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 (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.
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References[edit]
- 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. JSTOR 119264.
- 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.
- Tanabe, Masaharu (2009). "On a theorem of de Franchis (Analysis and Topology of Discrete Groups and Hyperbolic Spaces)" (PDF). RIMS Kôkyûroku. 1660: 139–143.
Well, that’s interesting to know that Psilotum nudum are known as whisk ferns. Psilotum nudum is the commoner species of the two. While the P. flaccidum is a rare species and is found in the tropical islands. Both the species are usually epiphytic in habit and grow upon tree ferns. These species may also be terrestrial and grow in humus or in the crevices of the rocks.
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