Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Recent advances in Brill--Noether theory and the geometry of Brill--Noether curves

Isabel Vogt

TL;DR

Brill-Noether theory addresses when a genus $g$ curve admits a map to $\mathbb{P}^r$ of degree $d$, encoded by the Brill-Noether loci $W^r_dC$ and governed by the Brill-Noether number $\rho(g,r,d)$. The survey synthesizes three pillars of progress: moduli aspects for special curves and the maximal Brill-Noether loci phenomenon, Hurwitz–Brill–Noether theory for fixed gonality, and the geometry/interpolation of BN-curves in projective space, all supported by degeneration, limit linear series, tropical geometry, and normal-bundle techniques. Major achievements include a complete analogue of BN for fixed gonality, irreducibility and dimension results for Brill-Noether loci, explicit descriptions of splitting types in Hurwitz–BN theory, and the Maximal Rank Theorem along with several cases of the Strong MR conjecture, yielding divisors of general type on certain moduli spaces. Together these results provide a robust, characteristic-free framework for understanding maps from curves to projective spaces and illuminate the birational geometry of the moduli spaces $\overline{\mathcal{M}}_g$.

Abstract

The first goal of this article is to survey recent progress in Brill--Noether theory, including both the study of the moduli space of maps from a curve to projective space and the geometry of the resulting curves in projective space. The second goal is to introduce newcomers to some of the important techniques that have been introduced or developed in the last decade that made these advances possible.

Recent advances in Brill--Noether theory and the geometry of Brill--Noether curves

TL;DR

Brill-Noether theory addresses when a genus curve admits a map to of degree , encoded by the Brill-Noether loci and governed by the Brill-Noether number . The survey synthesizes three pillars of progress: moduli aspects for special curves and the maximal Brill-Noether loci phenomenon, Hurwitz–Brill–Noether theory for fixed gonality, and the geometry/interpolation of BN-curves in projective space, all supported by degeneration, limit linear series, tropical geometry, and normal-bundle techniques. Major achievements include a complete analogue of BN for fixed gonality, irreducibility and dimension results for Brill-Noether loci, explicit descriptions of splitting types in Hurwitz–BN theory, and the Maximal Rank Theorem along with several cases of the Strong MR conjecture, yielding divisors of general type on certain moduli spaces. Together these results provide a robust, characteristic-free framework for understanding maps from curves to projective spaces and illuminate the birational geometry of the moduli spaces .

Abstract

The first goal of this article is to survey recent progress in Brill--Noether theory, including both the study of the moduli space of maps from a curve to projective space and the geometry of the resulting curves in projective space. The second goal is to introduce newcomers to some of the important techniques that have been introduced or developed in the last decade that made these advances possible.
Paper Structure (10 sections, 22 theorems, 60 equations, 1 figure)

This paper contains 10 sections, 22 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Important recent results
  4. Moduli aspects of Brill--Noether theory
  5. Hurwitz--Brill--Noether Theory
  6. Geometry of Brill--Noether curves
  7. Important Techniques
  8. Abstract degeneration and limit line bundles
  9. Embedded degeneration and deformation theory
  10. Modifications of normal bundles

Key Result

Theorem 2.2

Fix integers $g, r, d$ with $g \geq 2$. Let $C$ be a general curve of genus $g$ (i.e., $[C]$ is in a dense open in $\mathcal{M}_g$).

Figures (1)

  • Figure 1: For fixed $r$ (here $r=3$) the set of tuples $(g, d)$ such that $\rho(g, r, d) \geq 0$ (pictured here in black) can be obtained from the rational normal curve $(0, r)$ by successively applying one of (A) $(d, g) \mapsto (d+1, g)$, (B) $(d, g) \mapsto (d+1, g+1)$, or finally (C) $(d, g) \mapsto (d+r, g+r+1)$.

Theorems & Definitions (54)

  • Theorem 2.2: The Brill--Noether theorem
  • Remark 1: The Gieseker--Petri Theorem and the singularities of $W^r_dC$
  • Corollary 2.3
  • Theorem 2.4: The Embedding Theorem, Eisenbud--Harris EH83 for $p=1$, Farkas farkas_p
  • Definition 2.5
  • Theorem 3.1: Pflueger Pf_neg
  • Theorem 3.4: Auel--Haburcak--Knutsen AHK
  • Example 3.5: Trigonal genus $5$ curve
  • Theorem 3.6: The Hurwitz--Brill--Noether Theorem
  • Remark 2
  • ...and 44 more