Algebraic geometry

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.

Algebraic varieties are the central objects of study in algebraic geometry.

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.

In mathematics, a ringed space can be equivalently thought of either (a) a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space.

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety to include, among other things multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety and different schemes) and "varieties"defined over rings (for example Fermat curves are defined over the integers).

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.

In mathematics, properties that hold for "typical" examples are called generic properties.

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures; a complex structure, a Riemannian structure, and a symplectic structure.

In mathematics, a Hodge–Tate module is an analogue of a Hodge structure over p-adic fields.

In mathematics, categorical algebra is a subfield of algebra that approaches algebra from the categorical point of view.

In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives.

In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic.

In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold.

In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points.

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality.

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.

In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine n-space of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety.

In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.

In algebraic geometry, given a smooth projective curve X over a finite field and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by , is an algebraic stack given by: for any -algebra R, the category of principal G-bundles over the relative curve .

In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety.

In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of -modules that represents (or classifies) S-derivations in the sense: for any -modules F, there is an isomorphism that depends naturally on F.