TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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Proper flat genus-one fibrations have one-dimensional fibres

Statement

Assume full AC. Let $B$ be a connected integral base and let $π : Y \to B$ be a proper, flat, finitely presented morphism whose geometric generic fiber is a smooth geometrically integral genus-one curve. For $b \in B$ , write $Y_{b} = Y \times_{B} Spec κ (b)$ for the scheme-theoretic fiber. Then every $Y_{b}$ has Krull dimension one.

Statement references

Flat genus-one fibration

Fibre dimension of proper flat finitely presented families

Facts & Assumptions

Given: Full AC, connected $B$ , and a proper flat finitely presented morphism $π : Y \to B$ with geometric generic fiber a smooth geometrically integral genus-one curve.

[A1]

Full AC is the choice-function principle of The Axiom of Choice; the local-constancy supplier below assumes it.

[F1]

For a proper flat finitely presented morphism, the function $b \mapsto \dim Y_{b}$ is locally constant, with empty fibers assigned dimension $- \infty$ (Fibre dimension of proper flat finitely presented families).

[F2]

The generic fiber and genus-one condition have the meanings fixed in Flat genus-one fibration: after algebraic closure of $K (B)$ it is a smooth proper geometrically integral curve with $\dim H^{1} (Y_{\overset{ˉ}{η}}, O) = 1$ .

Proof

technique · reduce to the locally constant fiber-dimension function

1.1A1F1F2given

By [F2], the geometric generic fiber is a curve, hence has Krull dimension one. In particular the fiber-dimension function of [F1] takes value one at the generic point $η \in B$ .

2.1F1step 1.1given∎

By [F1], the fiber-dimension function is locally constant on $B$ . A locally constant function on a connected topological space is constant: the inverse image of each value is open, and its complement is the union of the other open level sets, hence also open; a nonempty level set is therefore all of $B$ . Since the value at $η$ is one by step 1.1, every fiber has dimension one, including that no fiber is empty.

Proper flat genus-one fibrations have one-dimensional fibres

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources