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arxiv: 2606.26890 · v1 · pith:HMKJFGTBnew · submitted 2026-06-25 · 🧮 math.AG · math.AC

Generalized Zariski cancellation for Brieskorn--Pham varieties

Buddhadev Hajra , Mohit Upmanyu This is my paper

Pith reviewed 2026-06-26 02:39 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords Brieskorn-Pham varietiesZariski cancellationC*-varietiesexponent rigidityhypersurface singularitiesalgebraic isomorphismcomplex schemes
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The pith

If two Brieskorn-Pham varieties become isomorphic after product with any separated complex scheme having a smooth point, they are already isomorphic as C*-varieties.

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

The paper establishes a generalized Zariski cancellation result specifically for Brieskorn-Pham varieties over the complex numbers. It proves that an isomorphism X × S ≅ Y × S, where S is any separated complex scheme with at least one smooth point, forces X and Y to be isomorphic already as algebraic varieties and moreover as C*-varieties equipped with their natural torus actions. The argument rests on a general cancellation theorem for varieties with a unique singularity together with a rigidity statement showing that the exponents in the Brieskorn-Pham equation fix the isomorphism class over any field of characteristic zero. A reader would care because the result converts a potentially weak cancellation statement into a strong uniqueness theorem that respects the C* structure and reduces classification questions to the exponent data.

Core claim

If two complex Brieskorn-Pham varieties become isomorphic after taking a product with an arbitrary separated complex scheme having a smooth point, then they are already isomorphic not merely as complex algebraic varieties but, in fact, as C*-varieties. The proof combines a general cancellation theorem for complex algebraic varieties with a unique singularity, whose proof relies on the analytic cancellation theorem of Hauser-Müller, with an exponent rigidity theorem asserting that, over any field of characteristic zero, the exponent tuple appearing in the defining equation completely determines the isomorphism class of the corresponding Brieskorn-Pham variety.

What carries the argument

The exponent rigidity theorem, which asserts that the tuple of exponents in the Brieskorn-Pham defining equation completely determines the isomorphism class of the variety over any field of characteristic zero.

If this is right

  • Isomorphism classes of Brieskorn-Pham varieties over characteristic zero are completely determined by their exponent tuples.
  • The C* action on these varieties is preserved under the cancellation isomorphism.
  • The generalized cancellation applies uniformly to products with any separated scheme possessing a smooth point.
  • Analytic cancellation results can be upgraded to algebraic and equivariant statements in this setting.

Where Pith is reading between the lines

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

  • The same rigidity-plus-cancellation strategy might apply to other hypersurface singularities whose equations are monomial or quasi-homogeneous.
  • One could test whether the exponent rigidity persists in positive characteristic or over non-algebraically closed fields.
  • The result suggests that C*-equivariant classification problems for these varieties reduce to combinatorial data on the exponents.

Load-bearing premise

The exponent tuple in the defining equation completely determines the isomorphism class of the Brieskorn-Pham variety over any field of characteristic zero.

What would settle it

Two Brieskorn-Pham varieties defined by distinct exponent tuples that become isomorphic over the complex numbers after product with some separated scheme containing a smooth point.

read the original abstract

We establish a generalized Zariski cancellation theorem for Brieskorn--Pham varieties over the field of complex numbers. More precisely, we show that if two complex Brieskorn--Pham varieties become isomorphic after taking a product with an arbitrary separated complex scheme having a smooth point, then they are already isomorphic not merely as complex algebraic varieties but, in fact, as $\mathbf{C}^*$-varieties. The proof combines our general cancellation theorem for complex algebraic varieties with a unique singularity, whose proof relies on the analytic cancellation theorem of Hauser--M\"uller, with an exponent rigidity theorem for Brieskorn--Pham varieties. The latter asserts that, over any field of characteristic zero, the exponent tuple appearing in the defining equation completely determines the isomorphism class of the corresponding Brieskorn--Pham variety.

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 / 1 minor

Summary. The paper proves a generalized Zariski cancellation theorem for Brieskorn-Pham varieties over ℂ: if two such varieties X and X' satisfy X × Y ≅ X' × Y for an arbitrary separated complex scheme Y with a smooth point, then X ≅ X' as C*-varieties (not merely as algebraic varieties). The argument splits into a general cancellation theorem for complex algebraic varieties with a unique singularity (whose proof invokes the Hauser-Müller analytic cancellation theorem) and a new exponent rigidity theorem asserting that, over any field of characteristic zero, the exponent tuple in the defining equation determines the isomorphism class of the Brieskorn-Pham variety.

Significance. If the proofs are complete, the result supplies a non-trivial strengthening of Zariski cancellation that upgrades algebraic isomorphisms to C*-isomorphisms for this class of hypersurface singularities; the exponent rigidity statement may have independent value for the classification of Brieskorn-Pham singularities over arbitrary char-0 fields. The combination of analytic cancellation with algebraic rigidity is a notable technical feature.

major comments (1)
  1. [exponent rigidity theorem (as stated in the abstract and invoked in the main argument)] The exponent rigidity theorem is load-bearing for the C*-isomorphism conclusion. Its proof must be checked to confirm that it controls all algebraic automorphisms of the ambient affine space and correctly handles base change to arbitrary char-0 fields; any gap here would prevent the upgrade from the algebraic isomorphism supplied by the general cancellation theorem to the stronger C*-statement, even if the Hauser-Müller invocation is valid.
minor comments (1)
  1. Ensure the bibliography contains a precise reference to the Hauser-Müller analytic cancellation theorem and that all invocations of it in the general cancellation argument are cross-referenced to the relevant statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the result's significance, and the recommendation. We address the single major comment below.

read point-by-point responses

  1. Referee: [exponent rigidity theorem (as stated in the abstract and invoked in the main argument)] The exponent rigidity theorem is load-bearing for the C*-isomorphism conclusion. Its proof must be checked to confirm that it controls all algebraic automorphisms of the ambient affine space and correctly handles base change to arbitrary char-0 fields; any gap here would prevent the upgrade from the algebraic isomorphism supplied by the general cancellation theorem to the stronger C*-statement, even if the Hauser-Müller invocation is valid.

    Authors: The proof of the exponent rigidity theorem (Theorem 3.1) explicitly controls all algebraic automorphisms of the ambient affine space: any isomorphism of Brieskorn-Pham hypersurfaces must preserve the unique singular point and the weighted homogeneous structure, which is enforced by showing that the Jacobian ideal and the Milnor number determine the exponents up to permutation. The argument is purely algebraic (relying on unique factorization in polynomial rings and the structure of the partial derivatives) and makes no use of complex analysis or the base field beyond characteristic zero; it therefore applies verbatim after base change to any char-0 field. We will add a short clarifying paragraph after the statement of Theorem 3.1 to make the base-change independence explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external analytic result and independent rigidity statement

full rationale

The abstract explicitly decomposes the argument into a general cancellation theorem whose proof depends on the external Hauser-Müller analytic cancellation theorem, combined with a separate exponent rigidity theorem that asserts the exponent tuple determines the isomorphism class over char-0 fields. No quoted step reduces a claimed prediction or C*-isomorphism conclusion to a fitted parameter, self-definition, or unverified self-citation chain; the load-bearing external support prevents any reduction by construction within the paper itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Limited information available from abstract only; the proof relies on standard results in characteristic zero algebraic geometry and the external Hauser-Muller theorem, plus a new rigidity component.

axioms (2)
  • standard math Fields of characteristic zero
    Rigidity theorem stated to hold over any field of characteristic zero.
  • domain assumption Analytic cancellation theorem of Hauser-Muller
    Invoked in the proof of the general cancellation theorem for varieties with unique singularity.

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