Elliptic surface

Summary

An elliptic surface is an algebraic surface fibred over a curve so that almost every fibre is a smooth curve of genus one. So it is a family of elliptic curves, or of genus-one curves, parametrized by a curve, degenerating over finitely many points. Kodaira classified the possible singular fibres. Elliptic surfaces include all minimal surfaces of Kodaira dimension $1$ and many K3 and rational surfaces. They connect geometry with the arithmetic of elliptic curves over function fields.