Noether's theorem on rationality for surfaces
Encyclopedia
In mathematics
, Noether's theorem on rationality for surfaces is a classical result of Max Noether
on complex algebraic surface
s, giving a criterion for a rational surface
. Let S be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from S to the projective line
, with general fibre also a projective line. Then the theorem states that S is rational.
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...
, Noether's theorem on rationality for surfaces is a classical result of Max Noether
Max Noether
Max Noether was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century".-Biography:...
on complex algebraic surface
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s, giving a criterion for a rational surface
Rational surface
In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two...
. Let S be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from S to the projective line
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
, with general fibre also a projective line. Then the theorem states that S is rational.