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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 Iarrobinoon the most singular point in $\mathrm{Hilb}^d \mathbf{P}^n$. For pointsparametrizing monomial subschemes, we consider a decomposition of the tangentspace into six distinguished subspaces, and show that a fat point exhibits anextremal behavior in this respect. This decomposition is also used tocharacterize smooth monomial points on the Hilbert scheme. We prove the firstBrian\c{c}on-Iarrobino conjecture up to a factor of 4/3, and improve the knownasymptotic bound on the dimension of $\mathrm{Hilb}^d \mathbf{P}^3$.Furthermore, we construct infinitely many counterexamples to the secondBrian\c{c}on-Iarrobino conjecture, and we also settle a weaker conjecture ofSturmfels in the negative.

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