On the natural nullcones of the symplectic and general linear groups

Consider a group acting on a polynomial ring $S$ over a field $K$ by degree-preserving $K$-algebra automorphisms. Several key properties of the invariant ring can be deduced by studying the nullcone of the action, that is, the vanishing locus of all nonconstant homogeneous invariant polynomials. These properties include the finite generation of the invariant ring and the purity of its embedding in $S$. In this article, we study the nullcones arising from the natural actions of the symplectic and general linear groups. For the natural representation of the symplectic group (via copies of the regular representation), the invariant ring is a generic Pfaffian ring. We show that the nullcone of this embedding is $F$-regular in positive characteristic. Independent of characteristic, we give a complete description of the divisor class group of the nullcone and determine precisely when it is Gorenstein. For the natural representation of the general linear group (via copies of the regular representation and copies of its dual), the invariant ring is a generic determinantal ring. The nullcone of this embedding is typically non-equidimensional; its irreducible components are the varieties of complexes introduced by Buchsbaum and Eisenbud. We show that each of these irreducible components are $F$-regular in positive characteristic. We also show that the Frobenius splittings of the varieties of complexes may be chosen compatibly so that the nullcone is $F$-pure.