Blow-ups of $\mathbb{P}^{n-3}$ at $n$ points and spinor varieties

TL;DR

This paper provides a geometric interpretation of the bijection between Cox ring generators of certain blow-up varieties and weights of spinor representations, generalizing known results from del Pezzo surfaces to higher dimensions.

Contribution

It introduces a geometric embedding of the Cox ring into the even spinor variety, extending previous results to higher-dimensional blow-ups of projective space.

Findings

Established a geometric explanation for the Cox ring and spinor weights bijection.

Embedded Cox rings into even spinor varieties via torus translates.

Generalized results from del Pezzo surfaces to higher-dimensional cases.

Abstract

Work of Dolgachev and Castravet-Tevelev establishes a bijection between the $2^{n-1}$ weights of the half-spin representations of $\mathfrak{so}_{2n}$ and the generators of the Cox ring of the variety $X_n$ which is obtained by blowing up $\mathbb{P}^{n-3}$ at $n$ points. We derive a geometric explanation for this bijection, by embedding ${\rm Cox}(X_n)$ into the even spinor variety (the homogeneous space of the even half-spin representation). The Cox ring of the blow-up $X_n$ is recovered geometrically by intersecting torus translates of the even spinor variety. These are higher-dimensional generalizations of results by Derenthal and Serganova-Skorobogatov on del Pezzo surfaces.