Blowups, Gale duality, and moduli spaces

2026/05/26 by Carolina Araujo, Ana-Maria Castravet, Inder Kaur +1
#math.AG

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Abstract

The goal of this paper is to describe the birational geometry of the blowup of ℙⁿ 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 ℙⁿ and ℙ². We first prove that the blowup X of ℙⁿ 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 ℙ² 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 Eff(X), describing to some extent the birational geometry of X.

Related

• The blow-up of ℙ⁴ at 8 points and its Fano model, via vector bundles on a del Pezzo surface
• Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane
• Birational geometry of blow-ups of projective spaces along points and lines
• A moduli scheme parametrizing blowups of smooth projective surfaces
• Compactifications of moduli of points and lines in the projective plane