(-1)-divisors, Cremona transformations, and Mukai's counterexample to Hilbert's 14th problem

Abstract

We explore Prof. Shigeru Mukai's work in his 2001 paper on a counterexample to Hilbert's 14th problem, which realizes a specific ring of invariants as a Cox ring of a blowup of projective space at some number of points, and then relates the finite generation of the ring to the infinitude of (-1)-divisors on the variety. We focus on the geometric arguments used to prove the infinitude of this class of divisors, which is an application of Cremona transformations, with an extension via deformation theory. This paper is intended to provide a more explicit and thorough exposition of the results presented in the original paper in order to make the arguments more accessible.