[Paper Review] On the Chow theory of Quot schemes of locally free quotients

Qingyuan Jiang|arXiv (Cornell University)|Oct 21, 2020

Algebraic Geometry and Number Theory20 references4 citations

TL;DR

This paper establishes a universal formula for the Chow groups of Quot schemes parameterizing locally free quotients of a coherent sheaf on a Cohen-Macaulay scheme, under expected dimension conditions. It generalizes known results on Grassmannian bundles, blowups, Cayley's trick, and flips/flops by showing that Chow groups decompose into sums of Chow groups of resolutions of degeneracy loci, weighted by Betti numbers of Grassmannians.

ABSTRACT

We prove a formula for Chow groups of $Quot$-schemes which resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This result provides a unified way to understand known formulae for various geometric situations such as blowups, Cayley's trick, projectivizations, Grassmannian bundles, flops from Springer type resolutions, as well as provide new phenomena such as formulae for Grassmannain type flips/flops and virtual flips. We also give applications to blowups of determinantal ideals, moduli spaces of linear series on curves, and Hilbert schemes of points on surfaces.

Motivation & Objective

• To provide a unified framework for computing Chow groups of Quot schemes resolving degeneracy loci of vector bundle maps.
• To generalize classical formulas—such as for Grassmannian bundles, blowups, and Cayley's trick—within a single structural formula.
• To establish a motivic decomposition of Quot schemes in terms of Lefschetz twists and Chow groups of dualizing sheaf resolutions.
• To verify a conjecture from derived algebraic geometry on the structure of Chow groups in the context of Quot schemes.
• To apply the formula to moduli spaces of linear series on curves and Hilbert schemes of points on surfaces.

Proposed method

• Derive a decomposition of integral Chow groups of Quot schemes using the rank of the sheaf and its dualizing sheaf.
• Use Betti numbers of Grassmannians as multiplicity factors in the Chow group decomposition.
• Apply the Quot–formula: $ CH^k(\mathfrak{Quot}_{X,d}(\mathscr{G})) \simeq \bigoplus_{j=0}^{\min\{d,\delta\}} \bigoplus_{\ell=0}^{j(\delta-j)} CH^{k-(d-j)(\delta-j)-\ell}(\mathfrak{Quot}_{X,d-j}(\mathscr{K}))^{\oplus b_{\ell}^{(j,\delta)}} $, where $\mathscr{K} = \mathcal{E}xt^1_{\mathcal{O}_X}(\mathscr{G}, \mathcal{O}_X)$.
• Construct motivic decompositions using Lefschetz twists $L = 1(-1)$, yielding $ h(\mathfrak{Quot}_{X,d}(\mathscr{G})) = \bigoplus_{j,\ell} \left( h(\mathfrak{Quot}_{X,d-j}(\mathscr{K})) \otimes L^{(d-j)(\delta-j)+\ell} \right)^{\oplus b_{\ell}^{(j,\delta)}} $.
• Utilize duality and Chern class computations to relate Chow groups of different Quot schemes, especially in the case of flips and flops.

Experimental results

Research questions

• RQ1 How can Chow groups of Quot schemes resolving degeneracy loci be uniformly described across different geometric contexts?
• RQ2 What is the precise structure of Chow groups when the degeneracy loci have expected dimension?
• RQ3 Can the Quot–formula unify known results such as blowup formulas, projectivization, and Cayley’s trick?
• RQ4 What is the relationship between $ \mathfrak{Quot}_{X,d}(\mathscr{G}) $ and $ \mathfrak{Quot}_{X,d-\delta}(\mathscr{K}) $ in the case of flips or flops?
• RQ5 How do Betti numbers of Grassmannians naturally arise as multiplicity factors in Chow group decompositions?

Key findings

• The Chow groups of $ \mathfrak{Quot}_{X,d}(\mathscr{G}) $ decompose as a direct sum indexed by $ j \in [0, \min\{d, \delta\}] $, with each summand weighted by the Betti numbers $ b_{\ell}^{(j,\delta)} $ of the Grassmannian $ \mathrm{Gr}_j(\delta) $.
• When $ d \leq \delta $, the Quot scheme is generically a Grassmannian bundle over $ X $, and the Chow group decomposition includes contributions from both the Grassmannian fiber and resolutions of degeneracy loci.
• When $ d > \delta $, the Quot scheme and $ \mathfrak{Quot}_{X,d-\delta}(\mathscr{K}) $ are birational via a flip, and the Chow group of $ \mathfrak{Quot}_{X,d-\delta}(\mathscr{K}) $ embeds into that of $ \mathfrak{Quot}_{X,d}(\mathscr{G}) $, with complement given by higher degeneracy locus resolutions.
• In the case $ \delta = 0 $, the birational map is a flop, and the Chow rings of $ \mathfrak{Quot}_{X,d}(\mathscr{G}) $ and $ \mathfrak{Quot}_{X,d}(\mathscr{K}) $ are isomorphic.
• The formula specializes to known results: for $ \mathscr{G} $ locally free, it recovers the Grassmannian bundle formula; for $ d=1 $, it recovers the projectivization formula; for $ d=\delta $, it gives the blowup formula of determinantal subschemes.
• Applications include Chow group computations for Hilbert schemes of points on surfaces and moduli spaces of linear series on curves, via the Quot–formula’s decomposition of motivic and Chow-theoretic structures.