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Seshadri constants on algebraic surfaces

Thomas Bauer

TL;DR

This work advances the understanding of Seshadri constants on algebraic surfaces by providing explicit bounds for one- and multi-point constants, introducing the canonical slope to relate global positivity to the nef cone, and analyzing sub-maximal curves. It delivers complete and computable results for abelian surfaces of Picard number one via Pell-type arithmetic, linking geometry to number theory. The results encompass both general bounds and sharp classifications for very ample line bundles, as well as detailed structure of the nef cone across Picard ranks, and extend to multi-point constants with elliptic-curve dichotomies on abelian surfaces. Collectively, the paper illuminates how local positivity interacts with global geometry and arithmetic on surfaces, with concrete algorithms for computing Seshadri constants in key cases.

Abstract

Seshadri constants are local invariants, introduced by Demailly, which measure the local positivity of ample line bundles. Recent interest in Seshadri constants stems on the one hand from the fact that bounds on Seshadri constants yield, via vanishing theorems, bounds on the number of points and jets that adjoint linear series separate. On the other hand it has become increasingly clear by now that Seshadri constants are highly interesting invariants quite in their own right. Except in the simplest cases, however, they are already in the case of surfaces very hard to control or to compute explicitly---hardly any explicit values of Seshadri constants are known so far. The purpose of the present paper is to study these invariants on algebraic surfaces. On the one hand, we prove a number of explicit bounds for Seshadri constants and Seshadri submaximal curves, and on the other hand, we give complete results for abelian surfaces of Picard number one. A nice feature of this result is that it allows to explicitly compute the Seshadri constants---as well as the unique irreducible curve that accounts for it---for a whole class of surfaces. It also shows that Seshadri constants have an intriguing number-theoretic flavor in this case.

Seshadri constants on algebraic surfaces

TL;DR

This work advances the understanding of Seshadri constants on algebraic surfaces by providing explicit bounds for one- and multi-point constants, introducing the canonical slope to relate global positivity to the nef cone, and analyzing sub-maximal curves. It delivers complete and computable results for abelian surfaces of Picard number one via Pell-type arithmetic, linking geometry to number theory. The results encompass both general bounds and sharp classifications for very ample line bundles, as well as detailed structure of the nef cone across Picard ranks, and extend to multi-point constants with elliptic-curve dichotomies on abelian surfaces. Collectively, the paper illuminates how local positivity interacts with global geometry and arithmetic on surfaces, with concrete algorithms for computing Seshadri constants in key cases.

Abstract

Seshadri constants are local invariants, introduced by Demailly, which measure the local positivity of ample line bundles. Recent interest in Seshadri constants stems on the one hand from the fact that bounds on Seshadri constants yield, via vanishing theorems, bounds on the number of points and jets that adjoint linear series separate. On the other hand it has become increasingly clear by now that Seshadri constants are highly interesting invariants quite in their own right. Except in the simplest cases, however, they are already in the case of surfaces very hard to control or to compute explicitly---hardly any explicit values of Seshadri constants are known so far. The purpose of the present paper is to study these invariants on algebraic surfaces. On the one hand, we prove a number of explicit bounds for Seshadri constants and Seshadri submaximal curves, and on the other hand, we give complete results for abelian surfaces of Picard number one. A nice feature of this result is that it allows to explicitly compute the Seshadri constants---as well as the unique irreducible curve that accounts for it---for a whole class of surfaces. It also shows that Seshadri constants have an intriguing number-theoretic flavor in this case.

Paper Structure

This paper contains 9 sections, 17 theorems, 71 equations, 1 table.

Table of Contents

  1. Introduction
  2. Seshadri constants
  3. Very ample line bundles
  4. Bounds on global Seshadri constants
  5. The degree of sub-maximal curves
  6. On the number of sub-maximal curves
  7. Seshadri constants of polarized abelian surfaces
  8. The nef cone of an abelian surface
  9. Multiple point Seshadri constants on abelian surfaces

Key Result

Proposition 1.1

Let $X$ be a smooth projective variety, $x\in X$ a point, and $L$ an ample line bundle on $X$.

Theorems & Definitions (23)

  • Proposition 1.1
  • Theorem 2.1
  • Lemma 2.2
  • Remark 2.3
  • Theorem 3.1
  • Proposition 3.3
  • Remark 3.4
  • Lemma 3.5
  • Theorem 4.1
  • Remark 4.2
  • ...and 13 more