On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

TL;DR

This paper investigates the geometry of the Hilbert scheme and related moduli spaces of stable vector bundles over algebraic curves, using gonality and Hecke morphisms to describe components and compute their dimensions.

Contribution

It introduces new descriptions of open sets and computes dimensions of components in the Hilbert scheme and morphism schemes for vector bundles over curves, utilizing gonality and Hecke morphisms.

Findings

Dimension of morphism space from P^2 to M(3,ξ) is 8g-7.

Provides a sufficient condition for non-emptiness of certain morphism spaces.

Describes a smooth open set in the Hilbert scheme of the moduli space.

Abstract

Let $M(n,\xi)$ be the moduli space of stable vector bundles of rank $n\geq 3$ and fixed determinant $\xi$ over a smooth projective algebraic curve $X$ over $\mathbb{C}$ of genus $g\geq 4.$ We use the gonality of the curve and $r$-Hecke morphisms to describe a smooth open set and to compute the dimension of a component of the Hilbert scheme $Hilb_{M(n,\xi)}$, of the scheme of morphisms $Mor(\mathbb{G},M(n,\xi))$ and of the moduli space $ M_{X \times \mathbb{G}}$ of stable bundles over $X\times \mathbb{G},$ where $\mathbb{G}$ is the Grassmannian $\mathbb{G}(n-r,\mathbb{C}^n)$. In particular, we prove that $\dim Mor_P(\mathbb{P}^2,M(3,\xi))=8g-7$ and we give a sufficient condition for $Mor_{2ns}(\mathbb{P}^1,M(n,\xi))$ to be non-empty with $s\geq 1.$