On the local negativity of surfaces with numerically trivial canonical class

TL;DR

This paper investigates the bounds of local negativity for configurations of rational curves on surfaces with trivial canonical class, providing theoretical bounds and explicit examples on K3 and Enriques surfaces.

Contribution

It establishes bounds for local Harbourne constants on such surfaces and presents explicit configurations with computed constants.

Findings

Bound established for local Harbourne constants

Explicit examples of rational curve configurations provided

Computed local Harbourne constants for K3 and Enriques surfaces

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.