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Positivity on Blow-up of hyperelliptic surfaces

Praveen Kumar Roy

TL;DR

This work studies positivity on blow-ups of hyperelliptic surfaces by first giving an ampleness criterion for line bundles on $X_r$ and contrasting it with Küchle's criterion. It then analyzes Seshadri constants, deriving multi-point bounds and exact single-point values in various settings for hyperelliptic surfaces of different types, and proving rationality of global Seshadri constants in several regimes. The results provide computable expressions for Seshadri constants in both general and special-point configurations, advancing understanding of local and global positivity on blow-ups of hyperelliptic surfaces. Overall, the paper bridges ampleness criteria with explicit Seshadri invariants, yielding practical tools for positivity questions on these special surfaces.

Abstract

Let $X_r$ denote the blow-up of the hyperelliptic surface $X$ at $r$ very general points. In this paper, we first provide a criterion for the ampleness of a line bundle on $X_r$ and compare it with an existing result. We then study the multi-point Seshadri constants of ample line bundles on hyperelliptic surfaces $X$. Next, we compute single-point Seshadri constants on $X_r$ for specific ample line bundles on odd types. Furthermore, we show that the global Seshadri constants for certain ample line bundles on blow-up of hyperelliptic surfaces are rational.

Positivity on Blow-up of hyperelliptic surfaces

TL;DR

This work studies positivity on blow-ups of hyperelliptic surfaces by first giving an ampleness criterion for line bundles on and contrasting it with Küchle's criterion. It then analyzes Seshadri constants, deriving multi-point bounds and exact single-point values in various settings for hyperelliptic surfaces of different types, and proving rationality of global Seshadri constants in several regimes. The results provide computable expressions for Seshadri constants in both general and special-point configurations, advancing understanding of local and global positivity on blow-ups of hyperelliptic surfaces. Overall, the paper bridges ampleness criteria with explicit Seshadri invariants, yielding practical tools for positivity questions on these special surfaces.

Abstract

Let denote the blow-up of the hyperelliptic surface at very general points. In this paper, we first provide a criterion for the ampleness of a line bundle on and compare it with an existing result. We then study the multi-point Seshadri constants of ample line bundles on hyperelliptic surfaces . Next, we compute single-point Seshadri constants on for specific ample line bundles on odd types. Furthermore, we show that the global Seshadri constants for certain ample line bundles on blow-up of hyperelliptic surfaces are rational.

Paper Structure

This paper contains 6 sections, 20 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Notation and convention:
  3. Ampleness of line bundle on blow-up of Hyperelliptic surfaces
  4. Seshadri constants
  5. Multi-point Seshadri constants on hyperelliptic surfaces
  6. Seshadri constants on blow-up of hyperelliptic surfaces

Key Result

Theorem 1.1

Let $X \cong (A\times B)/G$ be a hyperelliptic surface with two natural projections $\Phi$ and $\Psi$. Then the generators for ${\rm Num} (X)$ and the multiplicities $m_1,\ldots, m_s$, where each $m_i$ represents the multiplicity of a singular fibre of $\Psi$ and $s$ denote the number of such fibre

Theorems & Definitions (44)

  • Theorem 1.1: Serrano
  • Theorem 2.1: Corollary Kuchle
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • ...and 34 more