In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
group of ''X'' is finite (see though
Hurwitz's automorphisms theorem). More generally,
*the set of non-constant
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s 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 (1875–1946). It is sometimes referenced as the De Franchis-
Severi Severi may refer to:
*Severi (surname), Italian surname
*Severan dynasty, dynasty of Roman emperors, ruling in the late 2nd and early 3rd century
*Severi (tribe), tribe that participated in the formation of the First Bulgarian Empire in the 7th cen ...
theorem. It was used in an important way by
Gerd Faltings
Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry.
Education
From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathema ...
to prove the
Mordell conjecture
Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction.
Educati ...
.
See also
*
Castelnuovo–de Franchis theorem
In mathematics, the Castelnuovo–de Franchis theorem is a classical result on complex algebraic surfaces. Let ''X'' be such a surface, projective and non-singular, and let
:ω1 and ω2
be two differentials of the first kind on ''X'' which are lin ...
References
*M. De Franchis: ''Un teorema sulle involuzioni irrazionali'', Rend. Circ. Mat Palermo 36 (1913), 368
*
*
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Algebraic curves
Riemann surfaces
Theorems in algebraic geometry
Theorems in algebraic topology
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