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arxiv: 2606.18238 · v2 · pith:UF4W7TKFnew · submitted 2026-06-16 · 🧮 math.AG

Exceptional collections for canonical stacks of log del Pezzo surfaces with frac13(1,1) singularities

Alex Junior Gomez Saltachin This is my paper

Pith reviewed 2026-06-26 22:23 UTC · model grok-4.3

classification 🧮 math.AG
keywords log del Pezzo surfacesexceptional collectionsderived categoriesDeligne-Mumford stacks1/3(1,1) singularitiescanonical stacksquotient singularities
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The pith

If a complex log del Pezzo surface has only 1/3(1,1) singularities, then the derived category of its canonical smooth Deligne-Mumford stack admits a full exceptional collection.

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

The paper establishes that for log del Pezzo surfaces over the complex numbers whose singularities are exclusively of type 1/3(1,1), the bounded derived category of coherent sheaves on the associated canonical stack admits a full exceptional collection. This collection generates the category and consists of objects with vanishing higher Ext groups between them except in controlled degrees. A sympathetic reader would care because such a collection gives an explicit description of the category, allowing computations of its invariants and relations to the singular coarse space. The argument proceeds by constructing local objects at the stacky points and extending them globally.

Core claim

The central claim is that if X is a complex log del Pezzo surface with all singularities of type 1/3(1,1), then D^b(coh Χ) admits a full exceptional collection, where Χ is the canonical smooth Deligne-Mumford stack over X. The proof reduces to local computations around the stacky points together with a global generation argument. An explicit description of the local exceptional objects supported on the residual gerbes is supplied. As an application, the canonical stack of a general degree-10 hypersurface in P(1,2,3,5) carries a full exceptional collection of length 13, and the corresponding singular coarse category is discussed.

What carries the argument

The canonical smooth Deligne-Mumford stack Χ associated to X, together with the local exceptional objects supported on the residual gerbes at the stacky points of type 1/3(1,1).

If this is right

  • The derived category of the singular coarse surface X is related to that of the stack Χ via the pushforward functor.
  • The specific hypersurface X_10 in P(1,2,3,5) yields a stack whose derived category has a full exceptional collection of length 13.
  • Local exceptional objects at each stacky point can be described explicitly in terms of the residual gerbe.
  • The existence of the collection holds uniformly for all such surfaces, independent of the number of singular points.

Where Pith is reading between the lines

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

  • The same local-to-global strategy might adapt to other isolated quotient singularities once analogous local generators are identified.
  • Full exceptional collections on the stack imply that the numerical Grothendieck group is free of finite rank equal to the collection length.
  • The coarse-space category may inherit semiorthogonal decompositions or generation properties from the stack.

Load-bearing premise

The surface is defined over the complex numbers and all its singularities are exclusively of type 1/3(1,1).

What would settle it

An explicit computation, for some concrete log del Pezzo surface with only 1/3(1,1) singularities, showing that the Grothendieck group of D^b(coh Χ) has rank different from the length of any candidate exceptional collection.

read the original abstract

We study derived categories associated with log del Pezzo surfaces whose singularities are of type \(\frac{1}{3}(1,1)\). For such a surface \(X\), we consider the canonical smooth Deligne--Mumford stack \(\pi \colon \mathcal{X} \to X\) and also discuss the singular coarse surface \(X\). Our main result proves that, if \(X\) is a complex log del Pezzo surface whose singularities are all of type \(\frac{1}{3}(1,1)\), then \(\mathrm{D}^{\mathrm{b}}(\operatorname{coh} \mathcal{X})\) admits a full exceptional collection. We also give an explicit description of the local exceptional objects supported on the residual gerbes of the stacky points. As an application, we study a general degree \(10\) hypersurface \(X_{10} \subset \mathbb{P}(1,2,3,5)\), one of the sporadic Johnson--Koll\'ar examples. We show that its canonical stack \(\mathcal{X}_{10}\) has a full exceptional collection of length \(13\), and we discuss the corresponding singular coarse category.

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

0 major / 3 minor

Summary. The paper proves that if X is a complex log del Pezzo surface whose singularities are exclusively of type 1/3(1,1), then the bounded derived category of coherent sheaves on its canonical smooth Deligne-Mumford stack admits a full exceptional collection. It supplies an explicit description of the local exceptional objects supported on the residual gerbes at the stacky points and applies the result to the canonical stack of a general degree-10 hypersurface in P(1,2,3,5), obtaining a full exceptional collection of length 13; the singular coarse surface is also discussed.

Significance. If the central claim holds, the result supplies a concrete class of stacks whose derived categories are generated by exceptional collections, obtained via local computations at quotient singularities plus a global generation step. The restriction to 1/3(1,1) singularities permits an explicit, parameter-free construction that avoids additional fitting parameters. The application to the sporadic Johnson-Kollár hypersurface provides a verifiable, low-dimensional test case of length 13.

minor comments (3)
  1. [Abstract] Abstract: the sentence 'we also discuss the singular coarse surface X' is imprecise; clarify whether any statement is made about D^b(coh X) itself or only about the relation between the stack and its coarse space.
  2. The explicit local objects on residual gerbes are announced but their precise form (e.g., the rank or the Ext-vanishing conditions) should be stated in a numbered theorem or proposition for easy reference.
  3. In the application to X_10, confirm that the length-13 collection matches the expected rank coming from the Grothendieck group of the stack (or state the Euler characteristic computation that verifies this).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments were listed in the report, so we have no specific points to address point-by-point. We will incorporate any minor suggestions during revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes its main result via local computations around the residual gerbes of stacky points combined with a global generation argument on the canonical DM stack, without any self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the claim to its own inputs. The restriction to 1/3(1,1) singularities is an explicit hypothesis that enables the local-to-global passage but does not create circularity; the derivation remains self-contained and externally verifiable in the framework of derived categories of stacks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a pure-mathematics existence proof in algebraic geometry; no numerical parameters are fitted and no new entities are postulated beyond standard stack-theoretic constructions.

axioms (2)
  • domain assumption The base field is the complex numbers
    Explicitly stated in the main result as 'complex log del Pezzo surface'
  • domain assumption The canonical stack of a log del Pezzo surface with 1/3(1,1) singularities is a smooth Deligne-Mumford stack
    Invoked when the paper passes from the coarse surface X to the stack X

pith-pipeline@v0.9.1-grok · 5737 in / 1466 out tokens · 42501 ms · 2026-06-26T22:23:51.607066+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

15 extracted references · 12 canonical work pages

  1. [1]

    Nikulin.Del Pezzo and K3 Surfaces

    Valery Alexeev and Viacheslav V. Nikulin.Del Pezzo and K3 Surfaces. The Mathematical Society of Japan, 2006.isbn: 9784931469341.doi: 10 . 2969 / msjmemoirs / 015010000.url: http : / / dx . doi . org / 10 . 2969/msjmemoirs/015010000

    arXiv 2006

  2. [2]

    Coherent sheaves onPn and problems of linear algebra

    A. A. Beilinson. “Coherent sheaves onPn and problems of linear algebra”. In:Functional Analysis and Its Applications12.3 (1978), pp. 214–216.issn: 1573-8485.doi: 10.1007/bf01681436.url: http: //dx.doi.org/10.1007/BF01681436

    work page doi:10.1007/bf01681436.url: 1978

  3. [3]

    Brauer groups of singular del Pezzo surfaces

    Martin Bright. “Brauer groups of singular del Pezzo surfaces”. In: Michigan Mathematical Journal62.3 (2013).issn: 0026-2285.doi: 10.1307/mmj/1378757892.url: http://dx.doi.org/10.1307/mmj/ 1378757892

    work page doi:10.1307/mmj/1378757892.url: 2013

  4. [4]

    Del Pezzo surfaces with a single 1/k(1,1)singularity

    Daniel Cavey and Thomas Prince. “Del Pezzo surfaces with a single 1/k(1,1)singularity”. In:Journal of the Mathematical Society of Japan 72.2 (2020), pp. 465–505.doi:10.2969/jmsj/79337933.url: https: //doi.org/10.2969/jmsj/79337933

    work page doi:10.2969/jmsj/79337933.url: 2020

  5. [5]

    Del Pezzo surfaces with1 3(1,1) points

    Alessio Corti and Liana Heuberger. “Del Pezzo surfaces with1 3(1,1) points”. In:manuscripta mathematica153.1-2 (2016), pp. 71–118.issn: 1432-1785.doi: 10.1007/s00229-016-0870-y.url: http://dx.doi. org/10.1007/s00229-016-0870-y. REFERENCES 15

    work page doi:10.1007/s00229-016-0870-y.url: 2016

  6. [6]

    Exceptional sets on del Pezzo surfaces with one log- terminal singularity

    A. D. Elagin. “Exceptional sets on del Pezzo surfaces with one log- terminal singularity”. In:Mathematical Notes82.1-2 (2007), pp. 33– 46.issn: 1573-8876.doi: 10.1134/s000143460707005x.url: http: //dx.doi.org/10.1134/S000143460707005X

    work page doi:10.1134/s000143460707005x.url: 2007

  7. [7]

    Full Exceptional Collections for Anticanonical Log del Pezzo Surfaces

    Giulia Gugiatti and Franco Rota. “Full Exceptional Collections for Anticanonical Log del Pezzo Surfaces”. In:International Mathematics Research Notices2023.21 (Oct. 2023), pp. 18803–18855.issn: 1687-0247. doi: 10.1093/imrn/rnad228 .url: http://dx.doi.org/10.1093/ imrn/rnad228

    work page doi:10.1093/imrn/rnad228 2023

  8. [8]

    Exceptional sequences of invertible sheaves on rational surfaces

    Lutz Hille and Markus Perling. “Exceptional sequences of invertible sheaves on rational surfaces”. In:Compositio Mathematica147.4 (Mar. 2011),pp.1230–1280.issn:1570-5846.doi: 10.1112/s0010437x10005208. url:http://dx.doi.org/10.1112/S0010437X10005208

    work page doi:10.1112/s0010437x10005208 2011

  9. [9]

    The special McKay correspondence and exceptional collections

    Akira Ishii and Kazushi Ueda. “The special McKay correspondence and exceptional collections”. In:Tohoku Mathematical Journal67.4 (Dec. 2015).issn: 0040-8735.doi: 10.2748/tmj/1450798075 .url: http://dx.doi.org/10.2748/tmj/1450798075

    work page doi:10.2748/tmj/1450798075 2015

  10. [10]

    Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces

    Jennifer M. Johnson and János Kollár. “Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces”. In:Annales de l’Institut Fourier51.1 (2001), pp. 69–79.issn: 1777-5310.doi: 10.5802/aif.1815.url:http://dx.doi.org/10.5802/aif.1815

    work page doi:10.5802/aif.1815.url:http://dx.doi.org/10.5802/aif.1815 2001

  11. [11]

    De- rived categories of singular surfaces

    Joseph Karmazyn, Alexander Kuznetsov, and Evgeny Shinder. “De- rived categories of singular surfaces”. In:Journal of the European Mathematical Society24.2 (2021), pp. 461–526.issn: 1435-9863.doi: 10.4171/jems/1106.url:http://dx.doi.org/10.4171/JEMS/1106

    work page doi:10.4171/jems/1106.url:http://dx.doi.org/10.4171/jems/1106 2021

  12. [12]

    Fermat Curves on Weighted Projective Planes

    Jeremiah Mitchell Kermes. “Fermat Curves on Weighted Projective Planes”. NC State University Libraries Repository. PhD dissertation. North Carolina State University, May 3, 2007.url:http://www.lib. ncsu.edu/resolver/1840.16/5064

    2007

  13. [13]

    Koll´ ar and S

    Janos Kollár and Shigefumi Mori.Birational Geometry of Algebraic Varieties. Cambridge University Press, 1998.isbn: 9780511662560.doi: 10.1017/cbo9780511662560 .url: http://dx.doi.org/10.1017/ CBO9780511662560

    work page doi:10.1017/cbo9780511662560 1998

  14. [14]

    Projective Bundles, Monoidal Transformations, and Derived Categories of Coherent Sheaves

    D. O. Orlov. “Projective Bundles, Monoidal Transformations, and Derived Categories of Coherent Sheaves”. In:Russian Academy of Sciences. Izvestiya Mathematics41.1 (Feb. 1993), pp. 133–141.issn: 1064-5632.doi: 10.1070/im1993v041n01abeh002182.url: http:// dx.doi.org/10.1070/IM1993v041n01ABEH002182

    work page doi:10.1070/im1993v041n01abeh002182.url: 1993

  15. [15]

    Reflexive modules on quotient surface singularities

    Jürgen Wunram. “Reflexive modules on quotient surface singularities”. In:Mathematische Annalen279.4 (1988), pp. 583–598.doi: 10.1007/ BF01458530. Departamento de Matemática, PUC-Rio, Rua Marquês de São Vicente, 225, Rio de Janeiro, RJ 22451-900, Brazil Email address:alex.gomez@pucp.edu.pe

    1988