Riemann–Roch theorem

A theorem that relates the space of meromorphic functions satisfying prescribed pole conditions on a curve to the conditions’ degree and the curve’s genus.

Divisor: A finite formal sum of points on an algebraic curve or a higher-dimensional variety, with integer coefficients. Pole conditions are encoded as divisors, whose degree enters the theorem.
Riemann–Roch space: The vector space of rational functions whose poles are bounded by a specified divisor. Its dimension is the quantity the theorem computes.
Bernhard Riemann: A German mathematician whose work founded major parts of modern analysis, geometry, and number theory. His ideas on functions and surfaces gave rise to the theorem’s analytic form.
Linear system (algebraic geometry): A projective family of divisors linearly equivalent to a fixed divisor on an algebraic variety. Riemann–Roch computes the dimensions of the spaces defining linear systems on curves.
Riemann–Roch theorem for surfaces: A formula relating Euler characteristics of line bundles on smooth algebraic surfaces to intersection data and characteristic classes. It extends the curve theorem’s dimension-counting principle to surfaces.
Meromorphic function: A function holomorphic except at isolated poles. The theorem counts meromorphic functions whose poles are bounded by a chosen divisor.
Degree of a divisor: The sum of the coefficients of a divisor, weighted by residue-field degrees when necessary. The divisor’s degree gives the main linear contribution to the dimension.
Gustav Roch: A German mathematician who proved a key dimension relation for meromorphic differentials on compact Riemann surfaces. Roch supplied the contribution that completed the theorem’s classical formula.
Very ample line bundle: A line bundle whose global sections embed a variety into projective space. Riemann–Roch helps establish when divisors provide enough sections for embeddings.
Hirzebruch–Riemann–Roch theorem: A theorem expressing the Euler characteristic of a vector bundle on a smooth projective variety using characteristic classes. It generalizes Riemann–Roch from curves to higher-dimensional varieties.

Linked from 14 pages