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Abstract:The goal of this paper is to describe the birational geometry of the blowup of $\mathbb{P}^n$ at $n+4$ points in very general position. To achieve this, we follow an idea of Mukai and explore a special instance of Gale duality, namely, a correspondence between configurations of $n+4$ points in the projective spaces $\mathbb{P}^n$ and $\mathbb{P}^2$. We first prove that the blowup $X$ of $\mathbb{P}^n$ at $n+4$ general points is isomorphic to a certain Gieseker moduli space of rank $2$ vector bundles on the surface $S$ obtained by blowing up $\mathbb{P}^2$ at the $n+4$ Gale dual points. We then study the variation of these moduli spaces as we vary the polarization $L$ on $S$, and translate this variation into a partial Mori chamber decomposition of $\overline{Eff}(X)$, describing to some extent the birational geometry of $X$.

Submission history

From: Ana-Maria Castravet [view email]
[v1] Tue, 26 May 2026 15:14:27 UTC (45 KB)
[v2] Wed, 22 Jul 2026 10:54:48 UTC (45 KB)