) fails. Therefore, we must have $m = 2$ ; we assume this from now on.

As $m = 2$ , inequality (36) can be rewritten as

\begin{equation*}(rs - 1)(y_1^2x_2 + y_2^2x_1) + (x_1 + x_2) \ge ((r-1)^2s + 2)x_1x_2.\end{equation*}

Since $r \in \{2,3\}$ , one of the $y_i$ is 1 and the other is $r - 1$ . If we denote the $x_i$ that corresponds to $y_i = 1$ by x, so that the other is $r + s - x$ , then $y_1^2x_2 + y_2^2x_1 = (r-1)^2 x + (r + s - x)$ and the above inequality takes the form

\begin{equation*}f(x) \mathrel{:=} (rs - 1)((r-1)^2 x + (r + s - x)) + (r + s) - ((r-1)^2s + 2)x(r + s - x) \ge 0.\end{equation*}

Note that

\begin{align*}f(s) &= (rs - 1)((r - 1)^2 s + r) + (r + s) - ((r-1)^2s + 2)rs = 0,\\f(2) &= (rs - 1)(2(r-1)^2 + (r + s - 2)) + (r + s) - 2((r-1)^2s + 2)(r + s - 2)\\&= -s^2(r - 2)(2r - 1) - 5rs(r -2) - 2(r^2 - 4) \le 0,\end{align*}

and the second inequality is strict unless $r = 2$ . On the other hand, the function f is convex, hence the inequality $f(x) \ge 0$ implies

\begin{equation*}x \ge s\quad \text{or}\quad x \le 2\end{equation*}

and the second inequality is strict unless $r = 2$ . Now, if $x \ge s$ the slope of ${\mathcal{F}}_i$ is greater than or equal to $- 1/s$ , if $x \le 1$ then the slope of ${\mathcal{F}}_i$ is less than $ 1/r - 1$ , and if $x = 2 = r$ it is equal to $1/r - 1$ . In either case, (34) implies that the slopes of the ${\mathcal{F}}_i$ are $-1/s$ and $1/r - 1$ . Since these coincide with the slopes of the stable bundles ${\mathbf{L}}_S(\eta)$ and ${\mathbf{L}}_S(\xi)^\vee \otimes {\mathcal{O}}_S(-H)$ , we conclude that