Birational geometry of Beauville–Mukai systems I: the rank three and genus two case

Abstract

We study wall-crossing for the Beauville–Mukai system of rank three on a general genus two K3 surface. We show that such a system is related to the Hilbert scheme of ten points on the surface by a sequence of flops, whose exceptional loci can be described as Brill–Noether loci. We also obtain Brill–Noether type results for sheaves in the Beauville–Mukai system.

Appendix

Here we collect some results about wall-crossings for the moduli spaces which have appeared in previous sections.

1.1 Walls for $S^{[2]}$

We have ${\textbf{v}}=(1,0,-1)$.

Lemma 5.1

[2, Proposition 13.1] Let ${\tilde{H}}=\theta (0,-1,0)$ and $B=\theta (-1,0,-1)$. Then

$$\begin{aligned} \textrm{Mov}(S^{[2]})=\langle {\tilde{H}},{\tilde{H}}-B\rangle . \end{aligned}$$

The full list of walls is given in Table 3.

Table 3 Walls of $\textrm{Mov}(S^{[2]})$

The only other nontrivial birational model for $S^{[2]}$ is $M(0,1,-1)$, obtained by performing a flop of $S^{[2]}$ along the locus parametrizing $\xi _2\in S^{[2]}$ through which there passes a pencil of lines. From the $M(0,1,-1)$ side, the flopping wall is given by $\left( x+\frac{1}{2}\right) ^2+y^2=\left( \frac{\sqrt{5}}{2}\right) ^2$.

1.2 Walls for $S^{[3]}$

We have ${\textbf{v}}=(1,0,-2)$

Lemma 5.2

[2, Proposition 13.1] Let ${\tilde{H}}=\theta (0,-1,0)$ and $B=\theta (-1,0,-2)$. Then

$$\begin{aligned} \textrm{Mov}(S^{[3]})=\left\langle {\tilde{H}},{\tilde{H}}-\frac{1}{2}B\right\rangle . \end{aligned}$$

To understand the wall and chamber structure in $\textrm{Mov}(S^{[3]})$, we apply [2, Theorem 5.7]. The full list of walls is given in Table 4.

Table 4 Walls of $\textrm{Mov}(S^{[3]})$

As a result, $S^{[3]}$ has no other nontrivial birational models. Moreover, there are no other walls with radii larger than 1.

1.3 Walls for $S^{[4]}$

We have ${\textbf{v}}=(1,0,-3)$.

Lemma 5.3

[2, Proposition 13.1] Let ${\tilde{H}}=\theta (0,-1,0)$ and $B=\theta (-1,0,-3)$. Then

$$\begin{aligned} \textrm{Mov}(S^{[4]})=\left\langle {\tilde{H}},{\tilde{H}}-\frac{1}{2}B\right\rangle . \end{aligned}$$

The full list of walls is given in Table 5.

Table 5 Walls of $\textrm{Mov}(S^{[4]})$

As a result, there are two birational models of $S^{[4]}$. If we use ${}^{\sharp }S^{[4]}$ to denote the model not isomorphic to $S^{[4]}$, then ${}^{\sharp }S^{[4]}$ is obtained by performing a flop of $S^{[4]}$ along the locus $\{\xi _4\;|\;h^0({\mathcal {I}}_{\xi _4}(1))\ne 0\}$.

1.4 Walls for $S^{[8]}$

We let ${\textbf{v}}=(1,0,-7)$.

Lemma 5.4

[2, Proposition 13.1] Let ${\tilde{H}}=\theta (0,-1,0)$ and $B=\theta (-1,0,-7)$. Then

$$\begin{aligned} \textrm{Mov}(S^{[8]})=\left\langle {\tilde{H}},{\tilde{H}}-\frac{3}{8}B\right\rangle . \end{aligned}$$

The full list of walls is given in Table 6.

Table 6 Walls of $\textrm{Mov}(S^{[8]})$

As a result, there are eight birational models of $S^{[8]}$.

1.5 Walls for ${\textbf{v}}=(0,1,0)$

By [2, Theorem 5.7], there is no wall for (0, 1, 0) whose radius is larger than 1.

1.6 Walls for ${\textbf{v}}=(0,2,-2)$

By [29, Theorem 5.3], the only wall for $(0,2,-2)$ whose radius is larger than 1 is a flopping wall given by

$$\begin{aligned} \left( x+\frac{1}{2}\right) ^2+y^2=\left( \frac{\sqrt{5}}{2}\right) ^2. \end{aligned}$$

1.7 Walls for ${\textbf{v}}=(0,2,-1)$

These are computed in [18, Section 5]. There are two walls with radii larger than 1, given by

$$\begin{aligned} \left( x+\frac{1}{4}\right) ^2+y^2&=\left( \frac{5}{4}\right) ^2,\\ \left( x+\frac{1}{4}\right) ^2+y^2&=\left( \frac{\sqrt{17}}{4}\right) ^2. \end{aligned}$$