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Brill-Noether Theory of stable vector bundles on ruled surfaces

L. Costa, I. Macías Tarrío

Abstract

Let X be a ruled surface over a nonsingular curve C of genus $g\geq0$. Let $M_H:=M_{X,H}(2;c_1,c_2)$ be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes $c_i:=c_i(E)$ for $i=1,2$. The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space $M_H$ in terms of its Brill-Noether locus $W_H^k(2;c_1,c_2)$, whose points correspond to stable vector bundles in $M_H$ 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 $W_H^k(2;c_1,c_2)$ is non-empty.

Brill-Noether Theory of stable vector bundles on ruled surfaces

Abstract

Let X be a ruled surface over a nonsingular curve C of genus . Let be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes for . The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space in terms of its Brill-Noether locus , whose points correspond to stable vector bundles in 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 is non-empty.
Paper Structure (4 sections, 24 theorems, 111 equations)

This paper contains 4 sections, 24 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Walls and chambers structure in Brill-Noether loci
  4. Emptiness and non-emptiness of Brill-Noether loci

Key Result

Theorem 1.2

Let $X$ be a ruled surface over a nonsingular curve $C$ of genus $g\geq0$, $m\in\{0,1\}$, $c_2>>0$ an integer and $H\equiv\alpha C_0+\beta f$ an ample divisor on $X$ with Then, for any $k$ in the range the Brill-Noether locus $W_H^k(2;C_0+\mathfrak{m} f,c_2)\neq\emptyset$ with $m=\deg(\mathfrak{m})$.

Theorems & Definitions (47)

  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • ...and 37 more