Mathematics

Not much is known about the geometric properties of thepunctual Hilbert scheme of fat points of length n supported at the originof the affine spaceAk.In order to investigate them, a huge number ofinvariants, for fat points, has been introduced (e.g.multiplicity, order,type, blowup tree...).I will focus on the Behrend numberνZof a fatpoint Z inA2.Such invariant can be defined in terms of the blowup ofthe affine plane with center the subscheme Z. I will discuss the problemof computing the Behrend number of a monomial fat point following ajoint work with Andrea T. Ricolfi. In particular, I will explain, first in thenormal setting, how toric geometry methods apply in the construction ofthe blowup and in the computation ofνZ. Then, I will move to the non-normal setting, and I will show some examples of computation. Finally, iftime permits, I will show some difficulties that arise in higher dimension.