Surfaces of general type and sl_2-triples
Stefan Schröer, Nikolaos Tziolas
TL;DR
This work develops a general theory of sl_2-triples (i.e., height-one SL_2[F]-actions) on global vector fields of schemes in positive characteristic, with a focus on surfaces of general type. It shows that no faithful sl_2-triple can occur on a smooth surface of general type, via an inertia-map obstruction landing in a curve of subalgebras and a moduli-map rigidity argument, while also constructing abundant examples on singular (RDP) models using Lefschetz pencils and purely inseparable base-changes to realize faithful actions of PGL_2[F^n]. The paper further classifies rational double points that admit a tangent-module freeness or a surjective evaluation pairing, including characteristic two, and identifies precisely which RDPs support sl_2-triples. These results illuminate how infinitesimal SL_2-actions interact with the geometry of surfaces in positive characteristic and with moduli of curves, providing both obstructions and concrete constructions in the singular setting.
Abstract
The sl_2-triples play a fundamental role for the structure theory of Lie algebras, and representation theory in general. Here we investigate sl_2-triples of global vector fields on schemes X in positive characteristics p>0, and develop a general theory for actions of the corresponding height-one group scheme G=SL_2[F]. Sending a point to the Lie algebra of its stabilizer defines rational maps to various Grassmann varieties. For surfaces of general type, this yields fibrations in curves of genus g at least 2 over the projective line. Using properties of the corresponding moduli stack M_g, we prove that there are no smooth surfaces of general type with an sl_2-triple. On the other hand, employing Lefschetz pencils and Frobenius pullbacks we show that canonical surfaces of general type with such triples exist in abundance. In this connection, we classify the rational double points where the tangent sheaf is free or the evaluation pairing with Kähler differentials is surjetive, including characteristic two.