  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 Brian\c{c}on 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
Brian\c{c}on-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
Brian\c{c}on-Iarrobino conjecture, and we also settle a weaker conjecture of
Sturmfels in the negative.
