On the embeddings of universal torsors over del Pezzo surfaces in the cones over flag varieties

TL;DR

This paper constructs embeddings of universal torsors over del Pezzo surfaces into cones over flag varieties, introducing a new approach based on Mumford stability that extends to degree one surfaces.

Contribution

It presents a novel method for embedding universal torsors over del Pezzo surfaces into cones over flag varieties, especially for degree one surfaces, using Mumford stability.

Findings

Embeddings are constructed as closed orbits in projectivizations of quasiminiscule representations.

The approach differs from previous methods by employing Mumford stability.

Successfully extends embeddings to degree one del Pezzo surfaces.

Abstract

Following Skorobogatov and Serganova we construct the embeddings of universal torsors over del Pezzo surfaces in the cones over flag varieties, considered as the closed orbits in the projectivization of quasiminiscule representations. We give the approach that allows us to construct the embeddings of universal torsors over del Pezzo surfaces of the degree one. This approach uses the Mumford stability and slightly differs from the approach of the authors named above.