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.
Categories: math.AG math.AC