The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface

TL;DR

This paper computes the effective cones of moduli spaces of sheaves on a smooth quadric surface, demonstrating they are Mori dream spaces and identifying extremal divisors using exceptional bundles.

Contribution

It introduces a method to compute effective cones of moduli spaces of sheaves on a quadric surface, including explicit calculations for Hilbert schemes of points.

Findings

Effective cones are Mori dream spaces for all Chern characters.

Brill-Noether divisors span extremal rays of the effective cone.

Complete computation of effective cones for the first fifteen Hilbert schemes of points.

Abstract

Let $\xi$ be a stable Chern character on $\mathbb{P}^1 \times \mathbb{P}^1$, and let $M(\xi)$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^1 \times \mathbb{P}^1$ with Chern character $\xi$. In this paper, we provide an approach to computing the effective cone of $M(\xi)$ after showing that it is a Mori dream space for all $\xi$. We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(\xi)$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on $\mathbb{P}^1 \times \mathbb{P}^1$.