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

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

2009-06-27

arxiv.org/abs/0906.5096v1

Mathematics

Algebraic Geometry

Scientific paper

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.

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