Mathematical Research Letters

Volume 19 (2012)

Number 6

Murphy’s law for Hilbert function strata in the Hilbert scheme of points

Pages: 1277 – 1281



Daniel Erman (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)


An open question is whether the Hilbert scheme of points of a high-dimensional affine space satisfies Murphy’s Law, as formulated by Vakil. In this short note, we instead consider the loci in the Hilbert scheme parameterizing punctual schemes with a given Hilbert function, and we show that these loci satisfy Murphy’s Law. We also prove a related result for equivariant deformations of curve singularities with $\mathbb{G}_m$-action.

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