Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Covering gonality of hypersurfaces in a product of projective spaces

Raphaël Hiault

Abstract

In this work, we investigate the behaviour of the covering gonality of a very general hypersurface in a product of projective spaces. Inspired by the work of Bastianelli, Ciliberto, Flamini and Suppino in [BCFS19] which addresses the case of a hypersurface in a projective space, we establish a similar result for very general smooth hypersurfaces of sufficiently large bi-degree. More precisely, we show that the covering gonality of such hypersurfaces can be computed by viewing them as a family of hypersurfaces over a projective space. Then, the curves computing the covering gonality lie entirely within the fibres of one of the families. This rules out transversal curves from computing the covering gonality. In addition to this, we investigate the behaviour of the joint covering gonality as in [LM23] and establish a lower bound for bi-degree large enough.

Covering gonality of hypersurfaces in a product of projective spaces

Abstract

In this work, we investigate the behaviour of the covering gonality of a very general hypersurface in a product of projective spaces. Inspired by the work of Bastianelli, Ciliberto, Flamini and Suppino in [BCFS19] which addresses the case of a hypersurface in a projective space, we establish a similar result for very general smooth hypersurfaces of sufficiently large bi-degree. More precisely, we show that the covering gonality of such hypersurfaces can be computed by viewing them as a family of hypersurfaces over a projective space. Then, the curves computing the covering gonality lie entirely within the fibres of one of the families. This rules out transversal curves from computing the covering gonality. In addition to this, we investigate the behaviour of the joint covering gonality as in [LM23] and establish a lower bound for bi-degree large enough.
Paper Structure (4 sections, 14 theorems, 11 equations)

This paper contains 4 sections, 14 theorems, 11 equations.

Table of Contents

  1. Classical results around the covering gonality
  2. Lines over $\mathbb{P}^{n+1}\times \mathbb{P}^{m+1}$
  3. Lower bound for the covering gonality
  4. Measure of association for hypersurfaces

Key Result

Theorem 1

Let $X \subset \mathbb{P}^{n+1}\times \mathbb{P}^{m+1}$ be a very general irreducible smooth hypersurface of bi-degree $(a,b)$. Suppose that $a\geq 2n+m+1$, $b\geq 2m+n+1$, and $n,m \geq 2$. Then: Moreover, both sides of the inequality coincide whenever $m,n \in \{4\alpha^{2}+3\alpha, 4\alpha^{2}+5\alpha+1, \alpha \in \mathbb{N}\}$

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5: gonalitynoether
  • Lemma 1.6: gonalitynoether
  • Lemma 1.7: stapleton2020degree
  • Remark 1.8
  • ...and 19 more