The Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points

Abstract

For a fixed quasi-projective scheme $X$ we introduce a self-dual analogue of ${\mathrm{Hilb}}_d(X)$ which we call the Iarrobino scheme of $X$. It is a fine moduli space of zero-dimensional subschemes of $X$ together with a self-dual filtration. It contains an open subset that parameterises oriented Gorenstein subschemes. We provide also self-dual analogues of the Quot scheme of points and of the stacks of coherent sheaves and finite algebras. We prove that the resulting analogues of Hilbert and Quot schemes are smooth for $X$ a smooth curve. The Iarrobino scheme is a natural ambient space for enumerative invariants of algebras as introduced by Micha{\l}ek. When $X$ is an affine space, it also admits a commuting matrices description and related virtual structure.