Even Sets of Lines on Quartic Surfaces

Abstract

An effective divisor D on a smooth (compact complex) surface X is called even, if its class $[D] \in H^2(X,\Z)$ is divisible by 2. D may be assumed reduced w.l.o.g. Then D being even is equivalent to the existence of a double cover $Y \to X$ branched exactly over D. The aim of this note is to study arrangements of $n \leq 10$ distinct lines on a smooth quartic surface $X \subset ¶_3$, which form an even divisor in this sense. The result is that for $n \leq 8$ there are no unexpected ones (one type of six lines, four types of eight lines). And for n=10 a partial classification is given in the following sense: Each even set of ten lines on a smooth quartic surface is of one of eleven different types. At the moment I do not know which of these types do actually occur. The proof for these facts is messy, essentially checking cases.