Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces

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2024年5月11日

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L. Costa,
Irene Macías Tarrío

摘要

Let X be a ruled surface over a nonsingular curve C of genus $g\ge 0.$ $g \geq 0 .$ Let $M_H:=M_{X,H}(2;c_1,c_2)$ $M_{H} : = M_{X, H} (2; c_{1}, c_{2})$ be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes $c_i:=c_i(E)$ $c_{i} : = c_{i} (E)$ for $i=1,2.$ $i = 1, 2 .$ The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space $M_H$ $M_{H}$ in terms of its Brill–Noether locus $W_H^k(2;c_1,c_2),$ $W_{H}^{k} (2; c_{1}, c_{2}),$ whose points correspond to stable vector bundles in $M_H$ $M_{H}$ having at least k independent sections. We deal with the non-emptiness of this Brill–Noether locus, getting in most of the cases sharp bounds for the values of k such that $W_H^k(2;c_1,c_2)$ $W_{H}^{k} (2; c_{1}, c_{2})$ is non-empty.