On the motive of zero-dimensional Quot schemes on a curve

Barbara Fantechi; Fabio Perroni; Massimo Bagnarol

arxiv: 1907.00826 · v1 · pith:2QNG76RGnew · submitted 2019-07-01 · 🧮 math.AG

On the motive of zero-dimensional Quot schemes on a curve

Massimo Bagnarol , Barbara Fantechi , Fabio Perroni This is my paper

Pith reviewed 2026-05-25 11:51 UTC · model grok-4.3

classification 🧮 math.AG

keywords curvesheafclassquotquotientszero-dimensionalapplicationcoherent

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The pith

The Grothendieck class of zero-dimensional Quot schemes on a curve depends only on rank and length, with an explicit formula via symmetric products.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In algebraic geometry, Quot schemes are spaces that parametrize quotients of vector bundles or sheaves. This work looks at the case where the quotients are zero-dimensional, meaning they correspond to finite collections of points on a smooth projective curve. The authors work in the Grothendieck ring of varieties, an algebraic structure that records information about varieties so that one can perform calculations that capture their geometric essence. They show that the class of these particular Quot schemes in the ring is completely determined by just two pieces of data: the rank of the original sheaf and the length of the quotient. Because of this independence, they derive a concrete formula that writes the class in terms of the symmetric products of the curve, which are the spaces that parametrize unordered sets of points.

Core claim

We prove that this class depends only on the rank of the sheaf and on the length of the quotients.

Load-bearing premise

The base is a fixed smooth projective curve and the sheaf is locally free coherent; the Quot scheme parametrizes zero-dimensional quotients.

read the original abstract

For any locally free coherent sheaf on a fixed smooth projective curve, we study the class, in the Grothendieck ring of varieties, of the Quot scheme that parametrizes zero-dimensional quotients of the sheaf. We prove that this class depends only on the rank of the sheaf and on the length of the quotients. As an application, we obtain an explicit formula that expresses it in terms of the symmetric products of the curve.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper establishes that the Grothendieck class of zero-dimensional Quot schemes on a curve depends only on rank and length, with an explicit symmetric-product formula.

read the letter

The main takeaway is that for a locally free sheaf E on a smooth projective curve, the class [Quot(E,n)] in the Grothendieck ring of varieties is independent of the choice of E (beyond its rank) and equals a fixed expression built from symmetric products of the curve. This is the concrete simplification the abstract advertises, and the stress-test sketch explains why it holds: the local stalks are free of rank r, so the stratification by support partitions gives identical fibers for any two such sheaves, yielding a uniform generating function or recursion. That part looks clean and matches standard facts about Grassmannians over local rings on a curve. The explicit formula is the part that actually gets used downstream for motivic or enumerative calculations, so having it written down is the useful output. The argument does not appear to introduce new technical machinery beyond the usual stratification and the Grothendieck ring relations; it just applies them uniformly. The main soft spot is that the abstract alone does not let one check the precise recursive step or the base cases for small n, but the local-freeness reduction is standard and the stress-test finds no hidden dependence on degree or isomorphism class. No circularity or invented entities show up. This is a short, targeted note for people already working with motivic classes of moduli spaces on curves or with Quot schemes in low dimension. It is not a broad foundational paper, but the claim is narrow enough that a referee can verify it in a few pages. I would send it to peer review; the result is usable if the formula checks out, and the independence statement is worth having on record.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no free parameters, invented entities, or ad-hoc axioms are mentioned. The result relies on standard properties of the Grothendieck ring.

axioms (1)

standard math The Grothendieck ring of varieties is equipped with the usual addition and multiplication operations that allow classes of schemes to be compared and manipulated.
Used implicitly when assigning a class to the Quot scheme.

pith-pipeline@v0.9.0 · 5591 in / 1155 out tokens · 38634 ms · 2026-05-25T11:51:17.753543+00:00 · methodology

On the motive of zero-dimensional Quot schemes on a curve

Core claim

Load-bearing premise

discussion (0)