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

## TL;DR

This paper introduces the Iarrobino scheme, a self-dual analogue of the Hilbert scheme of points, which parametrizes Gorenstein subschemes with additional structure, revealing new geometric properties and applications in deformation and enumerative geometry.

## Contribution

It defines the Iarrobino scheme as a self-dual moduli space of Gorenstein subschemes, extending classical Hilbert schemes with new geometric and deformation-theoretic insights.

## Key findings

- Iarrobino schemes are smooth for smooth curves.
- They have rich geometric structures.
- Applications to deformation theory and enumerative geometry.

## 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 oriented Gorenstein zero-dimensional subschemes of $X$ together with some additional data (a self-dual filtration) which is vacuous over a big open set but non-trivial over the compactification. Via the link between Hilbert schemes and varieties of commuting matrices, Iarrobino schemes correspond to commuting symmetric matrices. We provide also self-dual analogues of the Quot scheme of points and of the stacks of coherent sheaves and finite algebras. A crucial role in the construction is played by the variety of completed quadrics. We prove that the resulting analogues of Hilbert and Quot schemes are smooth for $X$ a smooth curve and that they have very rich geometry. We give applications, in particular to deformation theory of (usual) Hilbert schemes of points on threefolds, and to enumerative geometry \`a la June Huh.