Published paper

Sur les critères d'équivalence en géométrie algébrique

André Weil

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Publication metadata

Journal citation: Mathematische Annalen 128, 95–127 (1954)
Publication date: 1954-12-01
DOI: 10.1007/BF01360127 ↗

Abstract

The article develops equivalence criteria for algebraic cycles. The selected lemmas make bad hyperplane-section loci algebraic and proper and provide the specialization input used directly in Weil's transformation-group construction.

Role in dependence graphs

Proof-critical source

The Dirichlet spectrum

This paper is included only for the following marked statements:

Section I.6 Chow-locus construction · exact GDZ scan physical PDF pp. 8–9 / printed pp. 101–102Shows that reducible and multiple-component cycle loci are closed and defined over the prime field.
Lemma 4 · exact GDZ scan physical PDF p. 9 / printed p. 102Makes the bad-hyperplane locus algebraic, descended to every field of definition, and of dimension strictly below the dual projective space.
Lemma 7 · exact GDZ scan printed pp. 104–105Supplies the specialization/open-set step directly imported by Weil 1955 Appendix Proposition 7.

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Exact GDZ digitized article scan · Mathematische Annalen 128 (1954), 95–127

André Weil. Sur les critères d'équivalence en géométrie algébrique. Mathematische Annalen 128 (1954), 95–127.

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Generated August 27, 2026

01Statements6 reported findingsNo statement issues found

Section I correctly supplies both selected consumer results: no. 6 identifies the closed Chow-coordinate and bad-hyperplane loci used by Lang–Weil, and no. 8, Lemma 7, proves the closed fiber-dimension locus used directly by Weil 1955. No incorrect selected statement was found.

Chow-coordinate lociNo issue found

Non-varieties and cycles with repeated components form closed loci

Exact GDZ article scan · physical PDF pp. 8–9 / printed pp. 101–102 · Section I, no. 6

For fixed ambient dimension, cycle dimension, and degree, associated forms embed positive cycles into a projective parameter space. Factorization into two positive-degree forms, and occurrence of a repeated factor, are projective algebraic conditions over the prime field; the finite union over degree splittings is closed.

Lemma 4No issue found

Bad hyperplane sections form algebraic loci with the stated dimension bounds

Exact GDZ article scan · physical PDF p. 9 / printed p. 102 · Lemma 4

If $V^n\subset P^N$ is not contained in a hyperplane, the locus $R$ of $w$ for which $V\cdot L_w$ is not a variety is closed and normally algebraic over every field of definition of $V$ , with component dimensions at most $N-n+1$ . The repeated-component locus has component dimensions at most $N-n$ .

Properness in Lang–WeilDependency match verified

The dimension bound makes the imported locus proper

Lang–Weil exact volume scan · physical PDF p. 824 / printed p. 820 · setup before Lemma 2

Lang–Weil applies the result when the variety dimension is $r\ge 2$ . Lemma 4 then gives $\dim R\le N-r+1\le N-1$ , so $R$ cannot be the entire dual projective space.

Prime-field equationsNo issue found

The universal bad-cycle equations depend only on the fixed numerical data

Exact GDZ article scan · physical PDF p. 9 / printed p. 102; Lang–Weil printed p. 821

The Chow loci are constructed over the prime field for fixed dimension and degree. Consequently a finite homogeneous equation system can be chosen independently of the particular $V$ , exactly as Lang–Weil requires for its uniform degree bound.

Lemma 7No issue found

Fiber-component dimension jumps form a closed locus

Exact GDZ article scan · physical PDF pp. 11–12 / printed pp. 104–105 · Lemma 7

For closed $B\subset V\times W$ with $W$ complete, the base points $M$ for which $B\cap(\{M\}\times W)$ has a component of dimension at least $r$ form a closed algebraic subset $B'_r$ over the stated field. If an $m$ -dimensional $Z'$ lies in $B'_r$ , a subvariety of $B$ of dimension at least $m+r$ projects onto $Z'$ .

Weil 1955 dependency matchDependency match verified

Appendix Proposition 7 is the threshold case of Lemma 7

Weil 1955 exact issue scan · proof of Proposition 3 on printed pp. 363–364 ('k-open by Prop. 7' on p. 364); physical PDF pp. 188–189 / printed pp. 380–381, Appendix Proposition 7, explicit Lemma 7 citation on p. 381

After completing a representative of the fiber variety, Weil takes $r=\dim V_{\mathrm{fiber}}$ . A fiber has a component of that dimension exactly when it contains the whole irreducible complete fiber, so Lemma 7 gives the $k$ -closed full-fiber containment locus asserted in Appendix Proposition 7.

02Proofs6 reported findingsNo proof issues found

The selected Chow-locus, hyperplane-section, and fiber-dimension proofs are correct and complete. The substitution map proves Lemma 4; the polynomial-coefficient induction proves Lemma 7. All external proof imports terminate in published books; older article references are historical attribution for a proof reproduced in full.

Chow-form constructionNo issue found in the reviewed proof

Projective factorization loci give the required closed subsets

Exact GDZ article scan · physical PDF pp. 8–9 / printed pp. 101–102 · first two paragraphs of no. 6

For each possible degree split, multiplication of associated forms is a morphism between projective parameter spaces and therefore has closed image. A finite union gives the reducible-cycle locus; imposing a repeated factor similarly gives the multiple-component locus. This also makes the prime-field descent transparent.

Pullback to hyperplanesNo issue found in the reviewed proof

Substituting the hyperplane coordinates is a global regular map

Exact GDZ article scan · physical PDF p. 9 / printed p. 102 · proof of Lemma 4

The associated form of $V\cdot L_w$ is obtained from the Chow form of $V$ by inserting $w$ in one variable block. Because $V$ lies in no hyperplane, this form is never identically zero, so its coefficients define a regular projective map for every $w$ . Preimages of the two closed Chow loci are therefore closed and inherit the stated fields of definition.

Lemma 3No issue found in the reviewed proof

Generic points of a bad-locus component force the dimension bounds

Exact GDZ article scan · physical PDF pp. 7–8 / printed pp. 100–101 · Lemma 3 and proof

Weil intersects once more with a hyperplane generic over $k(w)$ and applies induction to the resulting section. The projection argument proves that the induced parameter point has the same transcendence degree $m$ as $w$ . Thus $m>N-n+1$ forces irreducibility, and $m>N-n$ rules out repeated components, yielding the contraposed bounds in Lemma 4.

Lemma 1No issue found in the reviewed proof

The generic hyperplane irreducibility input is proved in the article

Exact GDZ article scan · physical PDF pp. 4–6 / printed pp. 97–99 · Lemma 1 and complete proof

The proof first excludes tangency at every simple point by a transcendence-degree count, then proves the generic section has a single component by showing the base field is algebraically closed in its function field. The separability and intermediate-field steps use explicitly cited results from Weil's Foundations and Bourbaki's Algèbre.

Lemma 7 proofNo issue found in the reviewed proof

Polynomial coefficients detect every fiber-dimension jump

Exact GDZ article scan · physical PDF pp. 11–12 / printed pp. 104–105 · complete Lemma 7 proof

For one irreducible component with generic fiber dimension $m$ , Weil chooses, for every $m+1$ fiber-coordinate tuple, a polynomial relation having a coefficient nonzero at the generic base point. If a specialized fiber has dimension at least $m+1$ , algebraic independence forces every coefficient of the relevant relation to vanish. Their finite product defines a proper closed exceptional set, and induction on the projection dimension supplies closedness and the lifted component of dimension at least $m+r$ .

Dependency closureClosed at published-book terminals

No unresolved non-book proof branch remains

Exact article printed pp. 97–105 and source-provenance record

The proof-critical imported results terminate in Weil's 1946 Foundations of Algebraic Geometry and Bourbaki's Algèbre, Chapter V. The footnotes naming Zariski, Matsusaka, and Nakai describe historical sources for Lemma 1, but Weil states and supplies the complete proof used here; they are not missing logical steps.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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