Max Noether's theorem
Encyclopedia
In mathematics
, Max Noether's theorem in algebraic geometry
may refer to at least six results of Max Noether
. Noether's theorem
usually refers to a result derived from work of his daughter Emmy Noether
.
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...
, Max Noether's theorem in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
may refer to at least six results 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:...
. Noether's theorem
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
usually refers to a result derived from work of his daughter Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
.
- There are several closely related results of Max Noether on canonical curves.
- Max Noether's residual intersection theorem (Fundamentalsatz or fundamental theorem) is a result on algebraic curveAlgebraic curveIn 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 in the projective planeProjective planeIn mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
, on the residual sets of intersections; see AF+BG theoremAF+BG theoremIn algebraic geometry, a field of mathematics, the AF+BG theorem is a result of Max Noether which describes when the equation of an algebraic curve in the complex projective plane can be written in terms of the equations of two other algebraic curves....
. - There is a Max Noether theorem on curves lying on algebraic surfaceAlgebraic surfaceIn 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, which are hypersurfaceHypersurfaceIn geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
s in P3, or more generally complete intersectionComplete intersectionIn mathematics, an algebraic variety V in projective space is a complete intersection if it can be defined by the vanishing of the number of homogeneous polynomials indicated by its codimension...
s. It states that, for degree at least four for hypersurfaces, the genericGeneric propertyIn mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic...
such surface has no curve on it apart from the hyperplane sectionHyperplane sectionIn mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H — in other words we look at the subset XH of those elements x of X that satisfy the single linear condition L = 0 defining H as a linear subspace...
. In more modern language, the Picard group is infinite cyclicCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
, other than for a short list of degrees. This is now often called the Noether-Lefschetz theorem. - There is Noether's theorem on rationality for surfacesNoether's theorem on rationality for surfacesIn mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, 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...
. - There is a Max Noether theorem on the generation of the Cremona groupCremona groupIn mathematics, in birational geometry, the Cremona group of order n over a field k is the group of birational automorphisms of the n-dimensional projective space over k...
by quadratic transformations.