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Seshadri constants and negative curves on blowups of ruled surfaces

Cyril J. Jacob, Bivas Khan, Ronnie Sebastian

TL;DR

The paper addresses local positivity on blowups of rational and ruled surfaces by computing Seshadri constants for ample line bundles on Blowups of $\mathbb{F}_e$ at up to $e+3$ very general points, and extends to certain decomposable ruled surfaces over curves of positive genus. The approach reduces the problem to a finite set of potential negative curves via effective anticanonical divisors and a carefully described set $\Delta$, yielding explicit formulas for $\varepsilon$ in terms of linear data $L=\alpha C_e+\beta f_e-\sum\mu_i E_i$, with refinements for the case $r=e+3$ (involving $A(L),B(L)$). The authors also establish fixed- and negative-curve analyses to bound $C^2$ on blowups, proving bounded negativity-type results for both blown-up Hirzebruch surfaces and more general ruled surfaces, and showing the effectiveness of the anticanonical divisor in broad regimes. Together, these results advance explicit local positivity calculations on blown-up ruled surfaces and contribute to bounded negativity conjectures in this setting.

Abstract

In this article we compute Seshadri constants of ample line bundles on the blowup of Hirzebruch surface $\mathbb{F}_e$ at $r\leqslant e+3$ very general points. Similarly, we compute Seshadri constants on the blowups of certain decomposable ruled surfaces over smooth curves of non-zero genus. We also prove some results related to bounded negativity of blowups of Hirzebruch surfaces and ruled surfaces.

Seshadri constants and negative curves on blowups of ruled surfaces

TL;DR

The paper addresses local positivity on blowups of rational and ruled surfaces by computing Seshadri constants for ample line bundles on Blowups of at up to very general points, and extends to certain decomposable ruled surfaces over curves of positive genus. The approach reduces the problem to a finite set of potential negative curves via effective anticanonical divisors and a carefully described set , yielding explicit formulas for in terms of linear data , with refinements for the case (involving ). The authors also establish fixed- and negative-curve analyses to bound on blowups, proving bounded negativity-type results for both blown-up Hirzebruch surfaces and more general ruled surfaces, and showing the effectiveness of the anticanonical divisor in broad regimes. Together, these results advance explicit local positivity calculations on blown-up ruled surfaces and contribute to bounded negativity conjectures in this setting.

Abstract

In this article we compute Seshadri constants of ample line bundles on the blowup of Hirzebruch surface at very general points. Similarly, we compute Seshadri constants on the blowups of certain decomposable ruled surfaces over smooth curves of non-zero genus. We also prove some results related to bounded negativity of blowups of Hirzebruch surfaces and ruled surfaces.
Paper Structure (7 sections, 26 theorems, 114 equations)

This paper contains 7 sections, 26 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Curves on blowups
  4. Fixed Components of anticanonical divisors
  5. Computation of Seshadri Constants
  6. Ruled Surfaces
  7. Negative curves on Blowups of Ruled Surfaces

Key Result

Theorem 1.2

Laz2004 Let $X$ be a smooth projective variety and $L$ be a nef line bundle on $X$. For a point $x\in X$, let $\pi: X_x \to X$ denote the blowup of $X$ at $x$ and let $E:=\pi^{-1}(\{x\})$ denote the exceptional divisor. Then, we have

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3: see Theorem \ref{['SC e+2']}
  • Theorem 1.4: see Theorem \ref{['SC e+3']}
  • Conjecture 1.5
  • Theorem 1.6: Theorem \ref{['Conj 4.8 HJNS']}
  • Theorem 1.7: Theorem \ref{['wbnc-hirz']}
  • Theorem 1.8: Theorem \ref{['wbnc-ruled']}
  • Proposition 2.1
  • proof
  • ...and 40 more