On the Behrend function and the blowup of some fat points

The Behrend function of a C-scheme X is a constructible function v(X) : 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 (sic) A(N) is a fat point, v(X) is the sum of the multiplicities of the irreducible components of the exceptional divisor E(X)A(N) in the blowup Bl(X) A(N). Moreover, we prove that v(X) can be computed explicitly through the normalisation of Bl(X) A(N).The proofs of our explicit formulas for the Behrend function of a fat point in A(2) rely heavily on toric geometry techniques. Along the way, we find a formula for the number of irreducible components of E(X)A(2), where X (sic) A(2) is a fat point such that Bl(X) A(2) is normal. (c) 2023 Elsevier Inc. All rights reserved.