On the automorphism group of a toral variety

• Published in: arXiv.org
• Main Authors: Shafarevich, Anton, Trushin, Anton
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 07.12.2023
• Subjects: Automorphisms; Toruses
• ISSN: 2331-8422

Let \(\mathbb{K}\) be an algebraically closed field of characteristic zero. An affine algebraic variety \(X\) over \(\mathbb{K}\) is toral if it is isomorphic to a closed subvariety of a torus \((\mathbb{K}^*)^d\). We study the group \(\mathrm{Aut}(X)\) of regular automorpshims of a toral variety \(X\). We prove that if \(T\) is a maximal torus in \(\mathrm{Aut}(X)\), then \(X\) is a direct product \(Y\times T\), where \(Y\) is a toral variety with a trivial maximal torus in the automorphism group. We show that knowing \(\mathrm{Aut}(Y)\), one can compute \(\mathrm{Aut}(X)\). In the case when the rank of the group \(\mathbb{K}[Y]^*/\mathbb{K}^*\) is \(\dim Y + 1\), the group \(\mathrm{Aut}(Y)\) can be described explicitly.