Max Noether's theorem

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

• There are several closely related results of Max Noether on canonical curves.
• 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; see AF+BG theorem.
• 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. This is now often called the Noether-Lefschetz theorem.
• 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.^[1]

Notes

1. Hazewinkel, Michiel, ed. (2001), "Cremona group", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

See also

• Noether inequality
• Special divisor
• Hirzebruch–Riemann–Roch theorem