Nested Hilbert schemes of the plane

Abstract: The Hilbert scheme of points Hilb^n(A^2), parametrizing finite subschemes of the plane of degree n, is a well studied and well behaved parameter space. A classical theorem of Fogarty states that it is a smooth variety of dimension 2n. By contrast, the nested Hilbert scheme Hilb^(n_1,n_2)(A^2), parametrizing nested pairs of subschemes of degrees n_1 and n_2, are usually singular, and very little is known about their singularities. Using techniques from commutative algebra, we prove that the nested Hilbert scheme Hilb^(n,2)(A^2) has rational singularities. This is a joint work with Ritvik Ramkumar.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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