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 $\mathbb{G}_a^m$ for $m$ greater than or equal to $3$. 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 $\mathbb{P}^n$. 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.

commutative algebraalgebraic geometrycombinatoricsrings and algebrasrepresentation theory

Audience: researchers in the topic

Geometry, Algebra, Singularities, and Combinatorics

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