On the Behrend function and the blowup of some fat points

Abstract

The Behrend function of a C-scheme X is a constructible function νX:X(C)→Z introduced by Behrend, intrinsic to the scheme structure of X. It is a (subtle) invariant of singularities of X, playing a prominent role in enumerative geometry. To date, only a handful of general properties of the Behrend function are known. In this paper, we compute it for a large class of fat points (schemes supported at a single point). We first observe that, if X↪AN is a fat point, νX is the sum of the multiplicities of the irreducible components of the exceptional divisor EXAN in the blowup BlXAN. Moreover, we prove that νX can be computed explicitly through the normalisation of BlXAN.The proofs of our explicit formulas for the Behrend function of a fat point in A2 rely heavily on toric geometry techniques. Along the way, we find a formula for the number of irreducible components of EXA2, where X↪A2 is a fat point such that BlXA2 is normal.