Hassett-Tschinkel correspondence: Modality and projective hypersurfaces

TL;DR

This paper extends the Hassett-Tschinkel correspondence to classify and analyze generically transitive actions of commutative unipotent groups on projective hypersurfaces, including quadrics, focusing on modality and specific classifications.

Contribution

It develops the Hassett-Tschinkel correspondence further, computes modality for G-actions on projective spaces, and classifies actions of modality one and on hypersurfaces.

Findings

Calculated modality of G-actions on projective spaces.

Classified G-actions of modality one.

Characterized G-actions on projective hypersurfaces, including quadrics.

Abstract

B. Hassett and Yu. Tschinkel (1999) introduced a remarkable correspondence between generically transitive actions of a commutative unipotent algebraic group G and finite-dimensional local algebras. In this paper we develop Hassett-Tschinkel correspondence and calculate modality of generically transitive G-actions on projective spaces, classify actions of modality one, and characterize generically transitive G-actions on projective hypersurfaces of given degree. In particular, actions on degenerate projective quadrics are studied.