A note on the Hilbert scheme of points on a cusp curve

摘要

We study the Hilbert scheme of points on a cusp curve, mostly the punctual Hilbert scheme Hilb _m ~0 which parameterizes m points supported at a cusp. We show that the reduced punctual scheme Hilb _m ~(0red) is isomorphic to p ~1 and the Hilbert scheme has one singularity along Hilb _m ~0.