Аннотация:
Let $G$ be a connected linear algebraic group. An equivariant embedding of the group $G$ into an algebraic variety $X$ is an open embedding of $G$ into $X$ such that the action of $G$ on itself by left translations extends to an action of $G$ on $X$. For example, equivariant embeddings of an algebraic torus are described by the theory of toric varieties. Equivariant embeddings of the group $\mathrm{SL}_2(С)$ into affine algebraic varieties were described by V. Popov.
Equivariant embeddings of the vector group $\mathbb{G}_a^n$ into projective space $\mathbb{P}^n$ were described by F. Knop and H. Lange, and independently by B. Hassett and Yu. Tschinkel. In particular, their results imply that for $n > 6$ there exist infinitely many pairwise inequivalent embeddings of $\mathbb{G}_a^n$ into $\mathbb{P}^n$ .
The closest analogue of the group $\mathbb{G}_a^n$ is the Heisenberg group. It is also a unipotent group, and its commutator subgroup is one-dimensional. Following the work of Cong Ding and Zhijun Luo, we will show that the Heisenberg group admits infinitely many equivariant embeddings into projective space.
|
