Gorenstein schemeIn algebraic geometry, a Gorenstein scheme is a locally Noetherian scheme whose local rings are all Gorenstein. The canonical line bundle is defined for any Gorenstein scheme over a field, and its properties are much the same as in the special case of smooth schemes. For a Gorenstein scheme X of finite type over a field, f: X → Spec(k), the dualizing complex f!(k) on X is a line bundle (called the canonical bundle KX), viewed as a complex in degree −dim(X).
Base change theoremsIn mathematics, the base change theorems relate the and the of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves: where is a of topological spaces and is a sheaf on X. Such theorems exist in different branches of geometry: for (essentially arbitrary) topological spaces and proper maps f, in algebraic geometry for (quasi-)coherent sheaves and f proper or g flat, similarly in analytic geometry, but also for étale sheaves for f proper or g smooth.
Toric varietyIn algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable.
Algebraic spaceIn mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.
Jacobian varietyIn 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 of C, hence an abelian variety. The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete version of the Abel–Jacobi theorem, making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized abelian variety, of dimension g, and hence, over the complex numbers, it is a complex torus.
Rational varietyIn mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to the field of all rational functions for some set of indeterminates, where d is the dimension of the variety. Let V be an affine algebraic variety of dimension d defined by a prime ideal I = ⟨f1, ..., fk⟩ in . If V is rational, then there are n + 1 polynomials g0, ..., gn in such that In order words, we have a of the variety.
Function field of an algebraic varietyIn algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions. In complex algebraic geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions.
Derived algebraic geometryDerived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ), simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements.
Residue fieldIn mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal. This construction is applied in algebraic geometry, where to every point x of a scheme X one associates its residue field k(x). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point.
Linear system of divisorsIn 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 parameters of the family. These arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors D on a general scheme or even a ringed space (X, OX).
