Glossary of algebraic geometry

See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry.

For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.

A generic point. For example, the point associated to the zero ideal for any integral affine scheme.
1.  Affine space is roughly a vector space where one has forgotten which point is the origin

Algebraic geometry occupied a central place in the mathematics of the last century. The deepest results of Abel, Riemann, Weierstrass, many of the most important papers of Klein and Poincare belong to this domain. At the end of the last and the beginning of the present century the attitude towards algebraic geometry changed abruptly. ... The style of thinking that was fully developed in algebraic geometry at that time was too far removed from the set-theoretical and axiomatic spirit, which then determined the development of mathematics. ... Around the middle of the present century algebraic geometry had undergone to a large extent such a reshaping process. As a result, it can again lay claim to the position it once occupied in mathematics.

Algebraic geometry is a branch of mathematics that studies solutions to algebraic equations.
A separated scheme of finite type over a field. For example, an algebraic variety is a reduced irreducible algebraic scheme.
A line bundle on a projective variety is ample if some tensor power of it is very ample.
0-dimensional and Noetherian. The definition applies both to a scheme and a ring.
A birational morphism between schemes is a morphism that becomes an isomorphism after restricted to some open dense subset. One of the most common examples of a birational map is the map induced by a blowup.
1.  The Calabi–Yau metric is a Kähler metric whose Ricci curvature is zero.
A scheme is catenary, if all chains between two irreducible closed subschemes have the same length. Examples include virtually everything, e.g. varieties over a field, and it is hard to construct examples that are not catenary.
2.  For the degree of a finite morphism, see morphism of varieties#Degree of a finite morphism.
2.  A divisorial scheme is a scheme admitting an ample family of invertible sheaves. A scheme admitting an ample invertible sheaf is a basic example.
3.  Some authors call a normal variety Gorenstein if the canonical divisor is Cartier; note this usage is inconsistent with meaning 1.
The Kempf vanishing theorem concerns the vanishing of higher cohomology of a flag variety.
2.  The Kodaira dimension of a normal variety X is the Kodaira dimension of its canonical sheaf.

While much of the early work on moduli, especially since [Mum65], put the emphasis on the construction of fine or coarse moduli spaces, recently the emphasis shifted towards the study of the families of varieties, that is towards moduli functors and moduli stacks. The main task is to understand what kind of objects form "nice" families. Once a good concept of "nice families" is established, the existence of a coarse moduli space should be nearly automatic. The coarse moduli space is not the fundamental object any longer, rather it is only a convenient way to keep track of certain information that is only latent in the moduli functor or moduli stack.

A quasi-projective variety is a locally closed subvariety of a projective space.
A Quot scheme parametrizes quotients of locally free sheaves on a projective scheme.
A coherent sheaf is reflexive if the canonical map to the second dual is an isomorphism.
A heuristic term, roughly equivalent to "killing automorphisms". For example, one might say "we introduce level structures to rigidify the geometric situation."

On Grothendieck's own view there should be almost no history of schemes, but only a history of the resistance to them: ... There is no serious historical question of how Grothendieck found his definition of schemes. It was in the air. Serre has well said that no one invented schemes (conversation 1995). The question is, what made Grothendieck believe he should use this definition to simplify an 80 page paper by Serre into some 1000 pages of Éléments de géométrie algébrique?

A sheaf with a set of global sections that span the stalk of the sheaf at every point. See Sheaf generated by global sections.

The higher-dimensional analog of étale morphisms are smooth morphisms. There are many different characterisations of smoothness. The following are equivalent definitions of smoothness of the morphism f : YX:

A toric variety is a normal variety with the action of a torus such that the torus has an open dense orbit.
A kind of a piecewise-linear algebraic geometry. See tropical geometry.
a scheme is weakly normal if any finite birational morphism to it is an isomorphism.
Another but more standard term for a "codimension-one cycle"; see divisor.
A Zariski–Riemann space is a locally ringed space whose points are valuation rings.