Hassett-Tschinkel correspondence: Modality and projective hypersurfaces
Let \(\mathbb K\) be an algebraically closed field of characteristic zero and \(\mathbb G_n^a\) an \(n\)-dimensional commutative unipotent algebraic group over \(\mathbb K\). In 1999, Hassett and Tschinkel discovered an interesting correspondence between generically transitive actions of \(\mathbb G_n^a\) on the projective space \(\mathbb P^n\) and finite-dimensional local \(\mathbb K\)-algebras (see [\textit{B. Hassett} and \textit{Y. Tschinkel}, Int. Math. Res. Not. 1999, No. 22, 1211--1230 (1999; Zbl 0966.14033)]).NEWLINENEWLINEThe goal of this paper is develop further the theory of this correspondence. Using the corresponding local algebra, the authors calculate modality (in the sense of Vinberg) of generically transitive \(\mathbb G_n^a\)-actions on projective spaces and classify actions of modality one. If \(n\geq 5\), then there are exactly \(n+1\) generically transitive \(\mathbb G_n^a\)-actions of modality one. The authors also characterize generically transitive \(\mathbb G_n^a\)-actions on projective hypersurfaces of a given degree. In particular, it is shown that families (of isomorphism classes) of generically transitive \(\mathbb G_n^a\)-actions on a degenerate quadric in \(\mathbb P^{n+1}\) admit moduli.
scientific article; zbMATH DE number 6029091
scientific article