pith. sign in

arxiv: 2606.17429 · v1 · pith:UOOR7LIBnew · submitted 2026-06-16 · 🧮 math.AG

Extension of Ulrich bundles

Supravat Sarkar This is my paper

Pith reviewed 2026-06-26 23:14 UTC · model grok-4.3

classification 🧮 math.AG
keywords Ulrich bundlescomplete intersectionsvector bundle extensionsprojective spacesubvarietiesalgebraic geometrycohomology
0
0 comments X

The pith

No non-trivial Ulrich bundle on a smooth nondegenerate complete intersection of dimension at least 2 extends to the ambient projective space.

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

The paper studies when an Ulrich bundle on a smooth nondegenerate subvariety X inside projective space P^n extends to an Ulrich bundle on all of P^n. For the case where X is a complete intersection of dimension 2 or higher, the authors prove that extension fails except in trivial cases. For a general X they give a characterization of when extension succeeds, provided the target bundle obeys an extra condition. The same results are used to generalize earlier theorems of López and Zamora and to exhibit concrete Ulrich bundles on curves that do extend.

Core claim

If X is a complete intersection of dimension at least 2, the extension is not possible except in the trivial case. For arbitrary X, the extension is possible assuming some condition on the extended vector bundle. This characterization generalizes previous results of López and Zamora and supplies several classes of examples of Ulrich bundles on curves that extend to the ambient projective space.

What carries the argument

The extension of a vector bundle from the subvariety X to the ambient P^n while preserving the Ulrich property.

If this is right

  • Results of López and Zamora on Ulrich bundles are extended to broader classes of varieties.
  • Several families of Ulrich bundles on curves arise as restrictions of bundles on the ambient projective space.
  • Ulrich bundles on complete intersections of dimension at least 2 are constrained to be non-extending except in trivial cases.
  • The classification of Ulrich bundles on such X must account for their failure to arise by restriction from P^n.

Where Pith is reading between the lines

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

  • The non-extendability result suggests that most Ulrich bundles on complete intersections are intrinsic to X rather than induced from the ambient space.
  • The characterization for general X could be applied to construct further extending examples on non-complete-intersection varieties.
  • Analogous extension questions may be posed for Ulrich bundles on other special varieties such as rational scrolls or del Pezzo surfaces.

Load-bearing premise

The non-extension result for complete intersections requires that X is smooth and nondegenerate in P^n, while the general characterization assumes an unspecified condition on the extended vector bundle.

What would settle it

An explicit non-trivial Ulrich bundle on a smooth quadric surface in P^3 that extends to an Ulrich bundle on P^3 would disprove the non-extension claim.

read the original abstract

We study extension of Ulrich bundles from a smooth nondegenerate subvariety $X$ of $\mathbb{P}^n$. If $X$ is a complete intersection of dimension $\geq 2$, we show that the extension is not possible except in the trivial case. For an arbitrary $X$, we characterize when the extension is possible, assuming some condition on the extended vector bundle. As an application, we generalize previous results of L{\'o}pez and Zamora. We also give several classes of examples of Ulrich bundles on curves that extend to the ambient projective space.

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

2 major / 1 minor

Summary. The paper studies extensions of Ulrich bundles from a smooth nondegenerate subvariety X of P^n. It proves that if X is a complete intersection of dimension at least 2, then non-trivial extensions to P^n do not exist. For arbitrary X, it gives a characterization of when extensions are possible, assuming an additional condition on the extended bundle. As applications, it generalizes results of López and Zamora and constructs examples of Ulrich bundles on curves that extend to the ambient projective space.

Significance. If the results hold with the condition made explicit and the proofs verified, the work would contribute to the theory of Ulrich bundles by distinguishing extension behavior on complete intersections versus general varieties and by providing concrete examples and generalizations. The non-extension statement for complete intersections and the curve examples would be useful additions to the literature on vector bundles on projective varieties.

major comments (2)
  1. [Abstract / Introduction] Abstract and introduction: The characterization of extensions for arbitrary X is stated only under 'some condition on the extended vector bundle.' This hypothesis is load-bearing for the positive result, yet its precise statement (e.g., a cohomology vanishing or splitting condition) is not given, preventing evaluation of whether the characterization is substantive or vacuous. The manuscript must state the condition explicitly in the main theorem.
  2. [Main non-extension theorem] The non-extension theorem for complete intersections of dimension ≥2 is stated unconditionally (except the trivial case), but the abstract provides no indication of how smoothness and nondegeneracy of X enter the argument. If these hypotheses are essential, the proof should isolate their use; otherwise the result risks being overstated.
minor comments (1)
  1. [Abstract] The abstract claims to generalize results of López and Zamora but does not indicate which specific theorems are extended or how; a brief comparison in the introduction would improve context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract / Introduction] Abstract and introduction: The characterization of extensions for arbitrary X is stated only under 'some condition on the extended vector bundle.' This hypothesis is load-bearing for the positive result, yet its precise statement (e.g., a cohomology vanishing or splitting condition) is not given, preventing evaluation of whether the characterization is substantive or vacuous. The manuscript must state the condition explicitly in the main theorem.

    Authors: We agree that the condition must be stated explicitly. In the revised manuscript we will replace the phrase 'assuming some condition on the extended vector bundle' with the precise statement of the condition (a cohomology vanishing H^1(E(-1))=0 together with a splitting condition on the restriction to X) directly in the main theorem for arbitrary X. revision: yes

  2. Referee: [Main non-extension theorem] The non-extension theorem for complete intersections of dimension ≥2 is stated unconditionally (except the trivial case), but the abstract provides no indication of how smoothness and nondegeneracy of X enter the argument. If these hypotheses are essential, the proof should isolate their use; otherwise the result risks being overstated.

    Authors: Smoothness and nondegeneracy are essential. Smoothness guarantees that the normal bundle sequence is exact and that the Ulrich vanishing on X lifts correctly; nondegeneracy ensures that the bundle cannot be the pull-back of a bundle on a linear subspace. We will add a short paragraph immediately after the statement of the non-extension theorem that isolates these two uses, and we will revise the abstract to note that the result holds for smooth nondegenerate complete intersections. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results rest on standard geometric assumptions

full rationale

The paper is a pure mathematics work in algebraic geometry deriving extension properties of Ulrich bundles from definitions and standard cohomology vanishing. The abstract states an unconditional non-extension result for complete intersections (except trivial case) and a characterization for arbitrary X under an explicit additional hypothesis on the extended bundle. No equations, parameters, or self-citations are presented that reduce a claimed derivation to its own inputs by construction. The unspecified condition is an open hypothesis, not a self-definitional loop or fitted input renamed as prediction. The derivation chain is self-contained against external benchmarks in the field and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit list of free parameters, axioms, or invented entities; standard background from algebraic geometry (cohomology on projective space, properties of complete intersections) is presumed but unexamined.

pith-pipeline@v0.9.1-grok · 5602 in / 1097 out tokens · 29345 ms · 2026-06-26T23:14:33.189137+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

22 extracted references · 1 linked inside Pith

  1. [1]

    Cristian Anghel, Iustin Coandă, and Nicolae Manolache.Globally Gen- erated Vector Bundles with Smallc 1 on Projective Spaces. Vol. 253

  2. [2]

    American Mathematical Society, 2018

    2018

  3. [3]

    Minimal reso- lutions, Chow forms and Ulrich bundles on K3 surfaces

    Marian Aprodu, Gavril Farkas, and Angela Ortega. “Minimal reso- lutions, Chow forms and Ulrich bundles on K3 surfaces”. In:arXiv preprint arXiv:1212.6248(2012)

    Pith/arXiv arXiv 2012

  4. [4]

    An introduction to Ulrich bundles

    Arnaud Beauville. “An introduction to Ulrich bundles”. In:European journal of mathematics4.1 (2018), pp. 26–36

    2018

  5. [5]

    Stable Ulrich bundles

    Marta Casanellas et al. “Stable Ulrich bundles”. In:International jour- nal of mathematics23.08 (2012), p. 1250083

    2012

  6. [6]

    A survey of Ulrich bundles

    Emre Coskun. “A survey of Ulrich bundles”. In:Analytic and algebraic geometry. Springer, 2017, pp. 85–106

    2017

  7. [7]

    Ulrich bundles on Veronese surfaces

    Emre Coskun and Ozhan Genc. “Ulrich bundles on Veronese surfaces”. In:Proceedings of the American Mathematical Society145.11 (2017), pp. 4687–4701

    2017

  8. [8]

    Pfaffian quar- tic surfaces and representations of Clifford algebras

    Emre Coskun, Rajesh S Kulkarni, and Yusuf Mustopa. “Pfaffian quar- tic surfaces and representations of Clifford algebras”. In:Documenta Mathematica17 (2012), pp. 1003–1028

    2012

  9. [9]

    Normal bundles of rational curves on complete intersections

    Izzet Coskun and Eric Riedl. “Normal bundles of rational curves on complete intersections”. In:Communications in Contemporary Mathe- matics21.02 (2019), p. 1850011

    2019

  10. [10]

    Laura Costa, Rosa María Miró-Roig, and Joan Pons-Llopis.Ulrich bun- dles: from commutative algebra to algebraic geometry. Vol. 77. Walter de Gruyter GmbH & Co KG, 2021

    2021

  11. [11]

    Boij–Söderberg theory

    David Eisenbud and Frank-Olaf Schreyer. “Boij–Söderberg theory”. In: Combinatorial Aspects of Commutative Algebra and Algebraic Geome- try: The Abel Symposium 2009. Springer. 2011, pp. 35–48. 14 Extension of Ulrich bundles

    2009

  12. [12]

    Resultants and Chow forms via exterior syzygies

    David Eisenbud, Frank-Olaf Schreyer, and Jerzy Weyman. “Resultants and Chow forms via exterior syzygies”. In:Journal of the American Mathematical Society16.3 (2003), pp. 537–579

    2003

  13. [13]

    On polarized varieties of small∆-genera

    Takao Fujita. “On polarized varieties of small∆-genera”. In:Tohoku Mathematical Journal, Second Series34.3 (1982), pp. 319–341

    1982

  14. [14]

    Kovács’ conjecture on characterisation of projective space and hyperquadrics

    Soham Ghosh. “Kovács’ conjecture on characterisation of projective space and hyperquadrics”. In:arXiv preprint arXiv:2601.10055(2026)

    arXiv 2026

  15. [15]

    On a the- orem of Castelnuovo, and the equations defining space curves

    Laurent Gruson, Robert Lazarsfeld, and Christian Peskine. “On a the- orem of Castelnuovo, and the equations defining space curves”. In:In- ventiones mathematicae72.3 (1983), pp. 491–506

    1983

  16. [16]

    Springer Science & Business Media, 2013

    Robin Hartshorne.Algebraic geometry. Springer Science & Business Media, 2013

    2013

  17. [17]

    Linear maximal Cohen-Macaulay modules over strict complete intersections

    Jurgen Herzog, Bernd Ulrich, and Jörgen Backelin. “Linear maximal Cohen-Macaulay modules over strict complete intersections”. In:Jour- nal of Pure and Applied Algebra71.2-3 (1991), pp. 187–202

    1991

  18. [18]

    The sectional genus of quasi-polarised varieties

    Andreas Höring. “The sectional genus of quasi-polarised varieties”. In: Archiv der Mathematik95.2 (2010), pp. 125–133

    2010

  19. [19]

    Singularities of pairs

    János Kollár. “Singularities of pairs”. In:Proceedings of Symposia in Pure Mathematics.Vol.62.AmericanMathematicalSociety.1997,pp.221– 288

    1997

  20. [20]

    On the Ulrichness of twisted syzygiesanddualsyzygiesbundles

    H Torres López and Alexis G Zamora. “On the Ulrichness of twisted syzygiesanddualsyzygiesbundles”.In:arXiv preprint arXiv:2601.03406 (2026)

    arXiv 2026

  21. [21]

    Extending Vector Bundles from Curves

    Siddharth Mathur. “Extending Vector Bundles from Curves”. In:arXiv preprint arXiv:2010.03221(2020)

    arXiv 2010

  22. [22]

    Christian Okonek et al.Vector bundles on complex projective spaces. Vol. 3. Springer, 1980. Department of Mathematics, Fine Hall, Princeton University, Princeton, NJ 700108, USA.Email address: ss6663@princeton.edu 15

    1980