Gale Duality, Blowups and Moduli Spaces

To understand the birational geometry of a projective variety $X$, one seeks to describe all rational contractions from $X$. From an algebraic perspective, information about all these contractions are encoded in the ring formed by all sections of all line bundles on $X$, the Cox ring of $X$. In this talk, we discuss the birational geometry and the Cox ring of blowups of projective spaces at points in general position. For that, we explore Gale duality, a correspondence between sets of $n=r+s+2$ points in projective spaces $\mathbb{P}^s$ and $\mathbb{P}^r$. For small values of $s$, this duality has a remarkable geometric manifestation: the blowup of $\mathbb{P}^r$ at $n$ points can be realized as a moduli space of vector bundles on the blowup of $\mathbb{P}^s$ at the Gale dual points.