Cox rings, linear blow-ups and the generalized Nagata action

Abstract: Nagata gave the first counterexample to Hilbert's 14th problem on the finite generation of invariant rings by actions of linear algebraic groups. His idea was to relate the ring of invariants to a Cox ring of a projective variety. Counterexamples of Nagata's type include the cases where the group is $G_a^m$ for $m=3, 6, 9$ or $13$. However, for $m=2$, the ring of invariants under the Nagata action is finitely generated. It is still an open problem whether counterexamples exist for $m=2$.

In this talk we consider a generalized version of Nagata's action by H. Naito. Mukai envisioned that the ring of invariants in this case can still be related to a cox ring of certain linear blow-ups of $P^n$. We show that when $m=2$, the Cox rings of this type of linear blow-ups are still finitely generated, and we can describe their generators. This answers the question by Mukai.

algebraic geometry

Audience: researchers in the discipline

American Graduate Student Algebraic Geometry Seminar

Series comments: The American Graduate Student Algebraic Geometry Seminar (AGSAGS) is a virtual seminar by and for algebraic geometry graduate students.

The goal of this seminar is for graduate students to share their research through online talks and to provide an algebraic geometry graduate networking system. Grad students, postdocs, and professors are welcome to attend.

Seminars will be held on Mondays at 4 p.m. Eastern on Zoom. We hope this time is convenient for graduate students in the Americas, hence the name AGSAGS. Prior registration is required and interested participants should register here: sites.google.com/view/agsags/registration. In addition to graduate talks, there will be occasional social events.