The tangent space of the punctual Hilbert scheme (Q2408346)

Let \(S\) be a smooth projective surface and denote with \(S^{[n]}\) the Hilbert scheme that parameterizes length \(n\) subschemes of \(S\). There exists a natural map from \(S^{[n]}\) to the symmetric power \(S^{(n)}\), whose fibers over multiplicity \(n\) cycles define the \textit{punctual Hilbert scheme} \(P_n\). In other words, \(P_n\) parameterizes subschemes of length \(n\) supported at one point. The authors study the tangent space \(T_\xi\) to \(P_n\) at a scheme \(\xi\in P_n\). The dimension \(\dim(T_\xi) \) is bounded below by the corank of the normal map \(\alpha_{n,\xi}\) which sends \(H^0(S, T_{S|\xi})\) to Hom\((\mathcal I_\xi,\mathcal O_\xi)\). The authors prove that when the ideal \(I_\xi\) of \(\xi\) is a monomial ideal, then \(\dim(T_\xi) \) is indeed equal to the corank of \(\alpha_{n,\xi}\). They also show how, for schemes \(\xi\) defined by monomials, the corank of \(\alpha_{n,\xi}\) can be computed from the Young diagram associated to \(I_\xi\).