1.1. r-point Seshadri constants

Let Y be a smooth projective surface and $\pi \colon X\longrightarrow Y$ the blowup of Y at r very general points $p_1$ ,…, $p_r\in Y$ . We denote by $E_1$ ,…, $E_r$ the exceptional divisors of $\pi $ , with $E_i$ lying over $p_i$ , and use $E=\sum _{i=1}^{r} E_i$ for their sum. Given an ample line bundle L on Y, the r-point Seshadri constant of L is defined to be

(1.1) $$ \begin{align} \epsilon_r(L):= \sup \left\{ \gamma \geqslant 0 \,\,|\,\, \pi^{*}L - \gamma E\,\,\text{is nef}\,\rule{0cm}{0.40cm}\right\}. \end{align} $$

Equivalently

(1.2) $$ \begin{align} \epsilon_r(L) = \inf_C \left\{\frac{C\cdot \pi^{*}L}{C\cdot E}\right\}, \end{align} $$

where the infimum is over the effective curves C in X which do not contain any $E_i$ as a component. We adopt the convention that C does not have to intersect any of the $E_i$ . In such cases $(C\cdot \pi ^{*}L)/(C\cdot E)$ , interpreted as $+\infty $ , does not affect the infimum, and this convention allows us to avoid many repetitions of “for those C which also intersect at least one of the $E_i$ ”.

To our knowledge r-point Seshadri constants were first introduced by Küchle [Reference Küchle7], for smooth projective varieties Y of arbitrary dimension. In general very few exact values of $\epsilon _r(L)$ are known. For instance, when $Y=\mathbb {P}^2$ , computing $\epsilon _r(\mathcal {O}_{\mathbb {P}^2}(1))$ is equivalent to the Nagata conjecture, a problem which is open for all $r\geqslant 10$ where r not a square.

In this paper we restrict to the surface $Y=\mathbb {P}^1\times \mathbb {P}^1$ . We use $L=\mathcal {O}_{Y}(e_1,e_2)$ for the line bundle on $\mathbb {P}^1\times \mathbb {P}^1$ of bidegree $(e_1,e_2)$ . Such a line bundle L is nef if and only if $e_1$ , $e_2\geqslant 0$ , and ample if and only if $e_1$ , $e_2\geqslant 1$ . By the slope of L we mean $e_2/e_1$ , allowing $\infty $ if $e_1=0$ and $e_2\neq 0$ .

1.2. Definition of $\alpha _r$ and $\beta _r$

For a positive integer r we set

$$ \begin{align*}\alpha_r:=\frac{(r-4)+\sqrt{r(r-8)}}{4} \rule{0.25cm}{0cm}\text{and}\rule{0.25cm}{0cm} \beta_r:=\frac{(r-4)-\sqrt{r(r-8)}}{4}.\end{align*} $$

The numbers $\alpha _r$ and $\beta _r$ are the roots of $t^2-\left ({(r-4)}/{2}\right )t+1=0$ . When r is even $\alpha _r$ and $\beta _r$ are mutually inverse units in the ring of integers of $\mathbb {Q}[\sqrt {r(r-8)}]$ . When $r\geqslant 10$ this ring is a real quadratic extension of $\mathbb {Q}$ , and $\alpha _r$ and $\beta _r$ are of infinite order. The numbers $\alpha _r$ and $\beta _r$ govern the problem of computing $\epsilon _r(L)$ on $\mathbb {P}^1\times \mathbb {P}^1$ in several ways. Here is the first.

1.3. Inner and outer bundles

We call a nef bundle $L=\mathcal {O}_{Y}(e_1,e_2)$ an inner bundle if $\frac {e_2}{e_1}\in [\beta _r,\alpha _r]$ , and an outer bundle otherwise. The motivation for this terminology comes from Figure 1 on page 10. We note that whether any particular L is an inner or outer bundle depends on the value of r.

Let $p_1$ ,…, $p_r\in Y$ be very general points, and $\pi \colon X\longrightarrow Y$ the blowup of Y at $p_1$ ,…, $p_r$ as in §1.1. For a line bundle L on $\mathbb {P}^1\times \mathbb {P}^1$ , and any $\gamma \geqslant 0$ we set

(1.3) $$ \begin{align} L_{\gamma} := \pi^{*}L-\gamma\sum_{i=1}^{r} E_i = \pi^{*}L-\gamma E. \end{align} $$

By definition of the r-point Seshadri constant, if $\gamma>\epsilon _r(L)$ then the class $L_{\gamma }$ is not nef, and therefore there is an irreducible curve $C\subset X$ such that $L_{\gamma }\cdot C<0$ . In §2.4 we give a heuristic argument that if L is an outer bundle, then one might expect that C is $K_X$ -negative. Thus by the fact that in such a case one must also have $C^2<0$ , it would follow that C is a $(-1)$ -curve.

One consequence of our analysis in the paper is that this guess is correct, and we are able to explicitly compute $\epsilon _r(L)$ for all outer bundles and all r. The answer appears in §1.5 after discussing (in §1.4) another appearance of $\alpha _r$ and $\beta _r$ .

A symmetrization procedure. Set $F_1$ and $F_2$ to be the pullback to X of the fibre classes $\mathcal {O}_{Y}(1,0)$ and $\mathcal {O}_{Y}(0,1)$ respectively, and let $V_r\subset H^2(X,\mathbb {R})$ be the subspace of the real Néron-Severi group spanned by $F_1$ , $F_2$ , and E. Thus $V_r$ is a three-dimensional real vector space, and for vectors $v=d_1F_1+d_2F_2-mE$ and $w=e_1F_1+e_2F_2-nE$ in $V_r$ the intersection pairing between v and w is given by

(1.4) $$ \begin{align} v\cdot w= d_1e_2+d_2e_1-mnr. \end{align} $$

By (1.3) the class of $L_{\gamma }$ is in $V_r$ . If $L_{\gamma }$ is not nef the following argument shows that there is always an effective curve C with class in $V_r$ such that $L_{\gamma }\cdot C<0$ .

Let $C'$ be an irreducible curve such that $L_{\gamma }\cdot C'<0$ , and let $d_1F_1+d_2F_2-\sum _{i=1}^{r} m_iE_i$ be the class of $C'$ . Since the points $p_1$ ,…, $p_r$ are general it follows that for any permutation $\sigma $ on $\{1,\ldots , r\}$ there is an irreducible curve $C^{\prime }_{\sigma }$ with class $d_1F_2+d_2F_2-\sum _{i=1}^{r} m_{\sigma (i)}E_i$ . Moreover, $L_{\gamma }\cdot C'=L_{\gamma }\cdot C^{\prime }_{\sigma }$ .

Let $\sigma $ be an r-cycle, and set C to be the sum $C := \sum _{i=1}^{r} C^{\prime }_{\sigma ^i}$ . Then C is an effective curve, of class $rd_1F_1+rd_1F_2-\left (\sum _{i=1}^{r} m_i\right )E\in V_{r}$ , and $L_{\gamma }\cdot C = r(L_{\gamma }\cdot C')<0$ .

More generally this symmetrization argument shows that the restriction of the nef and effective cones to $V_r$ are cones which are still dual in $V_r$ . Thus, to understand $\epsilon _{r}(L)$ we may restrict our attention to $V_r$ .

1.4. Automorphisms of the problem for even r

It is easy to verify that the linear transformation $T_{r}\colon V_{r}\longrightarrow V_{r}$ given, in the basis $F_1$ , $F_2$ , and E, by the matrix

(1.5) $$ \begin{align} \left[ \begin{array}{crr} 0 & 1 & 0 \\ 1 & \frac{r}{2} & r \\ 0 & -1 & -1 \rule{0cm}{0.45cm}\\ \end{array} \right] \end{align} $$

preserves the intersection form. When r is even we show in Theorem 3.1 that $T_r$ is an automorphism of the problem, in the sense that if $\xi \in V_r$ is any class, then $\xi $ is nef, or effective, or represents a curve with s irreducible components, if and only if $T_r(\xi )$ is respectively nef, effective, or represents a curve with s irreducible components. These statements are not true when r is odd.

The transformation $T_r$ has eigenvalues $\alpha _r$ , $\beta _r$ , and $1$ , with respective eigenvectors (in coordinates given by $F_1$ , $F_2$ , and E)

(1.6) $$ \begin{align} \rule{0.25cm}{0cm} v_{\alpha_r} = \left(\tfrac{1}{\alpha_r+1},\, \tfrac{\alpha_r}{\alpha_r+1},\, -\tfrac{2}{r}\right), \rule{0.25cm}{0cm} v_{\beta_r} = \left(\tfrac{\alpha_r}{\alpha_r+1},\, \tfrac{1}{\alpha_r+1},\, -\tfrac{2}{r}\right), \rule{0.25cm}{0cm}\text{and} \rule{0.15cm}{0cm} v_{1} = \left(-2,-2,1\right). \end{align} $$

We note that $v_1$ is the class of $K_{X}$ , that $v_{\alpha _r}$ and $v_{\beta _r}$ are exchanged by the automorphism exchanging $F_1$ and $F_2$ . Additionally, since $\alpha _r\beta _r=1$ , $v_{\beta _r}$ may also be written as $v_{\beta _r}=\left (\tfrac {1}{\beta _r+1},\,\tfrac {\beta _r}{\beta _r+1},\,-\tfrac {2}{r}\right ).$ Thus the Galois automorphism of $\mathbb {Q}[\sqrt {r(r-8)}]$ exchanging $\alpha _r$ and $\beta _r$ exchanges $v_{\alpha _r}$ and $v_{\beta _r}$ , although we will not use this fact.

If $r\geqslant 10$ then $\alpha _r$ and $\beta _r$ are units of infinite order, and thus $T_r$ also has infinite order. Starting with a nef or effective class and iterating $T_r$ then allows us to produce infinitely many other nef or effective classes. This is the key to our computation of $\epsilon _{r}(L)$ for even r and outer bundles L.

When $r\geqslant 10$ we have $0<\beta _r < 1 < \alpha _r$ . Thus, in forward iterations of $T_r$ vectors generally converge (modulo scaling) to $v_{\alpha _r}$ , and under backwards iterations to $v_{\beta _r}$ . As a consequence if r is even then both $v_{\alpha _r}$ and $v_{\beta _r}$ are limits of nef classes, and are therefore also nef. They are also square-zero classes, $v_{\alpha _r}^2=0=v_{\beta _r}^2$ , and so on the boundary of the nef cone.

A function on the square zero cone. Before proceeding to the results for $\epsilon _r(L)$ we make two more digressions. For a nef line bundle $L=\mathcal {O}_{Y}(e_1,e_2)$ we define the numerical bound, $\eta _{r}(L)$ by

(1.7) $$ \begin{align} \eta_r(L) := \sqrt{\tfrac{L^2}{r}} = \sqrt{\tfrac{2e_1e_2}{r}}. \end{align} $$

The value $\eta _{r}(L)$ is precisely the value of $\gamma $ so that $L_{\gamma }^2=0$ . In other words, $\eta _r(L)$ is the value of $\gamma $ which puts $L_{\gamma }$ on the cone of square-zero classes. The number $\eta _r(L)$ is therefore also an upper bound for the Seshadri constant: $\epsilon _r(L)\leqslant \eta _{r}(L)$ .

For a vector $v\in V_r$ with $v^2=0$ , and not a multiple of $v_{\alpha _r}$ or $v_{\beta _r}$ , we put

(1.8) $$ \begin{align} \varphi_{r}(v): = \frac{\log\left(\frac{v\cdot v_{\beta_r}}{v\cdot v_{\alpha_r}}\right)}{2\log(\alpha_r)}. \end{align} $$

This formula is justified by the following properties (see §6.2). For such a vector v, $\varphi _r(\lambda v)=\varphi _r(v)$ for any $\lambda \in \mathbb {R}$ , $\lambda \neq 0$ ; $\varphi _r(T_r^{n}(v)) = \varphi _r(v)+n$ for all $n\in \mathbb {Z}$ ; and if $\tilde {v}$ is the vector obtained from v by the automorphism switching $F_1$ and $F_2$ , then $\varphi _r(\tilde {v}) = - \varphi _r(v)$ . Thus, $\varphi _r$ is a map from the square-zero cone (up to scaling, and minus the lines spanned by $v_{\alpha _r}$ and $v_{\beta _r}$ ) to $\mathbb {R}$ which takes symmetries of the problem to similar symmetries on $\mathbb {R}$ .

1.5. Seshadri constants for outer bundles

Here we concentrate on the cases $r\geqslant 9$ . When $r\leqslant 7$ the blowup of $\mathbb {P}^1\times \mathbb {P}^1$ at r general points is Fano, and the answers in those cases have a different character than the general case. In addition, one minor aspect of our description below is not valid for $r=8$ . These cases are discussed in §5.

In order to describe the answers in the even case, here and in the symplectic packing problem, it will be convenient to define several sequences $\{ s_{n,r}\}_{n\in \mathbb {Z}}$ by giving the terms $s_{-1,r}$ , $s_{0,r}$ , and $s_{1,r}$ , and defining all other terms by the recursion equation coming from the characteristic polynomial of $T_r$ :

(1.9) $$ \begin{align} s_{n,r} = \tfrac{r-2}{2}\left(s_{n-1,r}-s_{n-2,r}\rule{0cm}{0.40cm}\right) + s_{n-3,r}. \end{align} $$

We define the sequence $\{p_{n,r}\}_{n\in \mathbb {Z}}$ by $p_{-1,r}=0$ , $p_{0,r}=0$ , $p_{1,r}=r$ , and determine all other $p_{n,r}$ by the recursion (1.9). Similarly we define $\{m_{n,r}\}_{n\in \mathbb {Z}}$ by $m_{-1,r}=1$ , $m_{0,r}=-1$ , $m_{1,r}=1$ , and (1.9). We note that $m_{n,r}=m_{-n,r}$ and $p_{n,r}=p_{-1-n,r}$ for all $n\in \mathbb {Z}$ .

Theorem 1.1 (Seshadri Constants for Outer Bundles).

Suppose that $L=\mathcal {O}_{Y}(e_1,e_2)$ with $e_1,e_2\geqslant 1$ , and that L is an outer bundle, that is, that $\frac {e_2}{e_1}\not \in [\beta _r,\alpha _r]$ .

If $\boldsymbol{r}$ is odd, $\boldsymbol {r\geqslant 9}$ . Then

(1.10) $$ \begin{align} \epsilon_r(L) = \left\{ \begin{array}{cl} e_2 & \text{if } \frac{e_2}{e_1}\leqslant \frac{2}{r+1}, \\ \frac{2e_1+(r-1)e_2}{2r} & \text{if } \frac{e_2}{e_1} \in [\frac{2}{r+1},\beta_r], \rule{0cm}{0.6cm}\\ \frac{(r-1)e_1+2e_2}{2r} & \text{if } \frac{e_2}{e_1} \in [\alpha_r,\frac{r+1}{2}], \rule{0cm}{0.6cm}\\ e_1 & \text{if } \frac{r+1}{2}\leqslant \frac{e_2}{e_1}. \rule{0cm}{0.6cm} \\ \end{array} \right. \end{align} $$

If $\boldsymbol {r}$ is even, $\boldsymbol {r\geqslant 10}$ . Set $v_{L}=(e_1,e_2,-\eta _{r}(L))\in V_r$ , that is, set $v_{L}$ to be the class of $L_{\gamma }$ with $\gamma =\eta _r(L)$ , and put $n=\lfloor \varphi _r(v_{L})+\frac {1}{2}\rfloor $ , where $\lfloor x \rfloor $ denotes the largest integer $\leqslant x$ . Then

(1.11) $$ \begin{align} \epsilon_r(L) = \frac{e_1p_{n,r}+e_2p_{n-1,r}}{rm_{n,r}}. \end{align} $$

The explanation for (1.11) is as follows. Consider the classes $C_{n,r}=(p_{n-1,r},p_{n,r},-m_{n,r})$ defined by the sequences $\{p_{n,r}\}_{n\in \mathbb {Z}}$ and $\{m_{n,r}\}_{n\in \mathbb {Z}}$ . Then $C_{0,r}=(0,0,1)$ is the class of E, and for all $n\in \mathbb {Z}$ , $C_{n,r}=T_{r}^{n}(C_{0,r})$ . In Theorem 3.3 we show that when r is even the $C_{n,r}$ generate the effective cone of curves in $V_{r}$ whose slopes lie outside of $[\beta _r,\alpha _r]$ . It follows that for an outer bundle L one of these curves determines $\epsilon _r(L)$ . The formula with $\varphi _{r}$ above is one possible method of locating the correct n, and intersecting with $C_{n,r}$ then gives (1.11).

For the proofs of the results in the even case see §3, and for the odd case, §4.

1.6. Applications to the symplectic packing problem

We recommend [Reference McDuff and Polterovich10, §1] and [Reference Biran2] for a discussion of the history of this problem and the reasons for its interest. Here we give a brief outline oriented towards our application of the previous results to the symplectic packing problem for $\mathbb {P}^1\times \mathbb {P}^1$ .

Let $(M,\omega _{M})$ be a closed symplectic manifold of real dimension $4$ , and let $B_{\lambda }\subset \mathbb {R}^4$ denote the ball of radius $\lambda $ centred at $0$ , equipped with the restriction of the standard symplectic form on $\mathbb {R}^4$ : $\omega _{\mathbb {R}^4} = dx_1\wedge dy_1 + dx_2\wedge dy_2$ .

For a given r, consider all possible symplectic embeddings of r disjoint copies of $B_{\lambda }$ into M, and denote by $\hat {\nu }_r(M)$ the supremum of the volumes which can be filled by such embeddings (i.e., the supremum of $r\pi ^2\lambda ^4$ , over those $\lambda $ for which there exists such a symplectic embedding of r disjoint $B_{\lambda }$ into M). Finally, set $\nu _{r}(M) = \hat {\nu }_{r}(M)/\operatorname {Vol}(M)$ , where the volume of M, like the volume of $B_{\lambda }$ , is computed using the volume form $\omega _{M}\wedge \omega _{M}$ .

Two basic questions are: () What is $\nu _r(M)$ for different values of r?; and () for which r does $\nu _r(M)=1$ ? If $\nu _r(M)<1$ one says that there is a packing obstruction, while if $\nu _{r}(M)=1$ one says that there is a full packing.

Let Y be a smooth projective surface, and L a real ample class on Y. The first Chern class $c_1(L)$ can be represented by a Kähler form $\omega _{L}$ , which, when written out in terms of the underlying real coordinates, is a real symplectic form. We consider the packing problem for the real manifold M underlying Y, with symplectic form $\omega _{L}$ . To align our notation with the notation in the rest of the paper, we will use $\nu _{r}(L)$ for the value of $\nu _{r}(M)$ in this situation.

A remarkable discovery of [Reference McDuff and Polterovich10] is that $(-1)$ -curves in X provide obstructions to full packings. Even more striking is that this obstruction looks much like the Seshadri constant, except with the test curves C in (1.2) limited to $(-1)$ -curves. To set this up we first extend definition (1.7) to any such pair $(Y,L)$ by setting $\eta _{r}(L) = \sqrt {L^2/r} = \sqrt {\operatorname {Vol}(M)/r}$ . Then we set

(1.12) $$ \begin{align} \tilde{\epsilon}_r(L) = \min\left(\inf_C \left\{\frac{C\cdot \pi^{*}L}{C\cdot E}\right\}, \eta_{r}(L)\right) \end{align} $$

where this time the infimum is over irreducible $(-1)$ -curves C in X distinct from the exceptional divisors $E_i$ . Following our convention in §1.1, if there is no such $(-1)$ -curve C which intersects any of the $E_i$ the infimum is interpreted as $\infty $ , and then $\tilde {\epsilon }_{r}(L) = \eta _{r}(L)$ .

The obstruction result of [Reference McDuff and Polterovich10] is that one always has $\nu _{r}(L) \leqslant \left (\frac {\tilde {\epsilon }_{r}(L)}{\eta _{r}(L)}\right )^2$ . The paper is concerned with the case $Y=\mathbb {P}^2$ , but the obstruction argument does not depend on this. Even more remarkably, a result of Biran, [Reference Biran1, Theorem 6.A], asserts that there is a class of surfaces, which includes $\mathbb {P}^2$ and ruled surfaces, where one has $\nu _{r}(L) = \left (\frac {\tilde {\epsilon }_{r}(L)}{\eta _{r}(L)}\right )^2$ for all L.

The Seshadri constants in Theorem 1.1 were obtained using $(-1)$ -curves, or their symmetrized versions. As a result we can compute $\tilde {\epsilon }_{r}(L)$ and so, thanks to the result of Biran cited above, $\nu _r(L)$ , for all real ample line bundles on $Y=\mathbb {P}^1\times \mathbb {P}^1$ ; equivalently, by [Reference Lalonde and McDuff8, Theorem 1.1], for all symplectic forms on the underlying real manifold. As before we list the results for $r\geqslant 9$ ; the results for $r\leqslant 8$ appear in §7.2.

Theorem 1.2 (Formulas for the symplectic packing constant).

Let L be a real ample line bundle of type $(e_1,e_2)$ (i.e., $e_1$ and $e_2$ are positive real numbers).

If $\boldsymbol {r}$ is odd, $\boldsymbol {r\geqslant 9}$ . Then

(1.13) $$ \begin{align} \nu_r(L) = \left\{ \begin{array}{cl} \frac{re_2}{2e_1} & \text{if } \frac{e_2}{e_1}\leqslant \frac{2}{r+1}, \\ \frac{(2e_1+(r-1)e_2)^2}{8re_1e_2} & \text{if } \frac{e_2}{e_1} \in [\frac{2}{r+1},\frac{2}{(\sqrt{r}-1)^2}], \\ 1 & \text{if } \frac{e_2}{e_1} \in [\frac{2}{(\sqrt{r}-1)^2},\frac{(\sqrt{r}-1)^2}{2}], \\ \frac{((r-1)e_1+2e_2)^2}{8e_1e_2} & \text{if } \frac{e_2}{e_1} \in [\frac{(\sqrt{r}-1)^2}{2},\frac{r+1}{2}], \\ \frac{re_1}{2e_2} & \text{if } \frac{r+1}{2}\leqslant\frac{e_2}{e_1}. \\ \end{array} \right. \end{align} $$

If $\boldsymbol {r}$ is even, $\boldsymbol {r\geqslant 10}$ . Then

(1.14) $$ \begin{align} \nu_r(L) = \left\{ \begin{array}{cl} 1 & \text{if } \frac{e_2}{e_1} \in [\beta_r,\alpha_r] \\ \frac{r\left(\epsilon_{r}(L)\right)^2}{2e_1e_2} & \text{if } \frac{e_2}{e_1}\not\in [\beta_r,\alpha_r] (\text{with } \epsilon_r(L) \text{ computed by the rule in } \mathrm{(1.11)}). \rule{0cm}{0.6cm} \\ \end{array} \right. \end{align} $$

1.7. Conditions for full packings

Define sequences $\{q_{n,r}\}_{n\in \mathbb {Z}}$ by $q_{-1,r}=1$ , $q_{0,r}=0$ , $q_{1,r}=1$ , and the recursion (1.9). For this sequence one has $q_{n,r}=q_{-n,r}$ for all n. Taking into account the cases $r\leqslant 8$ (see §5), and reversing the formulae in Theorem 1.2, we get the following answer to question ().

For a given $(e_1,e_2)$ , there is at most one value of r for which case (ii) occurs, see Theorem 6.4.

The lower bounds on r differ in the even and odd cases. Consequently by picking a line bundle with an extreme slope, we can find examples of line bundles L with ranges where full packings exist only for even r. Here are two examples with similar slopes.

This phenomenon seems very surprising to the authors, even knowing the proofs of the formulas. For instance, returning to the first example, there is a full packing when $r=410$ . If we look for a packing with $r=409$ , then we could start with a packing for $r=410$ and use $409$ of the balls. Admittedly, that is not yet a full packing, but surely it would be possible to increase the radius and move the centres just a little bit to make up for it, and not have the balls intersect …? Of course, the results above say that it is not possible. As a consistency check, Theorem 1.2 gives $\nu _{409}(L) = \frac {654481}{656036}\approx 0.9976297\ldots $ , larger than the ratio $\frac {409}{410} \approx 0.99756097\ldots $ obtained using $409$ out of the $410$ balls but without increasing the radius.

The proofs for the above results on symplectic packing, using the previous results about Seshadri constants, appear in §7.

1.8. Results for inner bundles

One implication of the SHGH conjecture (see, for example, [Reference Ciliberto, Harbourne, Miranda and Roé3, §1.4]) and our analysis of which $(-1)$ -curves affect Seshadri constants is that there should be a portion of the nef cone which is round. Specifically, for $r\geqslant 9$ , we should have $\epsilon _r(L)=\nu _{r}(L)$ for all L whose slopes are in, respectively, $[\beta _r,\alpha _r]$ if r is even, and $[\frac {2}{(\sqrt {r}-1)^2},\frac {(\sqrt {r}-1)^2}{2}]$ if r is odd.

If this description of the nef cone is correct, then the boundary of the nef cone, for slopes in the ranges indicated above, consists of classes $\xi $ which are nef, square-zero (i.e., $\xi ^2=0$ ) and $K_X$ -positive: $K_X\cdot \xi>0$ . In this paper we call such classes inner square-zero nef classes.

Finding such classes is quite useful. By definition, if $\xi $ is nef, then there are no effective classes on the half plane $\xi ^{<0}$ . If $\xi $ is $K_X$ -positive, this half plane will contain a large proportion of $K_X$ -positive classes, and it is exactly these classes whose existence we usually have the greatest difficulty in ruling out. In addition, if $\xi ^2=0$ it means that $\xi $ is on the boundary of the nef cone, and so provides the strongest condition on restricting effective classes.

One of the contributions of this paper is to construct such inner square-zero nef classes for all $r\geqslant 9$ . To our knowledge, this is the first construction of such classes on the blowup of $\mathbb {P}^1\times \mathbb {P}^1$ at r general points. These classes are constructed in §9, using a new “reflection method”. If r is even we obtain, using $T_r$ , infinitely many such classes, but if r is odd we only construct finitely many. In §10 we use pullback maps to construct other infinite families of such classes when r is even.

Given such a class $\xi $ , say of the form $\xi =(e_1,e_2,-\sqrt {{2e_1e_2}/{r}})$ (the last coordinate is determined by the condition that $\xi ^2=0$ ), then $\epsilon _r(L)=\eta _r(L)$ for the bundle $L=\mathcal {O}_{Y}(e_1,e_2)$ . We are thus able to exhibit classes which achieve the predicted value of the Seshadri constant.

1.9. Relation with other work, I

Seshadri constants on $\mathbb {P}^1\times \mathbb {P}^1$ and the related symplectic packing problem were studied in the 2005 Ph.D. thesis of W. Syzdek, the published version of which appears as [Reference Syzdek12].

In [Reference Syzdek12] Syzdek finds the same curves $C_{n,r}$ (from §1.5) which we use to compute the Seshadri constant in the even case. More precisely, our $C_{n,r}$ are each the disjoint union of $r (-1)$ -curves (for instance $C_{n,r}=T^{n}_{r}(E)$ , and E is the disjoint union of the r exceptional divisors), and Syzdek finds instead the classes of these $(-1)$ -curves. Specifically, our curve $C_{n,r}$ with $n\geqslant 1$ is the symmetric orbit of the curve called $M_{D_{n+2}}$ in [Reference Syzdek12, Proposition 3.9], with $l=\frac {r-6}{2}$ and with the bidegrees switched. These curves ( $C_{n,r}$ , or its components $D_{n+2}$ ) impose the same conditions on the r-point Seshadri constants.

The approach in [Reference Syzdek12] cannot rule out that there may be other curves which affect the Seshadri constant when $\frac {e_2}{e_1}\not \in [\beta _r,\alpha _r]$ , and so for the corresponding line bundles can only give an upper bound on $\epsilon _{r}(L)$ , and an upper bound on the symplectic packing number. In our argument we can conclude that these upper bounds are the actual values. The extra piece of information in our method is that we know that the iterates $T^{n}_r(F_2)$ are nef, and duality of the nef and effective cones then eliminates the possibility of other such curves.

In the case that r is odd, the $(-1)$ -curves we find also already appear in [Reference Syzdek12, Table 4]. For instance, our curve of bidegree $(\frac {r-1}{2},1)$ is the curve of bidegree $(k+n+3,1)$ in Table 4 when $r=2k+2n+7$ . Theorem 3.17 of [Reference Syzdek12] seems to claim, in the case r is odd, that the curves in [Reference Syzdek12, Table 4] compute the Seshadri constant for all $(e_1,e_2)$ with $\frac {e_2}{e_1}\not \in [\frac {2}{(\sqrt {r}-1)^2},\frac {(\sqrt {r}-1)^2}{2}]$ . We do not know how to justify this claim since we cannot rule out the possibility that there may be curves C with $C^2<0$ and $C\cdot K_{X}>0$ which could impose a stronger condition on the Seshadri constant. In fact, we have had to do some work in §4.2–§4.3 to show that if such a curve exists, it at least could not affect line bundles with $\frac {e_2}{e_1}\not \in [\beta _r,\alpha _r]$ .

As part of question () in §1.6, one may ask for an $r_0$ so that for $r\geqslant r_0$ one has $\nu _r(L)=1$ . In general, as the examples in §1.7 suggest, it is the odd r which determine $r_0$ . Our bound in Theorem 1.3 is sharp. Setting $s=\frac {e_2}{e_1}$ , and assuming $s\geqslant 2$ to simplify the discussion, we get that there is a full packing for all r greater than $(\sqrt {2s}+1)^2 = 2s+2\sqrt {2s}+1$ . In contrast, [Reference Syzdek12] (Definition 3.2, formula for $R_0$ with $a=1$ , $b=s$ ) gives a slightly worse estimate of

$$ \begin{align*}\frac{3+2s+3s^2}{2s} + \frac{(1+s)\sqrt{2(1+s^2)}}{s} = \left(\tfrac{3}{2}+\sqrt{2}\right)s+ (1+\tfrac{3}{\sqrt{2}})+O\left(\tfrac{1}{s}\right)\rule{0.25cm}{0cm}\text{as } s\to\infty.\end{align*} $$

Finally, we should note that the dichotomy of behaviour between even and odd r, one aspect of the problem which we find surprising, already appears in [Reference Syzdek12]. For instance, in the estimates $R_0$ and $r_0$ in [Reference Syzdek12, Definition 3.2].

In summary, the improvements in this paper over the results of [Reference Syzdek12] are: () When r is even to give exact values of $\epsilon _r(L)$ for outer bundles L and exact values of $\nu _r(L)$ for all ample L; () When r is odd to justify the calculation of $\epsilon _r(L)$ for outer bundles L and to give the exact region where $\nu _r(L)=1$ . We are thus able give a complete answer to the symplectic packing problem for $\mathbb {P}^1\times \mathbb {P}^1$ . () To produce inner square-zero nef classes for all $r\geqslant 9$ . Thus, to produce inner bundles where the Seshadri constant can be computed exactly.

The authors also think that the introduction of the numbers $\alpha _r$ and $\beta _r$ , the realization that the problem has an infinite order automorphism when r is even, $r\geqslant 8$ , and the graphical reasoning from §2.1–§2.2 greatly simplify the analysis of the problem.

1.10. Relation with other work, II

The paper [Reference De Volder and Tutaj-Gasińska5] gives lower bounds on $\epsilon _r(L)$ for those L whose Seshadri constant is not affected by $(-1)$ -curves. When r is odd, this means line bundles L with $\frac {e_2}{e_1}\in [\frac {2}{(\sqrt {r}-1)^2},\frac {(\sqrt {r}-1)^2}{2}]$ . Then [Reference De Volder and Tutaj-Gasińska5, Theorem 5] gives the lower bound $\epsilon _r(L)\geqslant \eta _{r}(L)\cdot (1-\frac {1}{5r})^{\frac {1}{2}}$ . When r is even, the results of [Reference De Volder and Tutaj-Gasińska5] apply to all inner bundles, and [Reference De Volder and Tutaj-Gasińska5, Theorem 4] gives the lower bound $\epsilon _r(L) \geqslant \eta _{r}(L)\cdot (1-\frac {2}{9r})^{\frac {1}{2}}$ .

As discussed in §1.8 in this case we are able to find inner bundles L where $\epsilon _r(L)=\eta _r(L)$ . Using these bundles and convexity of the Seshadri constant then gives lower bounds on $\epsilon _r$ which are better than the lower bound above on various regions of the intervals above. See the discussion in §10.2.

Limitations of this paper. In the study of Seshadri problems on blowups of rational surfaces, and in particular the Nagata conjecture, the sticking point is our inability to either rule out all $K_{X}$ -positive curves C with $C^2<0$ , or exhibit one which exists. Unfortunately this paper is no exception.

However, the construction of the inner square-zero nef classes in §9–§10 does eliminate a large range of such classes, and seems to the authors to be a useful step forward.

Second, the most precise results about Seshadri constants in this paper are for outer bundles, those bundles $L=\mathcal {O}_{Y}(e_1,e_2)$ with $\frac {e_2}{e_1}\not \in [\beta _r,\alpha _r]$ . As $r\to \infty $ we have $\beta _r\to 0$ and $\alpha _r\to \infty $ . Thus, as r increases, the region of our ignorance also increases, and the region of complete understanding shrinks to zero.

1.11. Organization of the paper

In §2 we describe a graphical way of representing and arguing about the problem and give a heuristic argument that irreducible curves affecting the Seshadri constant of outer bundles should be $(-1)$ -curves. This picture also explains the appearance of $\alpha _r$ and $\beta _r$ in the problem.

In §3 we show that $T_r$ is an automorphism of the problem when r is even, calculate the nef and effective cones for classes whose slope is outside of $[\beta _r,\alpha _r]$ , and compute the Seshadri constants for outer bundles for even $r\geqslant 10$ . In §4 we compute the Seshadri constants for outer bundles for odd $r\geqslant 9$ . In §5 we give the results for all r, even and odd, with $r\leqslant 8$ .

In §6 we study the slopes $\frac {q_{n+1,r}}{q_{n,r}}$ which show up in the exceptional case (ii) in the symplectic packing problem (§1.3), as well as establish the properties of the map $\varphi _r$ defined in (1.8).

In §7 we use the results of the previous sections to establish the results on symplectic packings, Theorems 1.2 and 1.3. In §8 we give the reflection theorem, a method of producing nef classes using certain types of specializations. In §9 we use the reflection theorem to construct inner square-zero nef classes in both the odd and even cases. Finally in §10 we use pullback maps to produce other families of inner square-zero nef classes when r is even, and an interesting family of bounds when $8\mid r$ .