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Abelian variety

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

Divisor (algebraic geometry)

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named fo

Linear system of divisors

In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of paramet

Theorem of Bertini

In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by E

Normal crossing singularity

In algebraic geometry a normal crossing singularity is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually norma

Jacobian variety

In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group

Ample line bundle

In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The m

Algebraic geometry of projective spaces

Projective space plays a central role in algebraic geometry. The aim of this article is to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective space

Hodge index theorem

In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanne

Nevanlinna invariant

In mathematics, the Nevanlinna invariant of an ample divisor D on a normal projective variety X is a real number connected with the rate of growth of the number of rational points on the variety with

Height zeta function

In mathematics, the height zeta function of an algebraic variety or more generally a subset of a variety encodes the distribution of points of given height.

Nef line bundle

In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the

Riemann–Roch theorem

The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions wit

Lefschetz pencil

In 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.

Picard group

In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product

Invertible sheaf

In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry

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