Blow-ups of P^n-3 at n points and spinor varieties

Work of Dolgachev and Castravet-Tevelev establishes a bijection between the 2^n-1 weights of the half-spin representations of so_2n and the generators of the Cox ring of the variety X_n which is obtained by blowing up P^n-3 at n points. We derive a geometric explanation for this bijection, by embedding 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-Skorobogatovon del Pezzo surfaces.