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Universal extension spaces and modular maps: unveiling irreducible components of Brill-Noether loci of stable bundles on a general $ν$-gonal curve

Youngook Choi, Flamino Flamini, Seonja Kim

Abstract

We investigate the Brill-Noether theory of rank-two, degree-$d$ stable vector bundles of speciality $3$ on a general $ν$-gonal curve of genus $g$, $3 \leq ν< \lfloor \frac{g+3}{2} \rfloor$. Our approach leverages universal extension spaces, modular maps, and recent advancements in rank-one Brill-Noether theory over Hurwitz spaces. We establish existence criteria for the corresponding Brill-Noether loci and provide a comprehensive description of their irreducible components. We moreover prove that these components exhibit diverse geometric behaviors, categorized by their regularity, superabundance, and the properties of their general points. Notably, for specific degrees $d$, we prove the coexistence of multiple superabundant components alongside a regular one. Using specialization techniques, we uncover a stratification into locally closed subschemes within these components and provide insights into their birational geometry and local structure. Furthermore, our results yield also consequences for Brill-Noether loci of stable, rank-two bundles with a fixed general determinant.

Universal extension spaces and modular maps: unveiling irreducible components of Brill-Noether loci of stable bundles on a general $ν$-gonal curve

Abstract

We investigate the Brill-Noether theory of rank-two, degree- stable vector bundles of speciality on a general -gonal curve of genus , . Our approach leverages universal extension spaces, modular maps, and recent advancements in rank-one Brill-Noether theory over Hurwitz spaces. We establish existence criteria for the corresponding Brill-Noether loci and provide a comprehensive description of their irreducible components. We moreover prove that these components exhibit diverse geometric behaviors, categorized by their regularity, superabundance, and the properties of their general points. Notably, for specific degrees , we prove the coexistence of multiple superabundant components alongside a regular one. Using specialization techniques, we uncover a stratification into locally closed subschemes within these components and provide insights into their birational geometry and local structure. Furthermore, our results yield also consequences for Brill-Noether loci of stable, rank-two bundles with a fixed general determinant.
Paper Structure (14 sections, 24 theorems, 251 equations)

This paper contains 14 sections, 24 theorems, 251 equations.

Table of Contents

  1. Preliminaries
  2. Brill-Noether theory of line bundles on general $\nu$-gonal curves
  3. Ruled surfaces and families of special unisecants
  4. Moduli spaces of rank-two (semi)stable bundles on curves and Brill-Nother loci
  5. Line bundle extensions and (semi)stability via Segre invariant
  6. Auxiliary initial results for the proof of Main Theorem
  7. Proof of Main Theorem
  8. No components for $4g-6 \leq d \leq 4g-4$
  9. Preliminary reductions for $2g-2 \leq d \leq 4g-7$
  10. Some obstructions to fill-up components
  11. Components from bundles of first type
  12. Components from modifications of bundles of second type
  13. First-type superabundant loci in components from second-type modifications
  14. Regular components from bundles of second type

Key Result

Proposition 1.1

Let $C$ be a general $\nu$-gonal curve of genus $g$, let $r$ be a non-negative integer and let $A \in {\rm Pic}^{\nu}(C)$ be the unique line bundle on $C$ associated to the unique base-point-free pencil on $C$ of degree $\nu$, i.e. $|A| = g^1_{\nu}$. If $g\geq r(\nu-1)$, then $\operatorname{dim} (|A

Theorems & Definitions (110)

  • proof
  • Remark 1
  • proof
  • Proposition 1.1: cf. Ball
  • Theorem 1.2
  • Definition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Definition 1.6
  • Remark 1.7
  • ...and 100 more