Hyperquot schemes on curves: virtual class and motivic invariants

Abstract

Let $C$ be a smooth projective curve, $E$ a locally free sheaf. Hyperquot schemes on $C$ parametrise flags of coherent quotients of $E$ with fixed Hilbert polynomial, and offer alternative compactifications to the spaces of maps from $C$ to partial flag varieties. Motivated by enumerative geometry, in this paper we construct a perfect obstruction theory (and hence a virtual class and a virtual structure sheaf) on these moduli spaces, which we use to provide criteria for smoothness and unobstructedness. Under these assumptions, we determine their motivic partition function in the Grothendieck ring of varieties, in terms of the motivic zeta function of $C$.

37 pages. Final version, to appear in Mathematische Annalen

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• The motive of the Hilbert scheme of points in all dimensions
• Invariants of nested Hilbert and Quot schemes on surfaces