Mathematics

The geometry of a quartic del Pezzo surface $S$ is very well understood.Embedded as a smooth complete intersection of two quadric hypersurfacesin $\mathbb P^4$, the surface $S$ contains exactly 16 lines. It canalso be described as the blowup of $\mathbb P^2$ at 5 points in generallinear position. In fact, there are 16 different ways to realize $S$ assuch blowup: for each line $\ell\subset S$, there is one such blowupunder which $\ell$ is the transform of the unique conic through theblown up points.In this talk, we will explain how this picture generalizes to arbitraryeven dimension. Given an even positive integer $n=2m$, we consider thevariety $G$ parametrizing $(m−1)$-planes in a smooth completeintersection of two quadrics in $\mathbb P^{2m+2}$. This is a Fanovariety of dimension $n$ that can also be described as a smallmodification of the blowup of $\mathbb P^n$ at $n+3$ points in generallinear position. We show that there are $2n+2$ different ways torealize $G$ in this manner, one for each of the $2n+2$ distinct$m$-planes contained in the complete intersection of two quadrics, anddescribe these birational maps explicitly. This is a joint work with Cinzia Casagrande.