De Franchis theorem

Short description: Finiteness statements applying to compact Riemann surfaces

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 Michele De Franchis (it) (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
• Tanabe, Masaharu (1999). "A Bound for the Theorem of de Franchis". Proceedings of the American Mathematical Society 127 (8): 2289–2295. doi:10.1090/S0002-9939-99-04858-3. 
• Howard, Alan; Sommese, Andrew J. (1983). "On the theorem of de Franchis". Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 10 (3): 429–436. http://eudml.org/doc/83912. 
• Tanabe, Masaharu (2009). "On a theorem of de Franchis (Analysis and Topology of Discrete Groups and Hyperbolic Spaces)". RIMS Kôkyûroku 1660: 139-143. https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1660-13.pdf.