Abstract

We describe the effect of ramified morphisms on Harbourne constants of reduced effective divisors. With this goal, we introduce the pullback of a weighted cluster of infinitely near points under a dominant morphism between surfaces, and describe some of its basic properties. As an application, we describe configurations of curves with transversal intersections and H-index arbitrarily close to -25/7simeq -3.571, smaller than any previously known result.

Highlights

  • The question whether, in every algebraic surface, self-intersections of irreducible and reduced curves are bounded from below has intrigued algebraic geometers for decades, and continues to do so

  • Is there a positive integer b(S, n) ∈ Z such that for every morphism Sπ →π S which is the composition of n point blowups, and every irreducible curve C ⊂ Sπ, one has C2 ≥ −b(S, n)?

  • In order to prove Theorems A and B, we study pullbacks of suitable curves by ramified morphisms; the effect of ramified morphisms on H-constants is the main theme of this work

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Summary

Introduction

The question whether, in every algebraic surface, self-intersections of irreducible and reduced curves are bounded from below has intrigued algebraic geometers for decades, and continues to do so. The so-called Bounded Negativity Conjecture (BNC for short) is an old folklore conjecture, formally posed by Bauer et al in [4] (where some of its history is explained), that asserts an affirmative answer: Conjecture 1.1 (BNC). There exists a positive integer b(S) ∈ Z such that for every irreducible and reduced curve C ⊂ S one has C2 ≥ −b(S). In recent years the question of bounded negativity has received considerable attention, especially via the approach of trying to determine classes of surfaces S which satisfy Conjecture 1.1 (see [4,8,9,12]). [3, Problem 1.2] raised the question whether bounded negativity is a birational property, which leads to the following question: 109 Page 2 of 24

Results
Preliminaries
Clusters and H-Constants
Singularities of Curves in Smooth Surfaces
Harbourne Constants Under Ramified Morphisms
The Pullback Cluster
Pullback of Multi-Clusters and H-Indices
Examples
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