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

TL;DR

This work extends coherent systems theory from curves to algebraic surfaces, focusing on $\alpha$-stable coherent systems of type $(n;c_{1},c_{2},k)$ with $k<n$ for large $\alpha$. It proves that $\alpha$ is bounded above and that, near this bound, the moduli space is a Grassmann bundle over the moduli space of $H$-stable torsion-free sheaves, via an exact sequence $0\to \mathcal{O}_{X}^{\oplus k}\to E\to F\to 0$ with $F$ $H$-stable. The section on large $\alpha$ yields a precise dimension formula for the Grassmann bundle, establishes irreducibility under suitable conditions, and provides a generically smooth example on $\mathbb{P}^{2}$. Overall, the paper advances higher-dimensional Brill-Noether theory on surfaces by describing the geometry of $\alpha$-stable coherent systems and their relation to stable bundles and $H$-stability, including wall-crossing aspects and explicit moduli descriptions.

Abstract

Let $X$ be a smooth, irreducible, projective algebraic surface, and let $α\in \mathbb{Q}[m]_{>0}$ be a polynomial. In this paper, we determine topological and geometric properties of the moduli space of $α$-stable coherent systems of type $(n; c_{1}, c_{2}, k)$ with $k < n$ on $X$, for sufficiently large values of $α$.

Theorems & Definitions (36)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.3