On the tangent space to the Hilbert scheme of points in P3

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 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 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.