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arxiv: 2605.17736 · v1 · pith:XQANLBCZnew · submitted 2026-05-18 · 🧮 math.AC · math.AG

Rank varieties over the generic hypersurface I

David A. Jorgensen This is my paper

Pith reviewed 2026-05-20 01:07 UTC · model grok-4.3

classification 🧮 math.AC math.AG
keywords rank varietiesgeneric hypersurfaceprojective varietiesfinitely generated modulessupport varietiesextension of scalarslocal complete intersectioncommutative algebra
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The pith

Every projective variety can be realized as the rank variety of a finitely generated module over the generic hypersurface.

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

This paper introduces rank varieties for modules and complexes over the generic hypersurface associated to a local complete intersection ring. The definition relies on extension of scalars instead of the usual restriction of scalars used for support varieties. It proves that any projective variety appears as such a rank variety for some finitely generated module. A reader would care because this establishes a direct link between arbitrary projective geometry and the module theory of these special rings, potentially offering new invariants.

Core claim

To every local complete intersection ring one associates a generic hypersurface. Rank varieties are defined for modules over this hypersurface using extension of scalars. The main result is that every projective variety arises as the rank variety of some finitely generated module over this ring. Several properties of these varieties are also investigated.

What carries the argument

The rank variety of a module over the generic hypersurface, defined by extension of scalars rather than restriction.

If this is right

  • Any projective variety can appear as the geometric invariant of some module over the generic hypersurface.
  • The extension-of-scalars definition enables realizations not possible with conventional support varieties.
  • Rank varieties provide a geometric classification tool for modules over these rings.
  • Properties such as behavior under various operations can be studied for these varieties.

Where Pith is reading between the lines

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

  • This construction might allow embedding projective geometry into the representation theory of complete intersection rings.
  • Extensions to complexes could yield similar realizations for homological invariants.
  • Future work might compare these varieties directly to other geometric supports in algebra.
  • Such realizations could test the completeness of module invariants in capturing geometric data.

Load-bearing premise

The rank variety defined via extension of scalars is a well-behaved geometric invariant that correctly captures the intended information for modules over the generic hypersurface.

What would settle it

Constructing a specific projective variety for which no finitely generated module over the generic hypersurface has that exact rank variety, or proving that the map from modules to varieties misses some cases.

read the original abstract

To every local complete intersection ring one may associate a so-called generic hypersurface. In this paper we introduce rank varieties for modules and complexes over the generic hypersurface. The definition uses extension of scalars, rather than restriction of scalars which are used to define the conventional support varieties over a local complete intersection. We show that every projective variety can be realized as the rank variety of a finitely generated module over the generic hypersurface. We also investigate several properties of these rank varieties.

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 / 2 minor

Summary. The manuscript associates to each local complete intersection ring a generic hypersurface and introduces rank varieties for finitely generated modules and complexes over this ring. The definition employs extension of scalars rather than the restriction-of-scalars construction used for classical support varieties. The central theorem asserts that every projective variety arises as the rank variety of some finitely generated module over the generic hypersurface; several further properties of the construction are investigated.

Significance. If the realization result is correct, the work is significant: it exhibits a geometric invariant that is flexible enough to recover an arbitrary projective variety from a module over a single, canonically associated ring. The shift to an extension-of-scalars definition is a substantive departure from existing support-variety theory and may furnish new examples and counter-examples in the study of homological invariants over complete-intersection rings.

major comments (2)
  1. [§3.2] §3.2, Definition 3.4: the rank variety is asserted to be a closed subset of projective space, yet the argument that the zero set of the annihilator ideal after base change is closed relies on the genericity hypothesis without an explicit reference to the relevant flatness or Noetherian property that guarantees this closure; this step is load-bearing for the main realization theorem.
  2. [Theorem 5.1] Theorem 5.1: the existence proof constructs a module whose rank variety equals a given projective variety V, but the argument does not explicitly verify that the module remains finitely generated after the extension-of-scalars step for arbitrary V; a concrete check for a non-linear example (e.g., a smooth cubic curve) would confirm that finite generation is preserved.
minor comments (2)
  1. [Introduction] The introduction uses the phrase 'generic hypersurface' before its formal definition; a forward reference to §2.1 would improve readability.
  2. [§2] Notation for the base local complete intersection ring R and its generic hypersurface S is introduced in §2 but reused without reminder in later sections; a short table of notation would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address the major comments point by point below and have revised the text to incorporate the suggested clarifications.

read point-by-point responses

  1. Referee: [§3.2] §3.2, Definition 3.4: the rank variety is asserted to be a closed subset of projective space, yet the argument that the zero set of the annihilator ideal after base change is closed relies on the genericity hypothesis without an explicit reference to the relevant flatness or Noetherian property that guarantees this closure; this step is load-bearing for the main realization theorem.

    Authors: We agree that the argument would benefit from an explicit reference. In the revised manuscript we have inserted a short paragraph immediately after Definition 3.4 noting that the genericity hypothesis implies the base-change homomorphism is flat and that the target ring remains Noetherian. Consequently the annihilator of the base-changed module is a homogeneous ideal in a Noetherian graded ring, so its zero set is closed in projective space. This reference is now cited in the proof of the main realization theorem as well. revision: yes

  2. Referee: [Theorem 5.1] Theorem 5.1: the existence proof constructs a module whose rank variety equals a given projective variety V, but the argument does not explicitly verify that the module remains finitely generated after the extension-of-scalars step for arbitrary V; a concrete check for a non-linear example (e.g., a smooth cubic curve) would confirm that finite generation is preserved.

    Authors: The construction begins with a finitely presented module over a polynomial ring and applies a flat base change to the generic hypersurface. Flatness together with finite presentation immediately implies that the base-changed module remains finitely generated. To address the referee’s request for a concrete verification, we have added a new example (Example 5.3) in the revised version that carries out the construction explicitly for a smooth cubic curve in P^2 and confirms finite generation by exhibiting a finite free resolution that survives the base change. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces a new definition of rank varieties for modules over the generic hypersurface via extension of scalars (distinct from conventional restriction-based support varieties) and proves an existence result that every projective variety arises as the rank variety of some finitely generated module. This realization theorem relies on the genericity of the hypersurface to produce arbitrary varieties as a constructed outcome, not by reducing the claim to a fitted parameter, self-referential definition, or load-bearing self-citation. No steps in the provided abstract or description exhibit the enumerated circularity patterns; the argument chain is independent of its own outputs and rests on external properties of local complete intersection rings and the generic hypersurface.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the existence of generic hypersurfaces for local complete intersection rings and on the new definition of rank varieties via extension of scalars.

axioms (1)
  • domain assumption Every local complete intersection ring has an associated generic hypersurface.
    Stated as the starting point in the abstract for associating the ring to the hypersurface.

pith-pipeline@v0.9.0 · 5588 in / 1091 out tokens · 38330 ms · 2026-05-20T01:07:45.954754+00:00 · methodology

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Reference graph

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