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A Brill-Noether Theorem for (toric) surfaces

Alessio Cela, Carl Lian

TL;DR

This paper extends the Brill-Noether paradigm from maps of a general curve to projective spaces, to maps from a general curve into smooth projective surfaces. It proves that a non-constant map f: C -> S deforms in a family of the expected dimension whenever the image C' satisfies $[C']\cdot(-K_S)\ge 4$, with the moduli space ${\mathcal M}_\beta(C,S)$ being pure of dimension $\beta\cdot(-K_S)+2(1-g)$ and generically smooth near f. The approach leverages Severi varieties and a dimension-comparison with moduli of maps, handling multiple covers through a detailed analysis: for toric surfaces, the loci causing non-pureness are described by explicit toric contractions, yielding a clean Brill-Noether-type theorem in this setting. The toric section yields a classification of low anti-canonical degree curves (degrees 2 and 3) in terms of contractions to $\mathbb{P}^1$ or to fake toric planes, validating the toric BN phenomenon with several canonical examples and connections to Farkas-type results. These results motivate potential higher-dimensional generalizations and highlight the role of toric geometry in understanding degeneracies of maps from curves.

Abstract

The classical Brill-Noether theorem states that a map from a general curve to a projective space deforms in a family of expected dimension as long as its image does not lie in any hyperplane. In this note, we observe, as a direct consequence of standard results on Severi varieties, an analogous statement for maps from a general curve to any smooth, projective surface. Namely, a non-constant map deforms in a family of expected dimension as long as its image has anti-canonical degree at least 4. In the case of toric surfaces, curves of anti-canonical degree at most 3 admit a particularly elegant description in terms certain toric contractions. We raise the question of whether a Brill-Noether theorem could hold for toric varieties of higher dimension.

A Brill-Noether Theorem for (toric) surfaces

TL;DR

This paper extends the Brill-Noether paradigm from maps of a general curve to projective spaces, to maps from a general curve into smooth projective surfaces. It proves that a non-constant map f: C -> S deforms in a family of the expected dimension whenever the image C' satisfies , with the moduli space being pure of dimension and generically smooth near f. The approach leverages Severi varieties and a dimension-comparison with moduli of maps, handling multiple covers through a detailed analysis: for toric surfaces, the loci causing non-pureness are described by explicit toric contractions, yielding a clean Brill-Noether-type theorem in this setting. The toric section yields a classification of low anti-canonical degree curves (degrees 2 and 3) in terms of contractions to or to fake toric planes, validating the toric BN phenomenon with several canonical examples and connections to Farkas-type results. These results motivate potential higher-dimensional generalizations and highlight the role of toric geometry in understanding degeneracies of maps from curves.

Abstract

The classical Brill-Noether theorem states that a map from a general curve to a projective space deforms in a family of expected dimension as long as its image does not lie in any hyperplane. In this note, we observe, as a direct consequence of standard results on Severi varieties, an analogous statement for maps from a general curve to any smooth, projective surface. Namely, a non-constant map deforms in a family of expected dimension as long as its image has anti-canonical degree at least 4. In the case of toric surfaces, curves of anti-canonical degree at most 3 admit a particularly elegant description in terms certain toric contractions. We raise the question of whether a Brill-Noether theorem could hold for toric varieties of higher dimension.

Paper Structure

This paper contains 5 sections, 8 theorems, 13 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Severi varieties and moduli of maps
  4. Toric surfaces
  5. Examples

Key Result

Theorem 1.1

Suppose that $f:C\to\mathbb{P}^r$ is a morphism of degree $d$ which is non-degenerate, that is, the image of $f$ does not lie in any hyperplane. Then, the map $f$ moves in a family of expected dimension. That is, the moduli space of maps ${\mathcal{M}}_d(C,\mathbb{P}^r)$ of maps $C \to \mathbb{P}^r$ the expected, in a neighborhood of $f$.

Figures (4)

  • Figure 1: The polygons $\mathcal{P}(F)$ and $\overline{\mathcal{P}}(F)$ for $F(x,y)=F(x,y)=1+x+y+xy$ in $\mathbb{P}^2$
  • Figure 2: Fan of the blow-up of $\mathbb{P}^2$ at three points, with the two contractions to the fan of $\mathbb{P}^2$
  • Figure 3: Fan $\Sigma'$ in black, and the six added rays to obtain the fan $\Sigma$ in blue
  • Figure 4: Newton Polygon for $F(x,y) = x^2y + xy^2 - 3xy + 1$

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.6
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 15 more