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 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 Castelnuovo–de Franchis theorem References M. De Franchis: Un teorema sulle involuzioni irrazionali, Rend. Circ. Mat Palermo 36 (1913), 368 hide vte Topics in algebraic curves Rational curves Five points determine a conicProjective lineRational normal curveRiemann sphereTwisted cubic Elliptic curves Analytic theory Elliptic functionElliptic integralFundamental pair of periodsModular form Arithmetic theory Counting points on elliptic curvesDivision polynomialsHasse's theorem on elliptic curvesMazur's torsion theoremModular elliptic curveModularity theoremMordell–Weil theoremNagell–Lutz theoremSupersingular elliptic curveSchoof's algorithmSchoof–Elkies–Atkin algorithm Applications Elliptic curve cryptographyElliptic curve primality Higher genus De Franchis theoremFaltings's theoremHurwitz's automorphisms theoremHurwitz surfaceHyperelliptic curve Plane curves AF+BG theoremBézout's theoremBitangentCayley–Bacharach theoremConic sectionCramer's paradoxCubic plane curveFermat curveGenus–degree formulaHilbert's sixteenth problemNagata's conjecture on curvesPlücker formulaQuartic plane curveReal plane curve Riemann surfaces Belyi's theoremBring's curveBolza surfaceCompact Riemann surfaceDessin d'enfantDifferential of the first kindKlein quarticRiemann's existence theoremRiemann–Roch theoremTeichmüller spaceTorelli theorem Constructions Dual curvePolar curveSmooth completion Structure of curves Divisors on curves Abel–Jacobi mapBrill–Noether theoryClifford's theorem on special divisorsGonality of an algebraic curveJacobian varietyRiemann–Roch theoremWeierstrass pointWeil reciprocity law Moduli ELSV formulaGromov–Witten invariantHodge bundleModuli of algebraic curvesStable curve Morphisms Hasse–Witt matrixRiemann–Hurwitz formulaPrym varietyWeber's theorem Singularities AcnodeCrunodeCuspDelta invariantTacnode Vector bundles Birkhoff–Grothendieck theoremStable vector bundleVector bundles on algebraic curves Categories: Algebraic curvesRiemann surfacesTheorems in algebraic geometryTheorems in algebraic topology

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