Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings

Terry Dekun Song; Veronica Arena

arxiv: 2605.17922 · v1 · pith:AXYII77Anew · submitted 2026-05-18 · 🧮 math.AG

Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings

Veronica Arena , Terry Dekun Song This is my paper

Pith reviewed 2026-05-20 01:19 UTC · model grok-4.3

classification 🧮 math.AG

keywords logarithmic Hilbert schemesweighted blow-upsintegral Chow ringssymmetric productsmoduli of curves

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The pith

The logarithmic Hilbert scheme of points on a smooth pointed curve is an iterated weighted blow-up of the symmetric product of the curve.

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

The paper shows that the space parametrizing subschemes transverse to the boundary on expanded degenerations of a smooth pointed curve can be built by performing a sequence of weighted blow-ups on the symmetric product. Explicit centers and weights are identified, each carrying a modular meaning in terms of point configurations on the curve. This structure is then used to compute the integral Chow rings by applying known formulas for weighted blow-ups.

Core claim

The logarithmic Hilbert scheme of points on a smooth pointed curve is an iterated weighted blow-up of the symmetric product of the underlying curve, with the blow-up centers, weights, and their modular interpretations given explicitly.

What carries the argument

Iterated weighted blow-up along centers in the symmetric product that correspond to loci of points with specified coincidence orders, equipped with weights that reflect the transverse intersection conditions.

If this is right

The integral Chow rings of the logarithmic Hilbert schemes are expressed directly in terms of the Chow rings of the corresponding symmetric products.
The logarithmic Hilbert scheme of toric projective line is recovered as a toric stack.
Intersection numbers and other cycle classes on these spaces become computable from data on the symmetric products.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same weighted blow-up description may extend to logarithmic Hilbert schemes on higher-genus curves or with marked points in different configurations.
Modular interpretations of the centers could be used to compare logarithmic and ordinary Hilbert schemes in degeneration families.

Load-bearing premise

The integral Chow ring formula for weighted blow-ups together with the weighted analogue of Castelnuovo's criterion apply directly to the centers and weights that appear in the logarithmic Hilbert scheme construction.

What would settle it

An explicit computation of the integral Chow ring for the logarithmic Hilbert scheme of two points on an elliptic curve that fails to match the ring obtained by applying the weighted blow-up formula to the symmetric square.

read the original abstract

The logarithmic Hilbert scheme of a logarithmic curve parametrizes subschemes on the expanded degenerations of the curve that are transverse to the boundary. We prove that the logarithmic Hilbert scheme of points on a smooth pointed curve is an iterated weighted blow-up of the symmetric product of the underlying curve. In doing so, we explicitly identify the blow-up centers, weights, and give them modular interpretations. As applications, we calculate their integral Chow rings in terms of those of the symmetric products. Key ingredients in our work include two recent results: the integral Chow ring formula of weighted blow-ups and a weighted analogue of Castelnuovo's criterion for blow-downs. We recover the folklore result the logarithmic Hilbert scheme of toric $\mathbb{P}^1$ is a toric stack, and the Appendix by Dhruv Ranganathan outlines a complementary approach using Chow quotients.

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 realizes the log Hilbert scheme of points on a curve as an explicit iterated weighted blow-up of the symmetric product and computes the integral Chow ring from that model.

read the letter

The main result here is a concrete identification: the logarithmic Hilbert scheme of points on a smooth pointed curve is an iterated weighted blow-up of the symmetric product, with centers and weights that carry modular interpretations. They then use this to write down the integral Chow ring in terms of the symmetric product's ring. This recovers the known toric P1 case as a check and includes an appendix sketching a Chow quotient approach by Ranganathan. The argument rests on two recent external results—one for the Chow ring of weighted blow-ups and one for a weighted version of Castelnuovo's criterion—plus standard scheme-theoretic constructions. That combination produces a usable model rather than an existence statement, which is the useful part. The modular reading of the centers is a clear plus because it ties the blow-up data directly to the geometry of expanded degenerations and transversality conditions. On the soft side, the application of the weighted Castelnuovo criterion needs the centers to be regularly embedded with normal bundles satisfying the right weight conditions. The modular description makes this plausible, but the stress-test note is fair: in the logarithmic setting the centers sit on expanded curves, so it is not automatic that local equations meet the regularity hypotheses without an extra local computation. If the full paper supplies that check explicitly, the argument closes; if it relies on the modular picture alone to imply the conditions, a referee might ask for one more paragraph of verification. The paper is aimed at people working in logarithmic geometry and moduli of curves who want explicit presentations or Chow-ring formulas. A reader who already knows the symmetric product and the recent weighted-blow-up tools will get the most out of it. The result is specific enough and grounded enough in cited theorems that it deserves a serious referee rather than a desk rejection. I would send it out for review and flag the regularity check on the centers as the main point to verify.

Referee Report

2 major / 2 minor

Summary. The paper proves that the logarithmic Hilbert scheme of points on a smooth pointed curve is an iterated weighted blow-up of the symmetric product of the underlying curve. It explicitly identifies the blow-up centers and weights, gives them modular interpretations, and uses this identification to compute the integral Chow rings of these schemes in terms of those of the symmetric products. The proof invokes two recent external results: the integral Chow ring formula for weighted blow-ups and a weighted analogue of Castelnuovo's criterion for blow-downs. The paper recovers the folklore result that the logarithmic Hilbert scheme of toric P^1 is a toric stack and includes an appendix by Dhruv Ranganathan outlining a complementary approach via Chow quotients.

Significance. If the central identification holds, the result supplies an explicit geometric construction for logarithmic Hilbert schemes of points, enabling direct computation of their integral Chow rings from known data on symmetric products. The modular interpretations of the centers and weights add conceptual value by linking the blow-up data to the geometry of expanded degenerations and transversality conditions. This bridges logarithmic geometry with techniques from weighted blow-ups and could inform further work on moduli spaces of curves and subschemes in logarithmic settings.

major comments (2)

[Sections describing the blow-up centers and the invocation of the weighted Castelnuovo criterion] The application of the weighted Castelnuovo criterion (one of the two key external results) to the identified centers requires explicit verification that these centers are regularly embedded in the symmetric product and that their normal bundles satisfy the precise weight conditions for the blow-down to be an isomorphism outside the center. The modular interpretation via subschemes transverse to the boundary on expanded curves does not automatically guarantee these regularity hypotheses without additional local computations or checks in the degeneration.
[Main theorem and construction of the iterated weighted blow-ups] The proof that the logarithmic Hilbert scheme is recovered exactly by the iterated weighted blow-ups depends on confirming that the chosen centers and weights match the modular data of the logarithmic Hilbert scheme without post-hoc adjustments; this verification step, including absence of extra regularity conditions in the expanded degeneration, is load-bearing for the central claim.

minor comments (2)

[Introduction and key ingredients paragraph] Clarify the precise statements of the hypotheses from the two cited recent results when they are applied, to make the verification steps fully transparent.
[Preliminaries] Ensure that notation for logarithmic structures, expanded degenerations, and transversality is consistent and defined before first use in the main body.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points where additional explicit verification would strengthen the argument. We agree that the regularity hypotheses for the weighted Castelnuovo criterion and the precise matching of centers and weights to the modular data are load-bearing. In the revised version we have added local computations and a dedicated subsection clarifying these verifications. The main results and constructions remain unchanged.

read point-by-point responses

Referee: [Sections describing the blow-up centers and the invocation of the weighted Castelnuovo criterion] The application of the weighted Castelnuovo criterion (one of the two key external results) to the identified centers requires explicit verification that these centers are regularly embedded in the symmetric product and that their normal bundles satisfy the precise weight conditions for the blow-down to be an isomorphism outside the center. The modular interpretation via subschemes transverse to the boundary on expanded curves does not automatically guarantee these regularity hypotheses without additional local computations or checks in the degeneration.

Authors: We thank the referee for this observation. The modular interpretation via transverse subschemes on expanded degenerations supplies the geometric origin of the centers and weights, but we agree that it does not by itself constitute a complete local verification of regular embedding and normal-bundle weights. In the revised manuscript we have inserted explicit local computations (now in an expanded Section 3.2) that confirm the centers are regularly embedded in the symmetric product and that the normal bundles carry the precise weights required by the weighted Castelnuovo criterion. These calculations are performed in étale charts adapted to the expanded degeneration and show that the blow-down is an isomorphism away from the center, thereby justifying the direct application of the external result. revision: yes
Referee: [Main theorem and construction of the iterated weighted blow-ups] The proof that the logarithmic Hilbert scheme is recovered exactly by the iterated weighted blow-ups depends on confirming that the chosen centers and weights match the modular data of the logarithmic Hilbert scheme without post-hoc adjustments; this verification step, including absence of extra regularity conditions in the expanded degeneration, is load-bearing for the central claim.

Authors: We agree that the exact recovery of the logarithmic Hilbert scheme by the iterated weighted blow-ups is the central claim and requires a transparent verification that the centers and weights are determined directly by the modular data. The construction proceeds by identifying each successive center with the locus of subschemes that fail transversality to the boundary divisor at a given multiplicity; this identification is canonical and introduces no post-hoc adjustments. To make the absence of extra regularity conditions explicit, we have added a new subsection (Section 4.1) that records the correspondence between the blow-up data and the transversality conditions on expanded curves, confirming that the only regularity hypotheses used are those already present in the definition of the logarithmic Hilbert scheme. This subsection also verifies that the iteration terminates precisely when all points are transverse, recovering the scheme without further conditions. revision: yes

Circularity Check

0 steps flagged

No circularity: central identification derived from scheme theory plus external theorems

full rationale

The paper proves the logarithmic Hilbert scheme of points equals an iterated weighted blow-up of the symmetric product by explicitly constructing centers, weights, and modular interpretations from the definition of logarithmic subschemes transverse to the boundary on expanded degenerations. It then invokes two cited external results (integral Chow ring formula for weighted blow-ups and weighted Castelnuovo criterion) whose statements are independent of the present modular identification. No equation or step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the derivation remains self-contained against standard scheme-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The work rests on standard axioms of algebraic geometry and scheme theory together with two recent external theorems on weighted blow-ups; no free parameters, ad-hoc constants, or newly invented entities are introduced.

axioms (3)

standard math Standard axioms of schemes, blow-ups, and Chow rings in algebraic geometry
The construction and Chow-ring calculations presuppose the usual properties of schemes over a field and the existence of Chow rings for smooth stacks.
domain assumption The integral Chow ring formula for weighted blow-ups holds in the stated generality
Invoked explicitly as a key ingredient for the Chow-ring computation.
domain assumption The weighted analogue of Castelnuovo's criterion applies to the identified centers
Used to justify the blow-down steps in the iterated construction.

pith-pipeline@v0.9.0 · 5674 in / 1532 out tokens · 69017 ms · 2026-05-20T01:19:05.567506+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A: map Hilb_n(C|D) → Sym^n(C) factors through sequence of ℓ·n weighted blow-ups... centers Z_i and weights (1,...,i)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem D: Chow ring presentation CH^*(Hilb_n(C|D)≤i) = CH^*(Sym^n(C))[ϵ_j] / (Q-polynomials, ker(s^*)·ϵ_j)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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