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

In this paper we study the tangent space to the Hilbert scheme H i l b d P 3 \mathrm {Hilb}^d \mathbf {P}^3 , motivated by Haiman’s work on H i l b d P 2 \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 H i l b d P n \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 \frac {4}{3} , and improve the known asymptotic bound on the dimension of H i l b d P 3 \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.