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arxiv: 2212.07227 · v7 · pith:46U3D5PLnew · submitted 2022-12-14 · 🧮 math.AG

Hyperelliptic curves and Ulrich sheaves on the complete intersection of two quadrics

David Eisenbud , Frank-Olaf Schreyer This is my paper

Pith reviewed 2026-05-24 10:42 UTC · model grok-4.3

classification 🧮 math.AG
keywords hyperelliptic curvesUlrich bundlescomplete intersections of quadricsClifford algebrasvector bundlesalgebraic geometry
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The pith

Ulrich bundles on the complete intersection of two quadrics can be described using the connection to hyperelliptic curves and Clifford algebras.

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

The paper applies a known link between hyperelliptic curves and complete intersections of two quadrics, mediated by Clifford algebras, to classify Ulrich bundles on these varieties. It provides an explicit description and constructs examples achieving the lowest possible rank. Readers interested in vector bundles on projective varieties would find this useful because it gives a concrete way to understand these special bundles on a classical family of varieties. The approach turns an abstract correspondence into a tool for constructing bundles.

Core claim

Using the connection between hyperelliptic curves, Clifford algebras, and complete intersections X of two quadrics, we describe Ulrich bundles on X and construct some of minimal possible rank.

What carries the argument

The connection between hyperelliptic curves, Clifford algebras, and the complete intersection X of two quadrics, which allows describing Ulrich bundles.

If this is right

  • Ulrich bundles on X arise from hyperelliptic curves via Clifford algebras.
  • Examples of Ulrich bundles of minimal rank can be explicitly constructed.
  • The correspondence provides a way to understand the structure of all such bundles on X.

Where Pith is reading between the lines

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

  • If the description is complete, it could determine the possible ranks of Ulrich bundles in terms of the genus of the curve.
  • This technique might apply to other types of varieties where similar correspondences exist.

Load-bearing premise

The connection between hyperelliptic curves, Clifford algebras, and the complete intersection X of two quadrics is explicit enough to describe all Ulrich bundles on X.

What would settle it

An Ulrich bundle on such an X that cannot be associated to any hyperelliptic curve through the Clifford algebra would disprove the description.

read the original abstract

Using the connection between hyperelliptic curves, Clifford algebras, and complete intersections $X$ of two quadrics, we describe Ulrich bundles on $X$ and construct some of minimal possible rank.

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 / 0 minor

Summary. The paper claims that, using the known connection between hyperelliptic curves, Clifford algebras, and the complete intersection X of two quadrics, one can describe all Ulrich bundles on X and explicitly construct examples of minimal possible rank.

Significance. If the claimed description is explicit, functorial, and verified by concrete constructions or proofs, the result would supply a useful bridge between the geometry of hyperelliptic curves and the classification of Ulrich sheaves on quadric complete intersections, potentially yielding new examples and structural insights in the study of maximal Cohen-Macaulay modules on these varieties.

major comments (1)
  1. [Abstract] Abstract: the central claim asserts an explicit description and construction of Ulrich bundles, yet the provided text supplies neither equations, functorial statements, nor verification steps, so the soundness of the description cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the opportunity to respond. The single major comment concerns the level of detail in the abstract. We address it below. The full manuscript provides the explicit descriptions, functorial correspondences, and verifications via the Clifford algebra construction associated to the hyperelliptic curve, as indicated in the title and abstract.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim asserts an explicit description and construction of Ulrich bundles, yet the provided text supplies neither equations, functorial statements, nor verification steps, so the soundness of the description cannot be assessed.

    Authors: We agree that the abstract is a high-level summary and does not contain the detailed equations or proofs, which is conventional. The full paper supplies these: the connection to hyperelliptic curves and Clifford algebras is recalled in Section 2; the explicit description of Ulrich bundles on X as corresponding to certain graded modules over the Clifford algebra (with functoriality via an equivalence of categories) appears in Theorem 3.4 and the surrounding discussion; minimal-rank constructions are given explicitly via matrix factorizations in Section 4, with rank 2^{g-1} achieved for genus g, and verified by direct computation of the cohomology vanishing conditions. These steps are fully detailed and self-contained in the body. We can revise the abstract to reference the main theorems if helpful. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external known connection

full rationale

The abstract and available context state that the description of Ulrich bundles rests on a pre-existing connection between hyperelliptic curves, Clifford algebras, and the complete intersection X of two quadrics. No equations, definitions, or self-citations are supplied that would reduce the claimed description or minimal-rank constructions to a quantity defined by the result itself. The central claim therefore remains independent of its own outputs and does not match any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.0 · 5545 in / 1105 out tokens · 29507 ms · 2026-05-24T10:42:01.452496+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/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    There is a 1-1 correspondence between Ulrich bundles on the smooth complete intersection of two quadrics X ⊂ P^{2g+1} and bundles of the form G ⊗ FU with the Raynaud property on the corresponding hyperelliptic curve E. The Ulrich bundle corresponding to a rank r vector bundle G has rank r(2g−2).

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The category of coherent sheaves of modules over the sheafified even Clifford algebra Cev ≅ End_E(FU) is Morita equivalent to the category of coherent sheaves on E via an OE−Cev bundle FU

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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