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On the local negativity of surfaces with numerically trivial canonical class

Roberto Laface, Piotr Pokora

Abstract

In this note we study the local negativity for certain configurations of smooth rational curves in smooth surfaces with numerically trivial canonical class. We show that for such rational curves there is a bound for the so-called local Harbourne constants, which measure the local negativity phenomenon. Moreover, we provide explicit examples of interesting configurations of rational curves in some K3 and Enriques surfaces and compute their local Harbourne constants.

On the local negativity of surfaces with numerically trivial canonical class

Abstract

In this note we study the local negativity for certain configurations of smooth rational curves in smooth surfaces with numerically trivial canonical class. We show that for such rational curves there is a bound for the so-called local Harbourne constants, which measure the local negativity phenomenon. Moreover, we provide explicit examples of interesting configurations of rational curves in some K3 and Enriques surfaces and compute their local Harbourne constants.

Paper Structure

This paper contains 3 sections, 3 theorems, 56 equations, 8 figures.

Table of Contents

  1. Introduction
  2. A bound on local Harbourne constants
  3. Examples

Key Result

Theorem 2.1

Let $X$ be a smooth complex projective surface with numerically trivial canonical class and let ${C} \subset X$ be a configuration of smooth rational curves having $n$ irreducible components and only transversal intersection points. Then

Figures (8)

  • Figure 1: Six lines in general position in $\mathbb{P}^2$.
  • Figure 2: The Petersen graph.
  • Figure 3: Dual graph of the six lines $L_{ij}$ on $Y$.
  • Figure 4: Singular fibers induced by a line of 1st kind.
  • Figure 5: Singular fibers induced by a line of 2nd kind.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Conjecture 1.1: Bounded Negativity Conjecture
  • Definition 1.2: The numbers $t_i$
  • Definition 1.3: Local Harbourne constants of a transversal configuration
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Corollary 2.4
  • proof
  • ...and 8 more