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Higher rank Clifford's theorem on the smooth quadric

Neelarnab Raha

TL;DR

This work extends Clifford-type bounds to higher-rank torsion-free sheaves on the quadric surface $\mathbb{P}^1\times\mathbb{P}^1$ with polarization $H=(1,1)$, introducing the functions $\mu_{\max}(E)$ and $\beta_{r,\mu}$ to bound $h^0(E)$ as $h^0(E)\le \beta_{r,\mu_{\max}(E)}$ (provided $\mu_{\max}(E)\ge -1$). It characterizes maximal $h^0$ cases via balanced twisted Steiner-like bundles and provides explicit resolutions; it also refines bounds when global generation is weak or the first Chern class is unbalanced. The paper develops a detailed stratification of bounds, proves nontrivial cohomology vanishing results, and shows that maximal balanced (and often unbalanced) bundles arise from Steiner-like constructions; it further proves semistability of general extensions on elliptic curves and del Pezzo surfaces and analyzes how these extensions behave under restriction to curves. Collectively, these results map out the Brill-Noether geometry of higher-rank sheaves on the quadric, offering sharp tools for understanding emptiness, dimension, and structure of Brill-Noether loci in this setting and informing similar questions on related surfaces.

Abstract

Brill-Noether theory of curves has played a crucial role in the study of curves and their moduli since the 19th century, and has been extensively studied by several authors. Clifford's theorem provides a starting point in determining the emptiness of Brill-Noether loci by providing an upper bound on $h^0(L)$ for a line bundle $L$ on a smooth curve $C$ in terms of the degree of $L$. It also characterizes the cases for which equality holds. In this paper, we prove an analogous result for higher rank sheaves on $\mathbb{P}^1\times\mathbb{P}^1$. Depending on how nice the first Chern class is, and whether the sheaf has global generation properties, we prove sharp upper bounds on $h^0(E)$ for slope semistable sheaves $E$ in terms of $\operatorname{rk}(E)$ and $c_1(E)$. We also find that any $E$ achieving the bound is a twist of a Steiner-like bundle, or closely related to such a bundle. As part of our investigation, we show that general extensions of stable vector bundles on elliptic curves and del Pezzo surfaces are semistable.

Higher rank Clifford's theorem on the smooth quadric

TL;DR

This work extends Clifford-type bounds to higher-rank torsion-free sheaves on the quadric surface with polarization , introducing the functions and to bound as (provided ). It characterizes maximal cases via balanced twisted Steiner-like bundles and provides explicit resolutions; it also refines bounds when global generation is weak or the first Chern class is unbalanced. The paper develops a detailed stratification of bounds, proves nontrivial cohomology vanishing results, and shows that maximal balanced (and often unbalanced) bundles arise from Steiner-like constructions; it further proves semistability of general extensions on elliptic curves and del Pezzo surfaces and analyzes how these extensions behave under restriction to curves. Collectively, these results map out the Brill-Noether geometry of higher-rank sheaves on the quadric, offering sharp tools for understanding emptiness, dimension, and structure of Brill-Noether loci in this setting and informing similar questions on related surfaces.

Abstract

Brill-Noether theory of curves has played a crucial role in the study of curves and their moduli since the 19th century, and has been extensively studied by several authors. Clifford's theorem provides a starting point in determining the emptiness of Brill-Noether loci by providing an upper bound on for a line bundle on a smooth curve in terms of the degree of . It also characterizes the cases for which equality holds. In this paper, we prove an analogous result for higher rank sheaves on . Depending on how nice the first Chern class is, and whether the sheaf has global generation properties, we prove sharp upper bounds on for slope semistable sheaves in terms of and . We also find that any achieving the bound is a twist of a Steiner-like bundle, or closely related to such a bundle. As part of our investigation, we show that general extensions of stable vector bundles on elliptic curves and del Pezzo surfaces are semistable.
Paper Structure (26 sections, 35 theorems, 172 equations)

This paper contains 26 sections, 35 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Upper bounds on $h^0$
  3. General bound
  4. Bound for the non-globally generated case
  5. Bound for the unbalanced case
  6. Preliminaries
  7. Global generation
  8. Slope stability
  9. Gieseker stability
  10. Moduli spaces
  11. The upper bounds
  12. General bound
  13. Bound for the unbalanced case
  14. Bound for the non-globally generated case
  15. Investigating maximality - balanced case
  16. ...and 11 more sections

Key Result

Theorem 1.2

(Theorem thm:bound_sections.) Let $E$ be a torsion-free sheaf of rank $r\geq1$ on $\mathbb{P}^1\times\mathbb{P}^1$ with $\mu_{\max}(E)\geq-1$. Then $h^0(E)\leq\beta_{r,\mu_{\max}(E)}$.

Theorems & Definitions (70)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 3.1
  • ...and 60 more