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The Behrend function of a $\mathbb C$-scheme $X$ is a constructible function$\nu_X\colon X(\mathbb C) \to \mathbb Z$ introduced by Behrend, intrinsic tothe scheme structure of $X$. It is a (subtle) invariant of singularities of$X$, playing a prominent role in enumerative geometry. To date, only a handfulof general properties of the Behrend function are known. In this paper, wecompute it for a large class of fat points (schemes supported at a singlepoint). We first observe that, if $X \hookrightarrow \mathbb A^N$ is a fatpoint, $\nu_X$ is the sum of the multiplicities of the irreducible componentsof the exceptional divisor $E_{X}\mathbb A^N$ in the blowup$\textrm{Bl}_{X}\mathbb A^N$. Moreover, we prove that $\nu_X$ can be computedexplicitly through the normalisation of $\textrm{Bl}_{X}\mathbb A^N$.  The proofs of our explicit formulas for the Behrend function of a fat pointin $\mathbb A^2$ rely heavily on toric geometry techniques. Along the way, wefind a formula for the number of irreducible components of $E_{X}\mathbb A^2$,where $X \hookrightarrow \mathbb A^2$ is a fat point such that$\textrm{Bl}_{X}\mathbb A^2$ is normal.

On the Behrend function and the blowup of some fat points