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Hyperquot schemes on curves: virtual class and motivic invariants

Sergej Monavari, Andrea T. Ricolfi

TL;DR

This work extends Grothendieck’s Quot scheme by introducing hyperquot schemes on curves, parameterizing nested quotients of a fixed vector bundle and generalizing maps to partial flag varieties. It develops a perfect obstruction theory for these moduli spaces, yielding a virtual fundamental class and a virtual structure sheaf, and derives a precise virtual dimension formula. Under smoothness/unobstructedness assumptions, it computes the motivic partition function in the Grothendieck ring in terms of the motivic zeta function of the curve, with explicit genus-zero factorizations and rationality results. The paper also analyzes torus fixed loci, provides a BB-type motivic decomposition, and explores consequences for irreducibility and specializations to ordinary Quot schemes, linking virtual enumerative geometry with motivic invariants and curve-dependent data.

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 $C$.

Hyperquot schemes on curves: virtual class and motivic invariants

TL;DR

This work extends Grothendieck’s Quot scheme by introducing hyperquot schemes on curves, parameterizing nested quotients of a fixed vector bundle and generalizing maps to partial flag varieties. It develops a perfect obstruction theory for these moduli spaces, yielding a virtual fundamental class and a virtual structure sheaf, and derives a precise virtual dimension formula. Under smoothness/unobstructedness assumptions, it computes the motivic partition function in the Grothendieck ring in terms of the motivic zeta function of the curve, with explicit genus-zero factorizations and rationality results. The paper also analyzes torus fixed loci, provides a BB-type motivic decomposition, and explores consequences for irreducibility and specializations to ordinary Quot schemes, linking virtual enumerative geometry with motivic invariants and curve-dependent data.

Abstract

Let be a smooth projective curve, a locally free sheaf. Hyperquot schemes on parametrise flags of coherent quotients of with fixed Hilbert polynomial, and offer alternative compactifications to the spaces of maps from 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 .
Paper Structure (35 sections, 22 theorems, 148 equations)

This paper contains 35 sections, 22 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Statements of the main results
  4. Relation with Enumerative Geometry
  5. Virtual invariants
  6. Higher dimensional varieties
  7. Cohomology
  8. Conventions
  9. Hyperquot scheme
  10. Partial flag varieties
  11. The hyperquot scheme
  12. Tangent space to hyperquot scheme
  13. Our setup: the curve case
  14. Perfect obstruction theory
  15. Perfect obstruction theories
  16. ...and 20 more sections

Key Result

Theorem A

Let $C$ be a smooth projective curve of genus $g$, $E$ a locally free sheaf of rank $r>0$, and $l$ a positive integer. Fix tuples $\boldsymbol{s}=(s_1\leqslant \dots \leqslant s_l)$ and $\boldsymbol{d}=(d_1, \dots, d_l)$. The hyperquot scheme $\mathop{\mathrm{\mathrm{Q}}}\nolimits = \mathop{\mathrm{ Setting $s_{l+1}=r$ and $s_0=0$, the virtual class lives in dimension

Theorems & Definitions (46)

  • Theorem A: \ref{['thm: pot 2 nest']}, \ref{['cor: virtual cycles']}
  • Theorem B: \ref{['prop: smooth nested flag']}
  • Theorem C: \ref{['thm: theorem motivic all g']}, \ref{['thm: fact genus 0']}
  • Theorem D: \ref{['thm:euler-top']}
  • Example 2.1
  • Lemma 3.1: Sin and Sca_pot
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • ...and 36 more