On the tangent space to the Hilbert scheme of points in $\mathbf {P}^3$

Abstract:

In this paper we study the tangent space to the Hilbert scheme $\mathrm {Hilb}^d \mathbf {P}^3$, motivated by Haiman’s work on $\mathrm {Hilb}^d \mathbf {P}^2$ and by a long-standing conjecture of Briançon and Iarrobino [J. Algebra 55 (1978), pp. 536–544] on the most singular point in $\mathrm {Hilb}^d \mathbf {P}^n$. For points parametrizing monomial subschemes, we consider a decomposition of the tangent space into six distinguished subspaces, and show that a fat point exhibits an extremal behavior in this respect. This decomposition is also used to characterize smooth monomial points on the Hilbert scheme. We prove the first Briançon-Iarrobino conjecture up to a factor of $\frac {4}{3}$, and improve the known asymptotic bound on the dimension of $\mathrm {Hilb}^d \mathbf {P}^3$. Furthermore, we construct infinitely many counterexamples to the second Briançon-Iarrobino conjecture, and we also settle a weaker conjecture of Sturmfels in the negative.