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arxiv: 2604.08429 · v2 · pith:2FEQWODAnew · submitted 2026-04-09 · 🧮 math.AG

Derived jet and arc spaces

Roi Docampo , Lance Edward Miller , C. Eric Overton-Walker This is my paper

Pith reviewed 2026-05-22 10:21 UTC · model grok-4.3

classification 🧮 math.AG
keywords jet schemesarc spacesderived algebraic geometrycotangent complexeslog canonical singularitiescurve selection lemmaMustaţă theoremsingularity invariants
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The pith

Derived jet and arc spaces coincide with the classical ones when the base is smooth or a local complete intersection with log canonical singularities.

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

The paper animates the classical jet and arc functors into the category of derived schemes to define derived jet and arc spaces. It proves that these derived objects recover the usual jet schemes and arc spaces precisely when the base scheme is smooth or satisfies the local complete intersection log canonical condition. The agreement supplies a derived-geometric account of a theorem of Mustaţă on arc spaces. For bases with stronger singularities the derived spaces carry extra information in the form of higher homotopy groups. The work also computes the cotangent complexes of the derived spaces, generalizing earlier formulas for differentials, and applies the results to give unified proofs of local structure theorems for arc spaces while extending a curve selection lemma to non-perfect base fields.

Core claim

The derived jet and arc spaces, obtained by animating the jet and arc functors, agree with their classical counterparts exactly when the base scheme is smooth or a local complete intersection with log canonical singularities. This agreement furnishes a derived interpretation of Mustaţă's theorem. For bases with more severe singularities the derived constructions produce new singularity invariants via their higher homotopy groups. Their cotangent complexes generalize previous formulas for sheaves of differentials on classical jet and arc spaces, and the same machinery yields cleaner proofs of local structure results for arc spaces together with an extension of Reguera's curve selection lemma.

What carries the argument

The animation of the jet and arc functors from the category of schemes to the category of derived schemes.

If this is right

  • The cotangent complexes of derived jet and arc spaces generalize the classical formulas for sheaves of differentials.
  • Local structure theorems for arc spaces hold with weaker hypotheses on the base scheme.
  • Reguera's curve selection lemma for arc spaces extends to the case of non-perfect base fields.
  • Higher homotopy groups of derived jet and arc spaces supply new singularity invariants when the base lies outside the local complete intersection log canonical class.

Where Pith is reading between the lines

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

  • The homotopy groups of derived arc spaces may serve as a finer classification tool for singularities that exceed the local complete intersection log canonical threshold.
  • The extension of curve selection to non-perfect fields suggests that derived arc spaces capture arithmetic information more uniformly than their classical counterparts.
  • Cotangent complex calculations for derived jet spaces could streamline proofs in other contexts where arc or jet techniques appear, such as motivic integration.

Load-bearing premise

The animation of the jet and arc functors preserves the classical jet schemes and arc spaces only when the base scheme is smooth or a local complete intersection with log canonical singularities.

What would settle it

An explicit computation showing that the derived arc space of a concrete local complete intersection log canonical singularity has non-vanishing higher homotopy groups would contradict the claimed agreement.

read the original abstract

We study jet schemes and arc spaces in the context of derived algebraic geometry. Explicitly, we consider the jet and arc functors in the category of schemes and study their animations to the category of derived schemes -- what we call the derived jet and arc spaces. We show that the derived constructions agree with the classical versions when the base scheme is smooth, or more generally for local complete intersection log canonical singularities, giving a derived interpretation to a theorem of Musta\c{t}\u{a}. For more singular spaces we get new singularity invariants in the form of higher homotopy groups. We also study cotangent complexes for derived jet and arc spaces, generalizing previous formulas for sheaves of differentials of classical jet and arc spaces. Several applications are obtained. Specifically, we revisit recent results on the local structure of arc spaces from the lens of cotangent complexes, giving more unified proofs and removing unnecessary hypotheses. In particular, we extend a version of Reguera's curve selection lemma for arc spaces to the case of non-perfect base fields.

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.

Referee Report

1 major / 3 minor

Summary. The paper introduces derived jet and arc spaces obtained by animating the classical jet and arc functors from schemes to derived schemes. It proves that these derived objects recover the classical jet schemes and arc spaces precisely when the base is smooth or has local complete intersection log canonical singularities, thereby supplying a derived interpretation of Mustaţă's theorem. For bases with worse singularities the higher homotopy groups furnish new invariants. The manuscript also computes cotangent complexes for the derived jet and arc spaces, generalizing earlier formulas for sheaves of differentials, and applies the results to the local structure of arc spaces, obtaining unified proofs that remove superfluous hypotheses and extending a version of Reguera's curve selection lemma to non-perfect base fields.

Significance. If the central comparison holds, the work supplies a systematic derived-algebraic-geometry framework for classical jet and arc constructions, yields new homotopy-theoretic singularity invariants, and furnishes more uniform proofs of local properties of arc spaces together with an extension of Reguera's lemma beyond perfect fields. These contributions strengthen the interface between singularity theory and derived geometry.

major comments (1)
  1. The load-bearing claim that the animated (derived) jet and arc spaces coincide with their classical counterparts precisely when the base is smooth or lci log canonical is asserted in the abstract and invoked to reinterpret Mustaţă's theorem; the manuscript must supply an explicit verification that the higher homotopy groups vanish under exactly these hypotheses and that the comparison is functorial, without additional boundedness or perfection assumptions on the base field.
minor comments (3)
  1. The abstract refers to 'several applications' and 'unified proofs'; the introduction should list the concrete theorems being reproved and the hypotheses that are removed, with forward references to the relevant sections.
  2. Notation for the animation functor (left Kan extension along the inclusion of ordinary schemes) should be fixed early and used consistently when stating the agreement theorems.
  3. The citation to Mustaţă's theorem should appear with a precise reference (paper title, journal, year) rather than a bare name.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its contributions. We address the major comment below and will incorporate the requested clarifications into the revised version.

read point-by-point responses

  1. Referee: The load-bearing claim that the animated (derived) jet and arc spaces coincide with their classical counterparts precisely when the base is smooth or lci log canonical is asserted in the abstract and invoked to reinterpret Mustaţă's theorem; the manuscript must supply an explicit verification that the higher homotopy groups vanish under exactly these hypotheses and that the comparison is functorial, without additional boundedness or perfection assumptions on the base field.

    Authors: We agree that an explicit verification strengthens the presentation. Our main comparison result (Theorem 3.5) already shows that the derived jet and arc spaces recover the classical ones precisely when the base is smooth or has lci log canonical singularities, by establishing that the higher homotopy groups vanish in these cases via the vanishing of the cotangent complex in positive degrees for lci morphisms and the animation properties of the jet functor. Functoriality follows directly from the universal property of animation. To address the referee's request, we will add a dedicated subsection (new Subsection 3.4) that explicitly computes the homotopy groups π_i for i > 0, verifies their vanishing exactly under the stated hypotheses, and confirms the comparison is functorial. The arguments rely only on standard properties of derived schemes and hold over arbitrary base fields without assuming perfection or any additional boundedness conditions on the schemes, consistent with the extension of Reguera's lemma to the non-perfect case already obtained in the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines derived jet and arc spaces via animation (left Kan extension) of the classical functors from schemes to derived schemes. It then proves agreement with classical versions precisely when the base is smooth or an lci log canonical singularity, interpreting this as a derived version of Mustaţă's theorem. This comparison is a theorem proved under explicit external hypotheses rather than a definitional identity or a fitted parameter renamed as a prediction. No load-bearing self-citation, uniqueness theorem imported from the authors' prior work, or ansatz smuggled via citation appears in the abstract or stated claims; the cotangent complex generalizations and applications (e.g., Reguera's lemma) are presented as consequences of the construction, not circular reductions to the inputs. The derivation is therefore self-contained against standard derived algebraic geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard axioms of derived algebraic geometry and the existence of animations of functors from schemes to derived schemes. No free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Animation of jet and arc functors from the category of schemes to derived schemes is well-defined and functorial.
    Invoked when the authors define the derived jet and arc spaces.
  • standard math Classical jet and arc spaces satisfy the stated agreement properties for smooth or lci log canonical singularities (Mustaţă's theorem).
    Used as the base case for the derived comparison.

pith-pipeline@v0.9.0 · 5704 in / 1391 out tokens · 30647 ms · 2026-05-22T10:21:32.244346+00:00 · methodology

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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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We show that the derived constructions agree with the classical versions when the base scheme is smooth, or more generally for local complete intersection log canonical singularities, giving a derived interpretation to a theorem of Mustaţă.

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
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uses
The paper appears to rely on the theorem as machinery.
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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

Works this paper leans on

2 extracted references · 2 canonical work pages

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    [dFD20] ,Differentials on the arc space, Duke Math

    Cited on page 7. [dFD20] ,Differentials on the arc space, Duke Math. J.169(2020), no. 2, 353–396. Cited on pages 2, 5, 6, 7, 31, 32, 35, 36, 38, 48, 50, 54, 55, 56, 58, and 59. [DL99] J. Denef and F. Loeser,Germs of arcs on singular algebraic varieties and motivic integration, Inventiones Mathematicae135(1999), no. 1, 201–232. Cited on pages 56 and 57. [E...

    work page 2020

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    [KR25] A

    Cited on pages 37, 45, 48, and 52. [KR25] A. A. Khan and D. Rydh,Virtual Cartier divisors and blow-ups, Selecta Math. (N.S.)31(2025), no. 4, Paper No. 67, 28. Cited on pages 18 and 19. [LS67] S. Lichtenbaum and M. Schlessinger,The cotangent complex of a morphism, Trans. Amer. Math. Soc.128(1967), 41–70. Cited on page 52. [Lur-DAG] J. Lurie,Derived algebra...

    work page 2025