Flat genus-one fibration

Let $B$ be a connected nonsingular algebraic variety over $\mathbb{C}$, and let $Y$ be a projective variety over $\mathbb{C}$. Here a variety is an integral separated scheme of finite type over $\mathbb{C}$; a curve has dimension one; geometrically integral means integral after extension to an algebraic closure of the ground field; nonsingular means smooth over $\mathbb{C}$; projective means admitting a closed immersion into some ${\mathbb{P}}_{\mathbb{C}}^N$. A morphism $\pi :Y\to B$ is proper if it is separated, of finite type, and universally closed; it is surjective if its image is all of $B$; and it is flat when every local ring ${\mathcal{O}}_{Y,y}$ is a flat module over ${\mathcal{O}}_{B,\pi (y)}$. Write $\eta$ for the generic point of the integral scheme $B$, $K(B)=\kappa (\eta )$ for its function field, and $Y_{\eta }=Y{\times }_B\mathrm{Spec}K(B)$ for the generic fiber. Its geometric generic fiber is $Y_{\eta }{\times }_{K(B)}\overline{K(B)}$. A smooth proper geometrically integral curve over a field is genus one when its first coherent cohomology of the structure sheaf has dimension one. A flat genus-one fibration is a proper, flat, surjective morphism $\pi :Y\to B$ whose geometric generic fiber is a smooth geometrically integral genus-one curve. It has no section as part of the definition. In the fourfold application, impose $\dim B=3$ and $\dim Y=4$.

Flat morphism of schemes

Proper morphisms: proper means separated, of finite type, and universally closed.

Schemes and morphisms over a base

Smooth morphism of schemes: smooth means locally of finite presentation and flat, with geometrically regular fibres at every source point.

T. Weigand, TASI lectures, §§2.2 and 3.1

V. Braun and D. R. Morrison, §§1–2

For a proper flat morphism of finite presentation over a connected base, the dimension of the fibers is locally constant; Proper flat genus-one fibrations have one-dimensional fibres proves that the generic curve hypothesis then forces every fiber to have dimension one. The definition itself does not infer fiberwise smoothness or a section from the generic-fiber condition.

The geometric generic fiber is a base change, so the defining square is Cartesian:

The bottom map is the composite of the generic-point morphism with the chosen field extension, and the top left object is the fiber product by definition. The diagram records only base change; it does not say the special fibers are smooth.