The Iskovskih Theorem for regular surfaces over imperfect fields
Andrea Fanelli, Stefan Schröer
TL;DR
This work generalizes Iskovskih's theorem to minimal regular surfaces over arbitrary fields, including imperfect ones, by replacing smoothness with regularity and deriving a trichotomy: $X\cong \mathbb{P}^2$ or a quadric in $\mathbb{P}^3$, or $X$ admits a fibration over a regular genus-zero base with regular fibers, or $ω_X$ generates $Pic(X)$. The authors develop new methods to handle inseparable phenomena (inseparable pencils) in characteristic two, study non-normal quartic models via branch curves of degree three, and classify indecomposable rank-two sheaves on genus-zero curves (twisted ribbons) to describe bend-and-break configurations. The results illuminate how constant-field extensions and self-intersection numbers behave in imperfect settings, yielding a cohesive structure for regular surfaces beyond the classical smooth case. The work connects techniques from the Minimal Model Program in positive characteristic to explicit geometric models such as twisted cubics and ruled surfaces over genus-zero curves, with potential implications for the birational classification of surfaces in 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.