De Franchis theorem
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
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. 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.
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Topics in algebraic curves
- Five points determine a conic
- Projective line
- Rational normal curve
- Riemann sphere
- Twisted cubic
Analytic theory |
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Arithmetic theory | |
Applications |
- De Franchis theorem
- Faltings's theorem
- Hurwitz's automorphisms theorem
- Hurwitz surface
- Hyperelliptic curve
Divisors on curves | |
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Moduli | |
Morphisms |
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Singularities | |
Vector bundles |
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