Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of α-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.

Nef vector bundles on a quadric surface with the first Chern class (2, 1)

We classify nef vector bundles on a smooth quadric surface with the first Chern class (2, 1) over an algebraically closed field of characteristic zero; we see in particular that such nef bundles are globally generated.

THE EFFECTIVE CONE OF MODULI SPACES OF SHEAVES ON A SMOOTH QUADRIC SURFACE

Let $\unicode[STIX]{x1D709}$ be a stable Chern character on $\mathbb{P}^{1}\times \mathbb{P}^{1}$, and let $M(\unicode[STIX]{x1D709})$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ with Chern character $\unicode[STIX]{x1D709}$. In this paper, we provide an approach to computing the effective cone of $M(\unicode[STIX]{x1D709})$. We find Brill–Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(\unicode[STIX]{x1D709})$ 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 $\mathbb{P}^{1}\times \mathbb{P}^{1}$.

Stable Sheaves on a Smooth Quadric Surface with Linear Hilbert Bipolynomials

We investigate the moduli spaces of stable sheaves on a smooth quadric surface with linear Hilbert bipolynomial in some special cases and describe their geometry in terms of the locally free resolution of the sheaves.