# On the tangent space to the Hilbert scheme of points in P^3

The Hilbert scheme of n points in P^2 is smooth of dimension 2n and the tangent space to any monomial subscheme admits a pleasant combinatorial description. On the other hand, the Hilbert scheme of n points in P^3 is almost always singular and there is a conjecture by Briançon and Iarrobino describing the monomial subscheme with the largest tangent space dimension. In this talk we will generalize the combinatorial description to the Hilbert scheme of points in P^3, revealing new symmetries in the tangent space to any monomial subscheme. We will use these symmetries to prove many cases of the conjecture and strengthen previous bounds on the dimension of the Hilbert scheme. In addition, we will also characterize smooth monomial points on the Hilbert scheme. This is joint work with Alessio Sammartano.