Accouplement de WeilEn géométrie algébrique et en théorie des nombres, l'accouplement de Weil est une relation mathématique entre certains points d'une courbe elliptique, plus spécifiquement une application bilinéaire fonctorielle entre ses points de torsion. Cet accouplement est nommé en l'honneur du mathématicien français André Weil, qui en a systématisé l'étude. Il s'agit d'un outil important dans l'étude de ces courbes.
Differential of the first kindIn mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms. Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf Ω1 of Kähler differentials.
Courbe stableEn géométrie algébrique, une courbe stable est une courbe algébrique dont les singularités sont les plus simples possibles. Elles ont été introduites par Deligne et Mumford pour construire une compactification de l'espace de modules de courbes projectives lisses. Soit un corps algébriquement clos. Un point fermé d'une courbe algébrique (c'est-à-dire variété algébrique de dimension 1) sur est appelé un point double ordinaire si le complété formel de l'anneau local est isomorphe à la -algèbre .
Barth surfaceNOTOC In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by . Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degree 10 with 345 double points. For degree 6 surfaces in P3, showed that 65 is the maximum number of double points possible. The Barth sextic is a counterexample to an incorrect claim by Francesco Severi in 1946 that 52 is the maximum number of double points possible.
Irrelevant idealIn mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal. The terminology arises from the connection with algebraic geometry. If R = k[x0, ..., xn] (a multivariate polynomial ring in n+1 variables over an algebraically closed field k) graded with respect to degree, there is a bijective correspondence between projective algebraic sets in projective n-space over k and homogeneous, radical ideals of R not equal to the irrelevant ideal.
Théorème de Riemann-RochEn mathématiques, le théorème de Riemann-Roch est un résultat de géométrie algébrique. Originellement, il répond au problème de la recherche de l'existence de fonctions méromorphes sur une surface de Riemann donnée, sous la contrainte de pôles de multiplicité imposée en certains points. Par exemple, sous sa forme faible, le théorème énonce que pour points donnés, l'espace (vectoriel) des fonctions méromorphes sur ayant au plus un pôle du premier ordre en ces points et holomorphes ailleurs est de dimension finie sur C plus grande que , où est le genre de la surface.
Folium de Descartesthumb|right|Le folium de Descartes (en vert) et son asymptote (en bleu) pour a = 1. Le folium de Descartes est une courbe algébrique mathématique en forme de nœud de ruban, définie par l’équation cartésienne Elle fut étudiée tout d'abord par Descartes et Roberval en 1638 (lors d'une correspondance avec Mersenne), puis par Huygens en 1672. Cette courbe met en évidence les faiblesses de la méthode de Fermat dans la recherche des extremums d'une courbe algébrique.
Torelli theoremIn mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field.
Diagonal formIn mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is for some given degree m. Such forms F, and the hypersurfaces F = 0 they define in projective space, are very special in geometric terms, with many symmetries. They also include famous cases like the Fermat curves, and other examples well known in the theory of Diophantine equations. A great deal has been worked out about their theory: algebraic geometry, local zeta-functions via Jacobi sums, Hardy-Littlewood circle method.
Bring's curveIn mathematics, Bring's curve (also called Bring's surface and, by analogy with the Klein quartic, the Bring sextic) is the curve in cut out by the homogeneous equations It was named by after Erland Samuel Bring who studied a similar construction in 1786 in a Promotionschrift submitted to the University of Lund. Note that the roots xi of the Bring quintic satisfies Bring's curve since for The automorphism group of the curve is the symmetric group S5 of order 120, given by permutations of the 5 coordinates.
