Surfaces of general type and sl_2-triples

TL;DR

This paper explores the role of sl_2-triples in the structure and classification of surfaces of general type in positive characteristic, revealing their existence and constraints through geometric and algebraic methods.

Contribution

It develops a general theory for sl_2-triples of vector fields on schemes in positive characteristic and classifies surfaces of general type with such triples, including in characteristic two.

Findings

No smooth surfaces of general type with sl_2-triples exist.

Canonical surfaces of general type with sl_2-triples are abundant.

Classification of rational double points with specific tangent sheaf properties.

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 K\"ahler differentials is surjetive, including characteristic two.