An elliptic surface is a type of algebraic surface that has a fibration structure, meaning it can be viewed as a family of elliptic curves. In more technical terms, an elliptic surface is a smooth projective surface \(S\) over a base scheme, typically taken to be the complex numbers, which admits a morphism to a base scheme \(B\) such that for every point in \(B\), the fiber over that point is an elliptic curve.
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An elliptic surface is a smooth projective surface equipped with a surjective morphism to a smooth curve whose generic fiber is a smooth curve of genus one.