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 {{ill|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
References
- M. De Franchis: Un teorema sulle involuzioni irrazionali, Rend. Circ. Mat Palermo 36 (1913), 368
- {{cite journal |jstor=119264 |last1=Tanabe |first1=Masaharu |title=A Bound for the Theorem of de Franchis |journal=Proceedings of the American Mathematical Society |date=1999 |volume=127 |issue=8 |pages=2289–2295 |doi=10.1090/S0002-9939-99-04858-3 |doi-access=free }}
- {{cite journal |url=http://eudml.org/doc/83912 |title=On the theorem of de Franchis |journal=Annali della Scuola Normale Superiore di Pisa - Classe di Scienze |date=1983 |volume=10 |issue=3 |pages=429–436 |last1=Howard |first1=Alan |last2=Sommese |first2=Andrew J. }}
- {{cite journal |last1=Tanabe |first1=Masaharu|url=https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1660-13.pdf| title=On a theorem of de Franchis (Analysis and Topology of Discrete Groups and Hyperbolic Spaces) |journal=RIMS Kôkyûroku |date=2009 |volume=1660 |pages=139–143}}
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