Éclatement (mathématiques)En mathématiques, un éclatement est un type d'application birationnelle entre ou algébriques qui est un isomorphisme en dehors de sous-variétés propres Le cas le plus simple est celui où D est un point ; E est alors un diviseur isomorphe à un espace projectif. L'éclatement de l'origine dans s'obtient de la façon suivante. Soit Pn – 1 l'espace projectif de dimension n – 1 muni de coordonnées . Soit le sous-ensemble de Cn × Pn – 1 défini par les équations pour i, j = 1, ..., n.
Lefschetz pencilIn mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V. A pencil is a particular kind of linear system of divisors on V, namely a one-parameter family, parametrised by the projective line. This means that in the case of a complex algebraic variety V, a Lefschetz pencil is something like a fibration over the Riemann sphere; but with two qualifications about singularity.
Adequate equivalence relationIn algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Pierre Samuel formalized the concept of an adequate equivalence relation in 1958. Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the of pure motives with respect to that relation.
Théorème des hyperplans de LefschetzEn mathématiques, et plus précisément en géométrie algébrique et en topologie algébrique, le théorème des hyperplans de Lefschetz est un énoncé précis de certaines relations entre la forme d'une variété algébrique et la forme de ses sous-variétés. Plus précisément, le théorème énonce que pour une variété X plongée dans l'espace projectif et une section hyperplane (i.e. une intersection de X à un hyperplan) Y, les groupes d'homologie, de cohomologie et d'homotopie de X déterminent ceux de Y.
Kähler differentialIn mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available. Let R and S be commutative rings and φ : R → S be a ring homomorphism.
Hilbert schemeIn algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes.
Rational normal curveIn mathematics, the rational normal curve is a smooth, rational curve C of degree n in projective n-space Pn. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For n = 2 it is the plane conic Z0Z2 = Z, and for n = 3 it is the twisted cubic. The term "normal" refers to projective normality, not normal schemes. The intersection of the rational normal curve with an affine space is called the moment curve.
Arithmetic genusIn mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Let X be a projective scheme of dimension r over a field k, the arithmetic genus of X is defined asHere is the Euler characteristic of the structure sheaf . The arithmetic genus of a complex projective manifold of dimension n can be defined as a combination of Hodge numbers, namely When n=1, the formula becomes . According to the Hodge theorem, .
Chow's lemmaChow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following: If is a scheme that is proper over a noetherian base , then there exists a projective -scheme and a surjective -morphism that induces an isomorphism for some dense open The proof here is a standard one. We can first reduce to the case where is irreducible.
Plongement de SegreEn géométrie algébrique, le plongement de Segre est un morphisme qui identifie le produit fibré de deux espaces projectifs à une variété projective. Une conséquence en est que le produit fibré de deux variétés projectives est une variété projective. On fixe un corps et deux entiers naturels et on considère le produit fibré des espaces projectifs de dimensions respectives . Alors il existe un morphisme de variétés algébriques qui est une immersion fermée (i.e. induit un isomorphe sur son image qui est une sous-variété fermée de ).
