ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Plane cubic structure-sheaf cohomology

Example

Let $k$ be a field, let $f \in k [x_{0}, x_{1}, x_{2}]$ be a nonzero homogeneous cubic, and let $C = V_{+} (f) \subseteq P_{k}^{2}$ be the closed subscheme cut out by $f$ , with closed immersion $i : C ↪ P_{k}^{2}$ and structure sheaf $O_{C}$ (Hypersurface cohomology sequence). Then $H^{0} (C, O_{C}) ≅ k, H^{1} (C, O_{C}) ≅ k, H^{q} (C, O_{C}) = 0 (q \geq 2),$ with sheaf cohomology as in Sheaf cohomology as right derived global sections. Neither smoothness nor irreducibility nor reducedness of $C$ is required, the field $k$ is arbitrary, and the groups do not depend on $f$ beyond $f \neq 0$ .

Facts & Assumptions

Given: A field $k$ , a nonzero homogeneous cubic $f \in k [x_{0}, x_{1}, x_{2}]$ , the closed subscheme $C = V_{+} (f) \subseteq P_{k}^{2}$ with its structure sheaf $O_{C}$ , and the Axiom of Choice inherited from the cited suppliers.

[F1]

The hypersurface sequence (Hypersurface cohomology sequence): for a commutative ring $A$ with $1$ , $n \geq 0$ , a homogeneous $f$ of degree $d > 0$ and $X = V_{+} (f) \subseteq P_{A}^{n}$ with closed immersion $i$ , such that each dehomogenisation $f / x_{i}^{d}$ is a nonzerodivisor of the chart ring $B_{(x_{i})}$ , $B = A [x_{0}, \dots, x_{n}]$ (automatic for $A = k$ a field and $f \neq 0$ ), the sequence $0 \to O_{P^{n}} (- d) \overset{\cdot f}{\to} O_{P^{n}} \overset{i^{♯}}{\to} i_{*} O_{X} \to 0$ is exact, the long exact sequence of sheaf cohomology reads $\dots \to H^{q} (P^{n}, O (- d)) \to H^{q} (P^{n}, O) \to H^{q} (X, O_{X}) \to H^{q + 1} (P^{n}, O (- d)) \to \dots$ with $H^{q} (P^{n}, i_{*} O_{X}) ≅ H^{q} (X, O_{X})$ , the connecting maps give $H^{q} (X, O_{X}) ≅ H^{q + 1} (P^{n}, O (- d))$ for every $q \geq 1$ , and in degree zero the sequence $0 \to H^{0} (P^{n}, O (- d)) \to A \to H^{0} (X, O_{X}) \to H^{1} (P^{n}, O (- d)) \to 0$ is exact, with $H^{0} (P^{n}, O (- d)) = 0$ for $n \geq 1$ and $H^{1} (P^{n}, O (- d)) = 0$ for $n \neq 1$ .

[F2]

Cohomology of twists on $P^{2}$ (Cohomology of O(d) on projective space): for every commutative ring $A$ , every $n \geq 0$ and every $d \in Z$ one has $H^{q} (P_{A}^{n}, O (d)) = 0$ unless $q = 0$ or $q = n$ ; for $n = 2$ and $A = k$ a field, $H^{0} (P^{2}, O) ≅ k$ , $H^{2} (P^{2}, O (- 3))$ is free on the negative triples summing to $- 3$ , namely on the single monomial $x_{0}^{- 1} x_{1}^{- 1} x_{2}^{- 1}$ , and $H^{2} (P^{2}, O) = 0$ because $0 > - n - 1 = - 3$ .

[A1]

The Axiom of Choice (The Axiom of Choice): every family of nonempty sets has a choice function, inherited here from [F1] and [F2].

Verification

Proof technique: direct: specialise the hypersurface short exact sequence to a plane cubic and read the cohomology of the structure sheaf off the long exact sequence and the known groups of the twists on $P^{2}$ .

1.1F1

Specialising [F1] to $n = 2$ , $d = 3$ , $A = k$ and the nonzero cubic $f$ meets its hypotheses, since over the field $k$ each dehomogenisation $f / x_{i}^{3}$ is a nonzero element of the domain $B_{(x_{i})}$ and hence a nonzerodivisor; so $0 \to O_{P^{2}} (- 3) \overset{\cdot f}{\to} O_{P^{2}} \overset{i^{♯}}{\to} i_{*} O_{C} \to 0$ is exact and its long exact sequence is $\dots \to H^{q} (P^{2}, O (- 3)) \to H^{q} (P^{2}, O) \to H^{q} (C, O_{C}) \to H^{q + 1} (P^{2}, O (- 3)) \to \dots$ , with $H^{q} (P^{2}, i_{*} O_{C}) ≅ H^{q} (C, O_{C})$ .

2.1F2step 1.1

By [F2] with $n = 2$ the relevant groups are $H^{0} (P^{2}, O (- 3)) = 0$ and $H^{0} (P^{2}, O) ≅ k$ , the intermediate groups $H^{1} (P^{2}, O (- 3)) = H^{1} (P^{2}, O) = 0$ , the top groups $H^{2} (P^{2}, O (- 3)) ≅ k$ on the unique negative monomial $x_{0}^{- 1} x_{1}^{- 1} x_{2}^{- 1}$ and $H^{2} (P^{2}, O) = 0$ , and all $H^{q}$ with $q > 2$ vanish.

3.1F1step 1.1step 2.1

The degree-zero part of the sequence of step 1.1 is $0 \to H^{0} (P^{2}, O (- 3)) \to H^{0} (P^{2}, O) \to H^{0} (C, O_{C}) \to H^{1} (P^{2}, O (- 3)) \to 0$ , which by step 2.1 reads $0 \to 0 \to k \to H^{0} (C, O_{C}) \to 0$ ; exactness gives $H^{0} (C, O_{C}) ≅ k$ .

3.2F1step 1.1step 2.1

For $q = 1 \geq 1$ the connecting map of step 1.1 is an isomorphism $H^{1} (C, O_{C}) ≅ H^{2} (P^{2}, O (- 3))$ , and step 2.1 identifies the target with $k$ ; hence $H^{1} (C, O_{C}) ≅ k$ .

3.3F1step 1.1step 2.1

For every $q \geq 2$ the isomorphism of step 1.1 gives $H^{q} (C, O_{C}) ≅ H^{q + 1} (P^{2}, O (- 3))$ with $q + 1 \geq 3 > 2$ , so the group vanishes by step 2.1; in particular $H^{2} (C, O_{C}) = 0$ and all higher groups vanish.

4.1A1F1F2step 3.3∎

Boundary and degenerate cases: the field $k$ is arbitrary, including $F_{2}$ ; the only hypothesis on $f$ is $f \neq 0$ , so $C$ may be smooth, nodal, cuspidal, a union of three lines or nonreduced, and the answer is independent of the choice of nonzero cubic; the zero polynomial would give $C = P_{k}^{2}$ and is excluded; the degree $d = 3 = n + 1$ is the endpoint at which the middle group has rank $(\binom{d - 1}{n}) = (\binom{2}{2}) = 1$ , matching the single negative monomial $x_{0}^{- 1} x_{1}^{- 1} x_{2}^{- 1}$ ; degrees $q \geq 2$ are killed by the dimension bound $q + 1 > 2$ ; and no choice is made beyond the inherited Axiom of Choice [A1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

48 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Cohomology of Schemes, Chapter 30, Sections 30.2-30.22 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), Sections 19.1, 19.6, 19.9, 28.1-28.2 (standard reference, not scraped)