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.

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.

Topics in algebraic curves

Rational curves

Five points determine a conic
Projective line
Rational normal curve
Riemann sphere
Twisted cubic

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem (Algebraic curves)
Singularities	Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves