Triple planes with p_g=q=0

Abstract

We show that general triple planes with p_g=q=0 belong to at most 12 families, that we call surfaces of type I,..., XII, and we prove that the corresponding Tschirnhausen bundle is direct sum of two line bundles in cases I, II, III, whereas is a rank 2 Steiner bundle in the remaining cases. We also provide existence results and explicit constructions for surfaces of type I,..., VII, recovering all classical examples and discovering several new ones. In particular, triple planes of type VII provide counterexamples to a wrong claim made in 1942 by Bronowski.