Some moduli spaces of $\alpha$-stable coherent systems on algebraic surfaces

TL;DR

This paper studies the geometric structure of moduli spaces of $ ext{alpha}$-stable coherent systems on algebraic surfaces, revealing their description as Grassmann bundles and analyzing their topological properties for large $ ext{alpha}$.

Contribution

It establishes a new description of the moduli space as a Grassmann bundle over stable sheaves and explores its irreducibility and dimension for large $ ext{alpha}$.

Findings

Moduli space is a Grassmann bundle over stable sheaves.

Results on irreducibility and dimension of the moduli space.

Correspondence between $ ext{alpha}$-stable systems and extensions of sheaves.

Abstract

Let $X$ be a smooth, irreducible, projective algebraic surface, and let $\alpha \in \mathbb{Q}[m]_{>0}$ be a polynomial. In this paper, we determine topological and geometric properties of the moduli space of $\alpha$-stable coherent systems of type $(n; c_{1}, c_{2}, k)$ with $k < n$ on $X$, for sufficiently large values of $\alpha$. We prove that, for $\alpha$ sufficiently large, the moduli space admits a description as a Grassmann bundle over a moduli space of $H$-stable torsion free sheaves. As a consequence, we obtain results on irreducibility, dimension. Our approach relies on establishing a correspondence between $\alpha$-stable coherent systems and extensions of $H$-stable torsion free sheaves by trivial bundles.