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

In this paper we study the tangent space to the Hilbert scheme Hilb^d(P^3), motivated by Haiman’s work on Hilb^d(P^2) and by a long-standing conjecture of Briancon and Iarrobino [J. Algebra 55 (1978), pp. 536–544] on the most singular point in Hilb^d(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 Briancon-Iarrobino conjecture up to a factor of 4/3 , and improve the known asymptotic bound on the dimension of Hilb^d(P^3). Furthermore, we construct infinitely many counterexamples to the second Briancon-Iarrobino conjecture, and we also settle a weaker conjecture of Sturmfels in the negative.