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Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces

Carlos Galindo, Francisco Monserrat, Carlos-Jesús Moreno-Ávila

Abstract

We consider flags $E_\bullet=\{X\supset E\supset \{q\}\}$, where $E$ is an exceptional divisor defining a non-positive at infinity divisorial valuation $ν_E$ of a Hirzebruch surface $\mathbb{F}_δ$ and $X$ the surface given by $ν_E,$ and determine an analogue of the Seshadri constant for pairs $(ν_E,D)$, $D$ being a big divisor on $\mathbb{F}_δ$. The main result is an explicit computation of the vertices of the Newton-Okounkov bodies of pairs $(E_\bullet,D)$ as above, showing that they are quadrilaterals or triangles and distinguishing one case from another.

Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces

Abstract

We consider flags , where is an exceptional divisor defining a non-positive at infinity divisorial valuation of a Hirzebruch surface and the surface given by and determine an analogue of the Seshadri constant for pairs , being a big divisor on . The main result is an explicit computation of the vertices of the Newton-Okounkov bodies of pairs as above, showing that they are quadrilaterals or triangles and distinguishing one case from another.

Paper Structure

This paper contains 7 sections, 18 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Seshadri-type constants for non-positive at infinity valuations of Hirzebruch surfaces
  3. Hirzebruch surfaces and valuations of Hirzebruch surfaces
  4. Seshadri-type constants for non-positive at infinity valuations of Hirzebruch surfaces
  5. Newton-Okounkov bodies of non-positive at infinity valuations
  6. Newton-Okounkov bodies with respect to non-positive at infi-nity special valuations
  7. Newton-Okounkov bodies with respect to non-positive at infinity non-special valuations

Key Result

Theorem \oldthetheorem

Let $\nu_n$ be a non-positive at infinity divisorial valuation of the quotient field of $\mathcal{O}_{\mathbb{F}_\delta,p}$ centered at $\mathcal{O}_{\mathbb{F}_\delta,p}$ and $D\sim aF+bM$ a big divisor on $\mathbb{F}_\delta$. Then,

Figures (1)

  • Figure 1: $\Delta_\nu(F+2M)$ and $\mathfrak{C}(\nu)\cap \mathfrak{H}_{F+2M}(\nu)$ in Example \ref{['Example_NOB']}.

Theorems & Definitions (45)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • ...and 35 more