  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.
