Abstract

For a curve

C

, the Hilbert scheme

{Hilb}^{d} (C)

parametrizes zero-dimensional subschemes of

C

of length

d

. We can package the classes of these Hilbert schemes in the Grothendieck ring of varieties into the motivic Hilbert zeta function, defined by

Z_{C} (t) = \sum_{d \geq 0} [{Hilb}^{d} (C)] t^{d} .

C

is smooth, then

{Hilb}^{d} (C) ≅ {Sym}^{d} (C)

, so this agrees with Kapranov’s motivic zeta function. Motivic zeta functions are not rational in general, however, for reduced curves, the motivic Hilbert zeta function is rational, with the denominator governed by the number of branches. A natural question is what happens once the curve is allowed to be nonreduced.

We will begin with some background on Hilbert schemes of points on the affine plane, focusing mainly on their combinatorial description following Haiman. We will then study the motivic Hilbert zeta function of a planar

n

-fold thickening of a smooth curve. Finally, if time permits, we will look at nonreduced but generically reduced curves and discuss, through an example, the main steps in establishing rationality.

Zoom Link

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Organizer

Weichung Chen (NTU)