De Franchis theorem
Encyclopedia
In mathematics
, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surface
s, or, more generally, algebraic curve
s, 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,
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surface
Compact Riemann surface
In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...
s, or, more generally, algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s, X and Y, in the case of genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
g > 1. The simplest is that the automorphism
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 of an object forms a group, called 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
Francesco Severi
Francesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...
theorem. It was used in an important way by Gerd Faltings
Gerd Faltings
Gerd Faltings is a German mathematician known for his work in arithmetic algebraic geometry.From 1972 to 1978, he studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathematics and in 1981 he got the venia legendi in mathematics, both from the University...
to prove the Mordell conjecture.