Schubert varietyIn algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces V, specified using linear algebra, inside a fixed vector subspace W. Here W may be a vector space over an arbitrary field, though most commonly over the complex numbers.
Variété complèteEn mathématiques, en particulier en géométrie algébrique, une variété algébrique complète est une variété algébrique X, telle que pour toute variété Y le morphisme de projection est une application fermée (c'est-à-dire qu'elle envoie les fermés sur des fermés). Cela peut être vu comme un analogue de la compacité en géométrie algébrique : en effet, un espace topologique X est compact si et seulement si l'application de projection ci-dessus est fermée par rapport aux produits topologiques.
Iitaka dimensionIn algebraic geometry, the Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L. This is 1 less than the dimension of the section ring of L The Iitaka dimension of L is always less than or equal to the dimension of X. If L is not effective, then its Iitaka dimension is usually defined to be or simply said to be negative (some early references define it to be −1).
Stack (mathematics)In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic, compatible geometrical objects (such as vector bundles on topological spaces) can be "glued together" within a restriction of the topological basis.
Quasi-projective varietyIn mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in scheme theory, where a quasi-projective scheme is a locally closed subscheme of some projective space. An affine space is a Zariski-open subset of a projective space, and since any closed affine subset can be expressed as an intersection of the projective completion and the affine space embedded in the projective space, this implies that any affine variety is quasiprojective.
Corps à un élémentEn mathématiques, et plus précisément en géométrie algébrique, le corps à un élément est le nom donné de manière quelque peu fantaisiste à un objet qui se comporterait comme un corps fini à un seul élément, si un tel corps pouvait exister. Cet objet est noté F1, ou parfois Fun. L'idée est qu'il devrait être possible de construire des théories dans lesquelles les ensembles et les lois de composition (qui constituent les bases de l'algèbre générale) seraient remplacés par d'autres objets plus flexibles.
Zariski's main theoremIn algebraic geometry, Zariski's main theorem, proved by , is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational. Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related.
Twisted cubicIn mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a projective variety that is not linear or a hypersurface, in fact not a complete intersection. It is the three-dimensional case of the rational normal curve, and is the of a Veronese map of degree three on the projective line.
Arakelov theoryIn mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. The main motivation behind Arakelov geometry is the fact there is a correspondence between prime ideals and finite places , but there also exists a place at infinity , given by the Archimedean valuation, which doesn't have a corresponding prime ideal. Arakelov geometry gives a technique for compactifying into a complete space which has a prime lying at infinity.
Geometric invariant theoryIn mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties.
