On the equations for universal torsors over del Pezzo surfaces

TL;DR

This paper describes the structure of universal torsors over del Pezzo surfaces of degrees 2 to 5, showing they can be represented as dense open subsets of intersections of dilated affine cones over Grassmannians, providing a concise algebraic description.

Contribution

It introduces a new geometric description of universal torsors over del Pezzo surfaces using dilatations of affine cones over Grassmannians, extending to non-split cases with rational points.

Findings

Universal torsors over certain del Pezzo surfaces are dense open subsets of intersections of dilated affine cones.

Provides a concise algebraic description of quadratic equations for these torsors.

Extends results to non-split del Pezzo surfaces with rational points.

Abstract

We show that every split del Pezzo surface of degree d=5,4,3 or 2 has a universal torsor which is a dense open subset of the intersection of 6-d dilatations of the affine cone over the corresponding generalized Grassmannian G/P. Here a dilatation is the linear transformation by an element of the 'diagonal' torus. This gives a concise description of the quadratic equations of universal torsors obtained by Popov and Derenthal. Any (possibly, non-split) del Pezzo surface with a rational point has a universal torsor which embeds into the same homogeneous space as a split surface of the same degree. The proof uses a recent result of Ph. Gille and Raghunathan.