Max Noether's theorem

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In mathematics, Max Noether's theorem in algebraic geometry may refer to at least six results of Max Noether. NB that Noether's theorem usually refers to a result derived from work of his daughter Emmy Noether.

Max Noether's residual intersection theorem (Fundamentalsatz or fundamental theorem, is a result on algebraic curves in the projective plane, on the residual sets of intersections.

There is a Max Noether theorem on curves lying on algebraic surfaces, which are hypersurfaces in P³. or more generally complete intersections. It states that, for degree at least four for hypersurfaces, the generic such surface has no curve on it apart from the hyperplane section. In more modern language, the Picard group is infinite cyclic, other than for a short list of degrees.

There is a Max Noether theorem on canonical curves.

There is Noether's theorem on rationality for surfaces.

There is a Max Noether theorem on the generation of the Cremona group by quadratic transformations.

There is a Max Noether theorem on the generation of the quadratic differentials on a curve by the Abelian differentials, for curves of genus > 2 that are not hyperelliptic curves.

See also: Brill-Noether theory, Noether's formula for algebraic surfaces.