Bialynicki-Birula decompositions, Hilbert schemes and combinatorics

For a smooth projective variety X with a C^*-action, every orbit compactifies to a map from P^1 and one can split the points of X according to where infinity is mapped. The obtained division is called the negative Bialynicki-Birula decomposition. In the talk I will explain how to generalize the BB decomposition and use it to find new components and prove non-reducedness and other pathologies for the Hilbert scheme of points. Surprisingly, the key role is played by the tangent space to the Hilbert scheme, which is easily computable, hence the search can be effectively conducted. I will also list some open questions.