Mukai's counterexamples to Hilbert's 14th problem rely on the key observation, due to Nagata, that invariant rings for certain vector groups can be identified with Cox rings of certain blow-ups of projective spaces. The Cox ring of a variety encodes a lot of information about the birational geometry of the variety and one can use geometry to answer questions about the finite generation of the Nagata invariant rings. After an introduction to these questions, we will discuss the birational geometry of blow-ups of projective spaces at points in general position. For that, we will explore Gale duality, a correspondence between sets of n = r + s+ 2 points in projective spaces $\mathbb{P}^{r}$ and $\mathbb{P}^{s}$. For small values of s, this duality has a remarkable geometric manifestation: the blow-up of $\mathbb{P}^{r}$ at n points can be realized as a moduli space of vector bundles on the blow-up of $\mathbb{P}^{s}$ at the Gale dual points. This observation together with variation of stability for a suitable moduli problem can lead to an understanding of the birational geometry of such blow-ups.
De Ana-Maria Castravet
De Ana-Maria Castravet
De Ana-Maria Castravet
