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A picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve $C$

Marc Coppens

TL;DR

The paper extends Larson's Hurwitz-Brill-Noether framework to describe the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve $C$ of genus $g$ by using splitting degeneracy loci on the open subset $ ext{Pic}^d_f(C)$. It shows that all components of $W^r_d(C)$ arise as translations by $M$ of closures of Brill-Noether-type loci $W^r_{d,f}(C)$ and classifies these components into three types (I, II, III) based on balanced splitting types; type I components are closures of $W^r_{d,f}(C)$ and type II/III come from degeneracy loci associated to maps of locally free sheaves. A detailed tangent-space analysis via Fitting ideals and the map $df$ demonstrates that these components are generically smooth for general $C$, completing a previously incomplete part of the generically smoothness claim and providing new smoothness results for splitting degeneracy loci on arbitrary $k$-gonal curves. The work thus connects the geometry of linear systems on special gonality curves with explicit degeneracy loci descriptions, yielding a concrete, component-wise understanding of $W^r_d(C)$ in the Hurwitz-Brill-Noether setting.

Abstract

Based on results on Hurwitz-Brill-Noether theory obtained by H. Larson we give a picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve of genus $g$. This picture starts from irreducible components of $W^r_d(C)$ restricted to an open subset of $Pic (C)$ satisfying Brill-Noether theory as in the case of a general curve of genus $g$. We obtain some degeneracy loci associated to a morphism of locally-free sheaves on them of the expected dimension. All the irreducible components of the schemes $W^r_d(C)$ are translates of their closures in $Pic (C)$. We complete the proof that the schemes $W^r_d(C)$ are generically smooth in case $C$ is a general $k$-gonal curve (claimed but not completely proved before). We obtain some results on the tangent spaces to the splitting degeneracy loci for an arbitrary $k$-gonal curve and we obtain some new smoothness results in case $C$ is a general $k$-gonal curve.

A picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve $C$

TL;DR

The paper extends Larson's Hurwitz-Brill-Noether framework to describe the irreducible components of for a general -gonal curve of genus by using splitting degeneracy loci on the open subset . It shows that all components of arise as translations by of closures of Brill-Noether-type loci and classifies these components into three types (I, II, III) based on balanced splitting types; type I components are closures of and type II/III come from degeneracy loci associated to maps of locally free sheaves. A detailed tangent-space analysis via Fitting ideals and the map demonstrates that these components are generically smooth for general , completing a previously incomplete part of the generically smoothness claim and providing new smoothness results for splitting degeneracy loci on arbitrary -gonal curves. The work thus connects the geometry of linear systems on special gonality curves with explicit degeneracy loci descriptions, yielding a concrete, component-wise understanding of in the Hurwitz-Brill-Noether setting.

Abstract

Based on results on Hurwitz-Brill-Noether theory obtained by H. Larson we give a picture of the irreducible components of for a general -gonal curve of genus . This picture starts from irreducible components of restricted to an open subset of satisfying Brill-Noether theory as in the case of a general curve of genus . We obtain some degeneracy loci associated to a morphism of locally-free sheaves on them of the expected dimension. All the irreducible components of the schemes are translates of their closures in . We complete the proof that the schemes are generically smooth in case is a general -gonal curve (claimed but not completely proved before). We obtain some results on the tangent spaces to the splitting degeneracy loci for an arbitrary -gonal curve and we obtain some new smoothness results in case is a general -gonal curve.
Paper Structure (6 sections, 20 theorems, 57 equations)

This paper contains 6 sections, 20 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Some notations
  3. Definitions from Hurwitz-Brill-Noether theory
  4. The irreducible components of $W^r_d(C)$ as sets
  5. Tangent spaces to the scheme structures
  6. A smoothness result

Key Result

Lemma 1

Assume $L \in \mathop{\mathrm{Pic}}\nolimits^d(C)$ with $L \in \overline{\Sigma _{\overrightarrow {e}}(C,f)}$ (the Zariski closure) then $f_*(L)\cong \mathcal{O}(\overrightarrow {e}')$ for some $\overrightarrow {e}' \leq \overrightarrow {e}$.

Theorems & Definitions (51)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Definition 3
  • Lemma 2
  • proof
  • Theorem 1
  • Theorem 2
  • Definition 4
  • Theorem 3
  • ...and 41 more