Even sets of $(-4)$-curves on rational surface

TL;DR

This paper investigates rational surfaces with even sets of disjoint (-4)-curves, analyzing the properties of the associated double cover surface and establishing bounds on the number of such curves, with classifications for certain Kodaira dimensions.

Contribution

It provides a classification of rational surfaces with even sets of (-4)-curves, including bounds on the number of curves and characterizations based on Kodaira dimension.

Findings

Number of (-4)-curves in an even set is at most 12.

The associated double cover surface has Kodaira dimension ≥ 0.

Complete classification for Kodaira dimension 0 and 1 cases.

Abstract

We study rational surfaces having an even set of disjoint $(-4)$-curves. The properties of the surface $S$ obtained by considering the double cover branched on the even set are studied. It is shown, that contrarily to what happens for even sets of $(-2)$-curves, the number of curves in an even set of $(-4)$-curves is bounded (less or equal to 12). The surface $S$ has always Kodaira dimension bigger or equal to zero and the cases of Kodaira dimension zero and one are completely characterized. Several examples of this situation are given.