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

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 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

Riemann surfaces
Belyi's theorem • Bolza surface • Dessin d'enfant • Differential of the first kind • Klein quartic • Riemann's existence theorem • Riemann-Roch theorem • Teichmüller space • Torelli theorem

Constructions
Dual curve • Polar curve

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 for algebraic curves • 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

Singularities

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

Игры ⚽ Нужен реферат?

Look at other dictionaries:

Castelnuovo-de Franchis theorem — In mathematics, the Castelnuovo de Franchis theorem is a classical result on complex algebraic surfaces. If X is such a surface, projective and non singular, suppose given two differentials of the first kind : omega; i with i = 1,2on X which are… … Wikipedia
List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… … Wikipedia
Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… … Wikipedia
List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… … Wikipedia
Plane curve — In mathematics, a plane curve is a curve in a Euclidean plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. A smooth plane curve is a curve in a … Wikipedia
Moduli of algebraic curves — In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on… … Wikipedia
Genus–degree formula — In classical algebraic geometry, the genus–degree formula relates the degree d of a non singular plane curve with its arithmetic genus g via the formula: A singularity of order r decreases the genus by .[1] Proofs The proof follows immediately… … Wikipedia
Castelnuovo — is a name common to several communes in Italy: *Castelnuovo (TN), Trento *Castelnuovo Belbo, Asti *Castelnuovo Berardenga, Siena *Castelnuovo Bocca d Adda, Lodi *Castelnuovo Bormida, Alessandria *Castelnuovo Bozzente, Como *Castelnuovo Calcea,… … Wikipedia
Italian school of algebraic geometry — In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885 1935) done internationally in birational geometry, particularly on algebraic surfaces.… … Wikipedia

Academic Dictionaries and Encyclopedias

De Franchis theorem

See also

References

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

De Franchis theorem

See also

References

Look at other dictionaries:

Share the article and excerpts

Direct link