De Franchis theorem

De Franchis theorem: 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 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.

See also

Castelnuovo–de Franchis theorem

References

M. De Franchis: Un teorema sulle involuzioni irrazionali, Rend. Circ. Mat Palermo 36 (1913), 368

v · d · eTopics in algebraic curves

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Analytic theory

Elliptic function • Elliptic integral • Fundamental pair of periods • Modular form

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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 testing • Elliptic curve primality proving

Higher genus
De Franchis theorem • Faltings' theorem • Hurwitz's automorphisms theorem • Hurwitz surface • Hyperelliptic curve

Plane curves
AF+BG theorem • Bézout's theorem • Bitangent • Cayley–Bacharach theorem • Conic section • Cramer's paradox • Cubic plane curve • Fermat curve • Genus–degree formula • Hilbert's sixteenth problem • Nagata's conjecture on curves • Plücker formula • Quartic plane curve • Real plane curve

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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

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Acnode • Crunode • Cusp • Delta invariant • Tacnode

Vector bundles

Birkhoff–Grothendieck theorem • Stable vector bundle • Vector bundles on algebraic curves

Categories:
Algebraic curves
Riemann surfaces
Theorems in geometry

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De Franchis theorem

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