  The projective variety of square-zero elements in the six-dimensional minimal
supersymmetry algebra is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^3$. We
use this fact, together with the pure spinor superfield formalism, to study
supermultiplets in six dimensions, starting from vector bundles on projective
spaces. We classify all multiplets whose derived invariants for the
supertranslation algebra form a line bundle over the nilpotence variety; one
can think of such multiplets as being those whose holomorphic twists have rank
one over Dolbeault forms on spacetime. In addition, we explicitly construct
multiplets associated to natural higher-rank equivariant vector bundles,
including the tangent and normal bundles as well as their duals. Among the
multiplets constructed are the vector multiplet and hypermultiplet, the family
of $\mathcal{O}(n)$-multiplets, and the supergravity and gravitino multiplets.
Along the way, we tackle various theoretical problems within the pure spinor
superfield formalism. In particular, we give some general discussion about the
relation of the projective nilpotence variety to multiplets and prove general
results on short exact sequences and dualities of sheaves in the context of the
pure spinor superfield formalism.
