Algebraic geometry

branch of mathematics dealing with algebraic varieties and their generalizations (schemes, etc.)

mathematical object studied in the field of algebraic geometry

function defined by a relation of the form 𝑅(𝑥,𝑦)=0, where 𝑅 is a function of several variables and there is a unique 𝑦 that satisfies the relation for every 𝑥

Riemannian manifold with SU(n) holonomy

conjecture in algebraic geometry that every Hodge class on a nonsingular complex projective manifold is a linear combination with rational coefficients of the cohomology classes of complex subvarieties

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally.

polynomial whose nonzero terms all have the same degree

geometric concept of a 2D space with a "point at infinity" adjoined

conjectured relation between pairs of Calabi–Yau manifolds; situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory

Cosines of the angles between the vector and the three coordinate axes.

algebra over a ring such that multiplication is associative

Various conjectures about the zeroes of some functions similar to the Riemann zeta function

function in algebra which generalises the concept of multiplicity for commutative rings

particular generating subset of an ideal in a polynomial ring

Noetherian local commutative ring, the minimal number of generators of whose maximal ideal equals its Krull dimension

smooth manifold carrying compatible complex, Riemannian, and symplectic structures

study of complex manifolds and several complex variables

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.

lemma

module such that taking the tensor product with it induces an exact functor

family of geometric objects with a common property

in algebra, expression of an ideal as the intersection of ideals of a specific type

three dimensions self-intersecting surface

conjecture asserting that, over a characteristic-zero field K, given a polynomial map f: Kⁿ → Kⁿ, if its Jacobian determinant J: Kⁿ → K is a nonzero constant map, then f admits a polynomial inverse g: Kⁿ → Kⁿ

commutative ring, named after Irvin Cohen and Francis Sowerby Macaulay (1862-1937)

group of Italian mathematicians who studied birational geometry (c. 1885–1935)

numbers expressible as integrals of algebraic functions

branch of algebraic geometry concerned with counting solutions

Mathematical concept

topological space with no infinite strictly ascending sequence of closed subsets

theorem

vector bundle, complementary to the tangent bundle, associated to an embedding

Mathematical function associated to algebraic varieties

Group in arithmetic geometry

set associated with a polynomial in one or more complex variables; the image of the zero locus of a complex polynomial under the logarithm of the absolute value

tangent space associated to a point in a locally ringed space in algebraic geometry

branch of homological algebra that assigns a series of 𝐾‐groups to commutative rings and other algebras

concept in algebraic geometry

conjectural objects in algebraic geometry that provide a universal cohomology theory of varieties

commutative ring admitting a good dimension function, i.e. that relative dimension between two prime ideals is well defined, in the sense that any maximal strictly increasing chain of prime ideals between the two has the same finite length

field arising from a quotient ring by a maximal ideal

scheme morphism such that, with respect to a suitable open cover, is locally of the form Spec(A)→Spec(B) where A is a finitely generated module over B

polynomial function with rational coefficients whose values agree, for sufficiently large argument, with the dimensions of graded components of a graded algebra

the statement that the number of points of intersection of two planar higher-order curves can be greater than the number of arbitrary points usually needed to define one such curve

set of curves

right|thumb|The complex exponential function mapping biholomorphically a rectangle to a quarter-annulus. In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.

part of algebraic geometry devoted to the elimination of variables between polynomials

finite-type sheaf F of modules over a ringed space such that the kernel of a surjective morphism from a finite direct sum of the structure sheaf onto it is also of finite type

invariant in symplectic topology and algebraic geometry

short exact sequence of sheaves on projective space

algebraically defined notion of differential form on an algebra over a ring or on a scheme

type of graph drawing used to study Riemann surfaces

mathematical theory

Analogs of homology groups for algebraic varieties

Weil cohomology theory for schemes over a base field, whose values are modules over the ring of Witt vectors over the base field, that replaces Zariski open sets by infinitesimal thickenings of Zariski open sets with divided power structures

a Noetherian universally catenary ring that is J-2 and Grothendieck

On Schubert's enumerative calculus

kind of partial function between algebraic varieties

object much like an algebraic variety but defined as the zero set of finitely many (real- or complex-)analytic functions

smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation