De Franchis theorem explained

In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism group of X is finite (see though Hurwitz's automorphisms theorem). More generally,

• the set of non-constant morphisms from X to Y is finite;
• fixing X, for all but a finite number of such Y, there is no non-constant morphism from X to Y.

These results are named for (1875–1946). It is sometimes referenced as the De Franchis-Severi theorem. It was used in an important way by Gerd Faltings to prove the Mordell conjecture.

See also

• Castelnuovo–de Franchis theorem

References

• M. De Franchis: Un teorema sulle involuzioni irrazionali, Rend. Circ. Mat Palermo 36 (1913), 368
• 119264 . Tanabe . Masaharu . A Bound for the Theorem of de Franchis . Proceedings of the American Mathematical Society . 1999 . 127 . 8 . 2289–2295 . 10.1090/S0002-9939-99-04858-3 . free .
• On the theorem of de Franchis . Annali della Scuola Normale Superiore di Pisa - Classe di Scienze . 1983 . 10 . 3 . 429–436 . Howard . Alan . Sommese . Andrew J. .
• Tanabe . Masaharu. On a theorem of de Franchis (Analysis and Topology of Discrete Groups and Hyperbolic Spaces) . RIMS Kôkyûroku . 2009 . 1660 . 139-143.