The Iskovskih Theorem for regular surfaces over imperfect fields

Abstract

We generalize Iskovskih's theorem about surfaces without irregularity and bigenus from the smooth case to regular surfaces over arbitrary fields, with special focus on the case of imperfect fields. This includes surfaces that are geometrically non-normal or geometrically non-reduced. Here the usual approach of Galois descent breaks down, and one relies entirely on the scheme theory over the ground field. Moreover, the degrees of closed points can be larger than expected, and certain curves might have purely inseparable constant field extension. To deal with the latter we establish a general theory for inseparable pencils, which is of independent interest. A crucial case not present in the classical proof for Iskovskih's theorem leads to non-normal quartic surfaces that are singular along a twisted cubic, or more exotic space curves of degree three.