# Glossary of algebraic geometry

(Redirected from Glossary of scheme theory)

This is a **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.

## !$@Edit

- A generic point. For example, the point associated to the zero ideal for any integral affine scheme.
*F*(*n*),*F*(*D*)- 1. If
*X*is a projective scheme with Serre's twisting sheaf and if*F*is an -module, then - 2. If
*D*is a Cartier divisor and*F*is an -module (*X*arbitrary), then If*D*is a Weil divisor and*F*is reflexive, then one replaces*F*(*D*) by its reflexive hull (and call the result still*F*(*D*).) - |
*D*| - The complete linear system of a Weil divisor
*D*on a normal complete variety*X*over an algebraically closed field*k*; that is, . There is a bijection between the set of*k*-rational points of |*D*| and the set of effective Weil divisors on*X*that are linearly equivalent to*D*.^{[1]}The same definition is used if*D*is a Cartier divisor on a complete variety over*k*. - [X/G]
- The quotient stack of, say, an algebraic space
*X*by an action of a group scheme*G*. - The GIT quotient of a scheme
*X*by an action of a group scheme*G*. *L*^{n}- An ambiguous notation. It usually means an
*n*-th tensor power of*L*but can also mean the self-intersection number of*L*. If , the structure sheaf on*X*, then it means the direct sum of*n*copies of . - The tautological line bundle. It is the dual of Serre's twisting sheaf .
- Serre's twisting sheaf. It is the dual of the tautological line bundle . It is also called the hyperplane bundle.
- 1. If
*D*is an effective Cartier divisor on*X*, then it is the inverse of the ideal sheaf of*D*. - 2. Most of the times, is the image of
*D*under the natural group homomorphism from the group of Cartier divisors to the Picard group of*X*, the group of isomorphism classes of line bundles on*X*. - 3. In general, is the sheaf corresponding to a Weil divisor
*D*(on a normal scheme). It need not be locally free, only reflexive. - 4. If
*D*is a ℚ-divisor, then is of the integral part of*D*. - 1. is the sheaf of Kähler differentials on
*X*. - 2. is the
*p*-th exterior power of . - 1. If
*p*is 1, this is the sheaf of logarithmic Kähler differentials on*X*along*D*(roughly differential forms with simple poles along a divisor*D*.) - 2. is the
*p*-th exterior power of . **P**(*V*)- The notation is ambiguous. Its traditional meaning is the projectivization of a finite-dimensional
*k*-vector space*V*; i.e.,*k*[*V*]) and its*k*-points correspond to lines in*V*. In contrast, Hartshorne and EGA write**P**(*V*) for the Proj of the symmetric algebra of*V*. - Q-factorial
- A normal variety is -factorial if every -Weil divisor is -Cartier.
- Spec(
*R*) - The set of all prime ideals in a ring
*R*with Zariski topology; it is called the prime spectrum of*R*. - Spec
_{X}(*F*) - The relative Spec of the
*O*_{X}-algebra*F*. It is also denoted by**Spec**(*F*) or simply Spec(*F*). - Spec
^{an}(*R*) - The set of all valuations for a ring
*R*with a certain weak topology; it is called the Berkovich spectrum of*R*.

## AEdit

- abelian
- 1. An abelian variety is a complete group variety. For example, consider the complex variety or an elliptic curve over a finite field .
- 2. An abelian scheme is a (flat) family of abelian varieties.
- adjunction formula
- 1. If
*D*is an effective Cartier divisor on an algebraic variety*X*, both admitting dualizing sheaves , then the adjunction formula says:- .

*X*and

*D*are smooth, then the formula is equivalent to saying:

- where are canonical divisors on
*D*and*X*.

**affine**if the preimage of any open affine subset is again affine. In more fancy terms, affine morphisms are defined by the global

**Spec**construction for sheaves of

*O*-Algebras, defined by analogy with the spectrum of a ring. Important affine morphisms are vector bundles, and finite morphisms.

_{X}*X*of a projective space is the Spec of the homogeneous coordinate ring of

*X*.

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.

From the preface to I.R. Shafarevich, Basic Algebraic Geometry.

^{[2]}See also the field with one element and Peña, Javier López; Lorscheid, Oliver (2009-08-31). "Mapping F_1-land:An overview of geometries over the field with one element". arXiv:0909.0069. as well as

^{[3]}

^{[4]}.

*k*is a reduced separated scheme of finite type over . An irreducible algebraic set is called an algebraic variety.

*k*is an integral separated scheme of finite type over . Note, not assuming

*k*is algebraically closed causes some pathology; for example, is not a variety since the coordinate ring is not an integral domain.

^{[5]}

*X*of dimension

*r*is .

## BEdit

- Behrend function
- The weighted Euler characteristic of a (nice) stack
*X*with respect to the Behrend function is the degree of the virtual fundamental class of*X*. - Behrend's trace formula
- Behrend's trace formula generalizes Grothendieck's trace formula; both formulas compute the trace of the Frobenius on
*l*-adic cohomology. - big
- A big line bundle
*L*on*X*of dimension*n*is a line bundle such that . - birational morphism
- 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.
- blow-up
- A blow-up is a birational transformation that replaces a closed subscheme with an effective Cartier divisor. Precisely, given a noetherian scheme
*X*and a closed subscheme , the blow-up of*X*along*Z*is a proper morphism such that (1) is an effective Cartier divisor, called the exceptional divisor and (2) is universal with respect to (1). Concretely, it is constructed as the relative Proj of the Rees algebra of with respect to the ideal sheaf determining*Z*.

## CEdit

- Calabi–Yau
- 1. The Calabi–Yau metric is a Kähler metric whose Ricci curvature is zero.
- canonical
- 1. The canonical sheaf on a normal variety
*X*of dimension*n*is where*i*is the inclusion of the smooth locus*U*and is the sheaf of differential forms on*U*of degree*n*. If the base field has characteristic zero instead of normality, then one may replace*i*by a resolution of singularities. - 2. The canonical class on a normal variety
*X*is the divisor class such that . - 3. The canonical divisor is a representative of the canonical class denoted by the same symbol (and not well-defined.)
- 4. The canonical ring of a normal variety
*X*is the section ring of the canonical sheaf. - canonical model
- 1. The canonical model is the Proj of a canonical ring (assuming the ring is finitely generated.)
- Cartier
- 1. An effective Cartier divisor
*D*on a scheme*X*over*S*is a closed subscheme of*X*that is flat over*S*and whose ideal sheaf is invertible (locally free of rank one). - Castelnuovo–Mumford regularity
- The Castelnuovo–Mumford regularity of a coherent sheaf
*F*on a projective space over a scheme*S*is the smallest integer*r*such that*i*> 0. - catenary
- 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.
- central fiber
- 1. A special fiber.
- Chow group
- The
*k*-th Chow group of a smooth variety*X*is the free abelian group generated by closed subvarieties of dimension*k*(group of*k*-cycles) modulo rational equivalences. - classifying stack
- An analog of a classifying space for torsors in algebraic geometry; see classifying stack.
- closed
**Closed subschemes**of a scheme*X*are defined to be those occurring in the following construction. Let*J*be a quasi-coherent sheaf of -ideals. The support of the quotient sheaf is a closed subset*Z*of*X*and is a scheme called the**closed subscheme defined by the quasi-coherent sheaf of ideals**.*J*^{[6]}The reason the definition of closed subschemes relies on such a construction is that, unlike open subsets, a closed subset of a scheme does not have a unique structure as a subscheme.- Cohen–Macaulay
- A scheme is called Cohen-Macaulay
if all local rings are Cohen-Macaulay.
For example, regular schemes, and
*Spec k*[*x,y*]/(*xy*) are Cohen–Macaulay, but is not. - coherent sheaf
- A coherent sheaf on a Noetherian scheme
*X*is a quasi-coherent sheaf that is finitely generated as*O*_{X}-module. - conic
- An algebraic curve of degree two.
- connected
- The scheme is
*connected*as a topological space. Since the connected components refine the irreducible components any irreducible scheme is connected but not vice versa. An affine scheme*Spec(R)*is connected iff the ring*R*possesses no idempotents other than 0 and 1; such a ring is also called a**connected ring**. Examples of connected schemes include affine space, projective space, and an example of a scheme that is not connected is*Spec*(*k*[*x*]×*k*[*x*]) - compactification
- See for example Nagata's compactification theorem.
- Cox ring
- A generalization of a homogeneous coordinate ring. See Cox ring.
- crepant
- A crepant morphism between normal varieties is a morphism such that .
- curve
- An algebraic variety of dimension one.

## DEdit

- deformation
- Let be a morphism of schemes and
*X*an*S*-scheme. Then a deformation*X*' of*X*is an*S*'-scheme together with a pullback square in which*X*is the pullback of*X*' (typically*X*' is assumed to be flat). - degeneracy locus
- Given a vector-bundle map over a variety
*X*(that is, a scheme*X*-morphism between the total spaces of the bundles), the degeneracy locus is the (scheme-theoretic) locus- .

*X*is said to degenerate to a scheme (called the limit of

*X*) if there is a scheme with generic fiber

*X*and special fiber .

*L*on a complete variety is an integer

*d*such that .

*x*is a cycle on a complete variety over a field

*k*, then its degree is .

*O*

_{X}(

*D*) for some Weil divisor

*D*.

*f*:

*X*→

*Y*is called

*dominant*, if the image

*f*(

*X*) is dense. A morphism of affine schemes

*Spec A*→

*Spec B*is dense if and only if the kernel of the corresponding map

*B*→

*A*is contained in the nilradical of

*B*.

*n*, the dualizing sheaf is a coherent sheaf on

*X*such that

- holds for any locally free sheaf
*F*on*X*; for example, if*X*is a smooth projective variety, then it is a canonical sheaf.

## EEdit

*Éléments de géométrie algébrique*- The EGA was an incomplete attempt to lay a foundation of algebraic geometry based on the notion of scheme, a generalization of an algebraic variety.
*Séminaire de géométrie algébrique*picks up where the EGA left off. Today it is one of the standard references in algebraic geometry. - elliptic curve
- An elliptic curve is a smooth projective curve of genus one.
- essentially of finite type
- Localization of a finite type scheme.
- étale
- A morphism
*f*:*Y*→*X*is étale if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties and over an algebraically closed field, étale morphisms are precisely those inducing an isomorphism of tangent spaces , which coincides with the usual notion of étale map in differential geometry. Étale morphisms form a very important class of morphisms; they are used to build the so-called étale topology and consequently the étale cohomology, which is nowadays one of the cornerstones of algebraic geometry. - Euler sequence
- The exact sequence of sheaves:
**P**^{n}is the projective space over a field and the last nonzero term is the tangent sheaf, is called the Euler sequence. - equivariant intersection theory
- See Chapter II of http://www.math.ubc.ca/~behrend/cet.pdf

## FEdit

*F*-regular- Related to Frobenius morphism.
^{[7]} - Fano
- A Fano variety is a smooth projective variety
*X*whose anticanonical sheaf is ample. - fiber
- Given between schemes, the fiber of
*f*over*y*is, as a set, the pre-image ; it has the natural structure of a scheme over the residue field of*y*as the fiber product , where has the natural structure of a scheme over*Y*as Spec of the residue field of*y*. - fiber product
- 1. Another term for the "pullback" in the category theory.
- 2. A stack given for : an object over
*B*is a triple (*x*,*y*, ψ),*x*in*F*(*B*),*y*in*H*(*B*), ψ an isomorphism in*G*(*B*); an arrow from (*x*,*y*, ψ) to (*x'*,*y*', ψ') is a pair of morphisms such that . The resulting square with obvious projections*does not*commute; rather, it commutes up to natural isomorphism; i.e., it 2-commutes. - final
- One of Grothendieck's fundamental ideas is to emphasize
*relative*notions, i.e. conditions on morphisms rather than conditions on schemes themselves. The category of schemes has a final object, the spectrum of the ring of integers; so that any scheme is*over*, and in a unique way. - finite
- The morphism
*f*:*Y*→*X*is**finite**if may be covered by affine open sets such that each is affine — say of the form — and furthermore is finitely generated as a -module. See finite morphism. Finite morphisms are quasi-finite, but not all morphisms having finite fibers are quasi-finite, and morphisms of finite type are usually not quasi-finite. - finite type (locally)
- The morphism
*f*:*Y*→*X*is**locally of finite type**if may be covered by affine open sets such that each inverse image is covered by affine open sets where each is finitely generated as a -algebra. The morphism*f*:*Y*→*X*is**of finite type**if may be covered by affine open sets such that each inverse image is covered by finitely many affine open sets where each is finitely generated as a -algebra. - finite fibers
- The morphism
*f*:*Y*→*X*has**finite fibers**if the fiber over each point is a finite set. A morphism is**quasi-finite**if it is of finite type and has finite fibers. - finite presentation
- If
*y*is a point of*Y*, then the morphism*f*is**of finite presentation at**(or*y***finitely presented at**) if there is an open affine neighborhood*y**U*of*f(y)*and an open affine neighbourhood*V*of*y*such that*f*(*V*) ⊆*U*and is a finitely presented algebra over . The morphism*f*is**locally of finite presentation**if it is finitely presented at all points of*Y*. If*X*is locally Noetherian, then*f*is locally of finite presentation if, and only if, it is locally of finite type.^{[8]}The morphism*f*:*Y*→*X*is**of finite presentation**(or) if it is locally of finite presentation, quasi-compact, and quasi-separated. If*Y*is finitely presented over*X**X*is locally Noetherian, then*f*is of finite presentation if, and only if, it is of finite type.^{[9]} - flag variety
- The flag variety parametrizes a flag of vector spaces.
- flat
- A morphism is flat if it gives rise to a flat map on stalks. When viewing a morphism
*f*:*Y*→*X*as a family of schemes parametrized by the points of , the geometric meaning of flatness could roughly be described by saying that the fibers do not vary too wildly. - formal
- See formal scheme.

## GEdit

- g
^{r}_{d}. - Given a curve
*C*, a divisor*D*on it and a vector subspace , one says the linear system is a g^{r}_{d}if*V*has dimension*r*+1 and*D*has degree*d*. One says*C*has a g^{r}_{d}if there is such a linear system. - Gabriel–Rosenberg reconstruction theorem
- The Gabriel–Rosenberg reconstruction theorem states a scheme
*X*can be recovered from the category of quasi-coherent sheaves on*X*.^{[10]}The theorem is a starting point for noncommutative algebraic geometry since, taking the theorem as an axiom, defining a noncommutative scheme amounts to defining the category of quasi-coherent sheaves on it. See also https://mathoverflow.net/q/16257 - G-bundle
- A principal G-bundle.
- generic point
- A dense point.
- genus
- See #arithmetic genus, #geometric genus.
- genus formula
- The genus formula for a nodal curve in the projective plane says the genus of the curve is given as
*d*is the degree of the curve and δ is the number of nodes (which is zero if the curve is smooth). - geometric genus
- The geometric genus of a smooth projective variety
*X*of dimension*n*is - geometric point
- The prime spectrum of an algebraically closed field.
- geometric property
- A property of a scheme
*X*over a field*k*is "geometric" if it holds for for any field extension . - geometric quotient
- The geometric quotient of a scheme
*X*with the action of a group scheme*G*is a good quotient such that the fibers are orbits. - gerbe
- A gerbe is (roughy) a stack that is locally nonempty and in which two objects are locally isomorphic.
- GIT quotient
- The GIT quotient is when and when .
- good quotient
- The good quotient of a scheme
*X*with the action of a group scheme*G*is an invariant morphism such that

- where
*Z*is a closed subvariety of a variety*X*and equipped with the multiplication

## HEdit

- Hilbert polynomial
- The Hilbert polynomial of a projective scheme
*X*over a field is the Euler characteristic . - Hodge bundle
- The Hodge bundle on the moduli space of curves (of fixed genus) is roughly a vector bundle whose fiber over a curve
*C*is the vector space . - hyperelliptic
- A curve is hyperelliptic if it has a
*g*^{1}_{2}(i.e., there is a linear system of dimension 1 and degree 2.) - hyperplane bundle
- Another term for Serre's twisting sheaf . It is the dual of the tautological line bundle (whence the term).

## IEdit

- image
- If
*f*:*Y*→*X*is any morphism of schemes, the**scheme-theoretic image**of*f*is the unique*closed*subscheme*i*:*Z*→*X*which satisfies the following universal property:*f*factors through*i*,- if
*j*:*Z*′ →*X*is any closed subscheme of*X*such that*f*factors through*j*, then*i*also factors through*j*.^{[11]}^{[12]}

*f*,*f*(*Y*). For example, the underlying space of*Z*always contains (but is not necessarily equal to) the Zariski closure of*f*(*Y*) in*X*, so if*Y*is any open (and not closed) subscheme of*X*and*f*is the inclusion map, then*Z*is different from*f*(*Y*). When*Y*is reduced, then*Z*is the Zariski closure of*f*(*Y*) endowed with the structure of reduced closed subscheme. But in general, unless*f*is quasi-compact, the construction of*Z*is not local on*X*. - immersion
**Immersions***f*:*Y*→*X*are maps that factor through isomorphisms with subschemes. Specifically, an**open immersion**factors through an isomorphism with an open subscheme and a**closed immersion**factors through an isomorphism with a closed subscheme.^{[13]}Equivalently,*f*is a closed immersion if, and only if, it induces a homeomorphism from the underlying topological space of*Y*to a closed subset of the underlying topological space of*X*, and if the morphism is surjective.^{[14]}A composition of immersions is again an immersion.^{[15]}Some authors, such as Hartshorne in his book*Algebraic Geometry*and Q. Liu in his book*Algebraic Geometry and Arithmetic Curves*, define immersions as the composite of an open immersion followed by a closed immersion. These immersions are immersions in the sense above, but the converse is false. Furthermore, under this definition, the composite of two immersions is not necessarily an immersion. However, the two definitions are equivalent when*f*is quasi-compact.^{[16]}Note that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not: and may be homeomorphic but not isomorphic. This happens, for example, if*I*is the radical of*J*but*J*is not a radical ideal. When specifying a closed subset of a scheme without mentioning the scheme structure, usually the so-called*reduced*scheme structure is meant, that is, the scheme structure corresponding to the unique radical ideal consisting of all functions vanishing on that closed subset.- ind-scheme
- An ind-scheme is an inductive limit of closed immersions of schemes.
- invertible sheaf
- A locally free sheaf of a rank one. Equivalently, it is a torsor for the multiplicative group (i.e., line bundle).
- integral
- A scheme that is both reduced and irreducible is called
*integral*. For locally Noetherian schemes, to be integral is equivalent to being a connected scheme that is covered by the spectra of integral domains. (Strictly speaking, this is not a local property, because the disjoint union of two integral schemes is not integral. However, for irreducible schemes, it is a local property.) For example, the scheme*Spec k*[*t*]/*f*,*f*irreducible polynomial is integral, while*Spec A*×*B*. (*A*,*B*≠ 0) is not. - irreducible
- A scheme
*X*is said to be*irreducible*when (as a topological space) it is not the union of two closed subsets except if one is equal to*X*. Using the correspondence of prime ideals and points in an affine scheme, this means*X*is irreducible iff*X*is connected and the rings A_{i}all have exactly one minimal prime ideal. (Rings possessing exactly one minimal prime ideal are therefore also called irreducible.) Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called its irreducible components. Affine space and projective space are irreducible, while*Spec**k*[*x,y*]/(*xy*) = is not.

## JEdit

- Jacobian variety
- The Jacobian variety of a projective curve
*X*is the degree zero part of the Picard variety .

## KEdit

- Kempf vanishing theorem
- The Kempf vanishing theorem concerns the vanishing of higher cohomology of a flag variety.
- klt
- Abbreviation for "kawamata log terminal"
- Kodaira dimension
- 1. The Kodaira dimension (also called the Iitaka dimension) of a semi-ample line bundle
*L*is the dimension of Proj of the section ring of*L*. - 2. The Kodaira dimension of a normal variety
*X*is the Kodaira dimension of its canonical sheaf. - Kodaira vanishing theorem
- See the Kodaira vanishing theorem.
- Kuranishi map
- See Kuranishi structure.

## LEdit

- Lelong number
- See Lelong number.
- level structure
- see http://math.stanford.edu/~conrad/248BPage/handouts/level.pdf
- linearization
- Another term for the structure of an equivariant sheaf/vector bundle.
- local
- Most important properties of schemes are
*local in nature*, i.e. a scheme*X*has a certain property*P*if and only if for any cover of*X*by open subschemes*X*, i.e._{i}*X*=*X*, every_{i}*X*has the property_{i}*P*. It is usually the case that is enough to check one cover, not all possible ones. One also says that a certain property is*Zariski-local*, if one needs to distinguish between the Zariski topology and other possible topologies, like the étale topology. Consider a scheme*X*and a cover by affine open subschemes*Spec A*. Using the dictionary between (commutative) rings and affine schemes local properties are thus properties of the rings_{i}*A*. A property_{i}*P*is local in the above sense, iff the corresponding property of rings is stable under localization. For example, we can speak of*locally Noetherian*schemes, namely those which are covered by the spectra of Noetherian rings. The fact that localizations of a Noetherian ring are still noetherian then means that the property of a scheme of being locally Noetherian is local in the above sense (whence the name). Another example: if a ring is reduced (i.e., has no non-zero nilpotent elements), then so are its localizations. An example for a non-local property is*separatedness*(see below for the definition). Any affine scheme is separated, therefore any scheme is locally separated. However, the affine pieces may glue together pathologically to yield a non-separated scheme. The following is a (non-exhaustive) list of local properties of rings, which are applied to schemes. Let*X*=*Spec A*be a covering of a scheme by open affine subschemes. For definiteness, let_{i}*k*denote a field in the following. Most of the examples also work with the integers**Z**as a base, though, or even more general bases. Connected, irreducible, reduced, integral, normal, regular, Cohen-Macaulay, locally noetherian, dimension, catenary, - local complete intersection
- The local rings are complete intersection rings. See also: regular embedding.
- local uniformization
- The local uniformization is a method of constructing a weaker form of resolution of singularities by means of valuation rings.
- locally factorial
- The local rings are unique factorization domains.
- locally of finite type
- The morphism
*f*:*Y*→*X*is**locally of finite type**if may be covered by affine open sets such that each inverse image is covered by affine open sets where each is finitely generated as a -algebra. - locally Noetherian
- The
*A*are Noetherian rings. If in addition a finite number of such affine spectra covers_{i}*X*, the scheme is called*noetherian*. While it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false. For example, most schemes in finite-dimensional algebraic geometry are locally Noetherian, but is not. - logarithmic geometry
- log structure
- See log structure. The notion is due to Fontaine-Illusie and Kato.
- loop group
- See loop group (the linked article does not discuss a loop group in algebraic geometry; for now see also ind-scheme).

## MEdit

- moduli
- See for example moduli space.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.
Kollár, János, Chapter 1, "Book on Moduli of Surfaces".

- Mori's minimal model program
- The minimal model program is a research program aiming to do birational classification of algebraic varieties of dimension greater than 2.
- morphism
- 1. A morphism of algebraic varieties is given locally by polynomials.
- 2. A morphism of schemes is a morphism of locally ringed spaces.
- 3. A morphism of stacks (over, say, the category of
*S*-schemes) is a functor such that where are structures maps to the base category.

## NEdit

- nef
- See nef line bundle.
- nonsingular
- An archaic term for "smooth" as in a smooth variety.
- normal
- 1. An integral scheme is called
*normal*, if the local rings are integrally closed domains. For example, all regular schemes are normal, while singular curves are not. - 2. A smooth curve is said to be
*k*-normal if the hypersurfaces of degree*k*cut out the complete linear series . It is projectively normal if it is*k*-normal for all*k*> 0. One thus says that "a curve is projectively normal if the linear system that embeds it is complete." The term "linearly normal" is synonymous with 1-normal. - 3. A closed subvariety is said to be projectively normal if the affine cover over
*X*is a normal scheme; i.e., the homogeneous coordinate ring of*X*is an integrally closed domain. This meaning is consistent with that of 2. - normal
- 1. If
*X*is a closed subscheme of a scheme*Y*with ideal sheaf*I*, then the normal sheaf to*X*is . If the embedded of*X*into*Y*is regular, it is locally free and is called the normal bundle. - 2. The normal cone to
*X*is . if*X*is regularly embedded into*Y*, then the normal cone is isomorphic to , the total space of the normal bundle to*X*. - normal crossings
- See normal crossings.
- normally generated
- A line bundle
*L*on a variety*X*is said to be normally generated if, for each integer*n*> 0, the natural map is surjective.

## OEdit

- open
- 1. A morphism
*f*:*Y*→*X*of schemes is called*open*(*closed*), if the underlying map of topological spaces is open (closed, respectively), i.e. if open subschemes of*Y*are mapped to open subschemes of*X*(and similarly for closed). For example, finitely presented flat morphisms are open and proper maps are closed. - 2. An
**open subscheme**of a scheme*X*is an open subset*U*with structure sheaf .^{[14]} - orbifold
- Nowadays an orbifold is often defined as a Deligne–Mumford stack over the category of differentiable manifolds.
^{[17]}

## PEdit

*p*-divisible group- See
*p*-divisible group (roughly an analog of torsion points of an abelian variety). - pencil
- A linear system of dimension one.
- Picard group
- The Picard group of
*X*is the group of the isomorphism classes of line bundles on*X*, the multiplication being the tensor product. - Plücker embedding
- The Plücker embedding is the closed embedding of the Grassmannian variety into a projective space.
- plurigenus
- The
*n*-th plurigenus of a smooth projective variety is . See also Hodge number. - Poincaré residue map
- See Poincaré residue.
- point
- A scheme is a locally ringed space, so
*a fortiori*a topological space, but the meanings of*point of*are threefold:- a point of the underlying topological space;
- a -valued point of is a morphism from to , for any scheme ;
- a
*geometric point*, where is defined over (is equipped with a morphism to) , where is a field, is a morphism from to where is an algebraic closure of .

*e.g.*complex points, line at infinity) to simplify the geometry by refining the basic objects. The -valued points were a massive further step. As part of the predominating Grothendieck approach, there are three corresponding notions of*fiber*of a morphism: the first being the simple inverse image of a point. The other two are formed by creating fiber products of two morphisms. For example, a**geometric fiber**of a morphism is thought of as- .

- polarization
- an embedding into a projective space
- Proj
- See Proj construction.
- projection formula
- The projection formula says that, for a morphism of schemes, an -module and a locally free -module of finite rank, there is a natural isomorphism
- projective
- 1. A projective variety is a closed subvariety of a projective space.
- 2. A projective scheme over a scheme
*S*is an*S*-scheme that factors through some projective space as a closed subscheme. - 3. Projective morphisms are defined similarly to affine morphisms:
*f*:*Y*→*X*is called**projective**if it factors as a closed immersion followed by the projection of a projective space to .^{[18]}Note that this definition is more restrictive than that of EGA, II.5.5.2. The latter defines to be projective if it is given by the global**Proj**of a quasi-coherent graded*O*-Algebra such that is finitely generated and generates the algebra . Both definitions coincide when is affine or more generally if it is quasi-compact, separated and admits an ample sheaf,_{X}^{[19]}e.g. if is an open subscheme of a projective space over a ring . - projective bundle
- If
*E*is a locally free sheaf on a scheme*X*, the projective bundle**P**(*E*) of*E*is the global Proj of the symmetric algebra of the dual of*E*:*Intersection theory*) but differs from EGA and Hartshorne (they don't take a dual). - projectively normal
- See #normal.
- proper
- A morphism is
**proper**if it is separated,*universally closed*(i.e. such that fiber products with it are closed maps), and of finite type. Projective morphisms are proper; but the converse is not in general true. See also complete variety. A deep property of proper morphisms is the existence of a*Stein factorization*, namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism. - property P
- Let
**P**be a property of a scheme that is stable under base change (finite-type, proper, smooth, étale, etc.). Then a representable morphism is said to have property**P**if, for any with*B*a scheme, the base change has property**P**. - pure dimension
- A scheme has pure dimension
*d*if each irreducible component has dimension*d*.

## QEdit

- quasi-coherent
- A quasi-coherent sheaf on a Noetheiran scheme
*X*is a sheaf of*O*_{X}-modules that is locally given by modules. - quasi-compact
- A morphism
*f*:*Y*→*X*is called*quasi-compact*, if for some (equivalently: every) open affine cover of*X*by some*U*=_{i}*Spec B*, the preimages_{i}*f*^{−1}(*U*) are quasi-compact._{i} - quasi-finite
- The morphism
*f*:*Y*→*X*has**finite fibers**if the fiber over each point is a finite set. A morphism is**quasi-finite**if it is of finite type and has finite fibers. - quasi-projective
- A quasi-projective variety is a locally closed subvariety of a projective space.
- quasi-separated
- A morphism
*f*:*Y*→*X*is called**quasi-separated**or () if the diagonal morphism*Y*is quasi-separated over*X**Y*→*Y*×_{X}*Y*is quasi-compact. A scheme*Y*is called**quasi-separated**if*Y*is quasi-separated over Spec(**Z**).^{[20]} - Quot scheme
- A Quot scheme parametrizes quotients of locally free sheaves on a projective scheme.
- quotient stack
- Usually denoted by [
*X*/*G*], a quotient stack generalizes a quotient of a scheme or variety.

## REdit

- rational
- 1. Over an algebraically closed field, a variety is rational if it is birational to a projective space. For example, rational curves and rational surfaces are those birational to .
- 2. Given a field
*k*and a relative scheme*X*→*S*, a*k*-rational point of*X*is an*S*-morphism . - rational function
- An element in the function field where the limit runs over all coordinates rings of open subsets
*U*of an (irreducible) algebraic variety*X*. See also function field (scheme theory). - rational normal curve
- A rational normal curve is the image of
- .

*d*= 3, it is also called the twisted cubic. - rational singularities
- A variety
*X*over a field of characteristic zero has rational singularities if there is a resolution of singularities such that and . - reduced
- 1. A commutative ring is reduced if it has no nonzero nilpotent elements, i.e., its nilradical is the zero ideal, . Equivalently, is reduced if is a reduced scheme.
- 2. A scheme X is reduced if its stalks are reduced rings. Equivalently X is reduced if, for each open subset , is a reduced ring, i.e., has no nonzero nilpotent sections.
- reflexive sheaf
- A coherent sheaf is reflexive if the canonical map to the second dual is an isomorphism.
- regular
- A regular scheme is a scheme where the local rings are regular local rings. For example, smooth varieties over a field are regular, while
*Spec k*[*x,y*]/(*x*^{2}+*x*^{3}-*y*^{2})= is not. - regular embedding
- A closed immersion is a regular embedding if each point of
*X*has an affine neighborhood in*Y*so that the ideal of*X*there is generated by a regular sequence. If*i*is a regular embedding, then the conormal sheaf of*i*, that is, when is the ideal sheaf of*X*, is locally free. - regular function
- A morphism from an algebraic variety to the affine line.
- representable morphism
- A morphism of stacks such that, for any morphism from a scheme
*B*, the base change is an algebraic space. If "algebraic space" is replaced by "scheme", then it is said to be strongly representable. - resolution of singularities
- A resolution of singularities of a scheme
*X*is a proper birational morphism such that*Z*is smooth. - Riemann–Hurwitz formula
- Given a finite separable morphism between smooth projective curves, if is tamely ramified (no wild ramification); for example, over a field of characteristic zero, then the Riemann–Hurwitz formula relates the degree of π, the genera of
*X*,*Y*and the ramification indices:- .

- Riemann–Roch formula
- 1. If
*L*is a line bundle of degree*d*on a smooth projective curve of genus*g*, then the Riemann–Roch formula computes the Euler characteristic of*L*:- .

- For example, the formula implies the degree of the canonical divisor
*K*is 2*g*- 2.

*X*,

*S*and if

*E*is a vector bundle on

*X*, then as equality in the rational Chow group

*S*is a point,

*X*is a smooth curve of genus

*g*and

*E*is a line bundle

*L*, then the left-hand side reduces to the Euler characteristic while the right-hand side is

## SEdit

- scheme
- A scheme is a locally ringed space that is locally a prime spectrum of a commutative ring.
- Schubert
- 1. A Schubert cell is a
*B*-orbit on the Grassmannian where*B*is the standard Borel; i.e., the group of upper triangular matrices. - 2. A Schubert variety is the closure of a Schubert cell.
- secant variety
- The secant variety to a projective variety is the closure of the union of all secant lines to
*V*in . - section ring
- The section ring or the ring of sections of a line bundle
*L*on a scheme*X*is the graded ring . - Serre's conditions
*S*_{n} - See Serre's conditions on normality. See also https://mathoverflow.net/q/22228
- Serre duality
- See #dualizing sheaf
- separated
- A separated morphism is a morphism such that the fiber product of with itself along has its diagonal as a closed subscheme — in other words, the diagonal morphism is a
*closed immersion*. - sheaf generated by global sections
- A sheaf with a set of global sections that span the stalk of the sheaf at every point. See Sheaf generated by global sections.
- simple
- The term "simple point" is an old term for a "smooth point".
- smooth
- 1.
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*:*Y*→*X*:- 1) for any
*y*∈*Y*, there are open affine neighborhoods*V*and*U*of*y*,*x*=*f*(*y*), respectively, such that the restriction of*f*to*V*factors as an étale morphism followed by the projection of affine*n*-space over*U*. - 2)
*f*is flat, locally of finite presentation, and for every geometric point of*Y*(a morphism from the spectrum of an algebraically closed field to*Y*), the geometric fiber is a smooth*n*-dimensional variety over in the sense of classical algebraic geometry.

- 1) for any

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*?*k*is a scheme

*X*that is of locally of finite type and regular over

*k*.

*k*is a scheme

*X*that is geometrically smooth: is smooth.

*D*on a smooth curve

*C*is special if , which is called the index of speciality, is positive.

*G*-variety (

*G*connected reductive) with an open dense orbit by a Borel subgroup of

*G*.

*Z*and a morphism , the strict transform of

*Y*(also called proper transform) is the blow-up of

*Y*along the closed subscheme . If

*f*is a closed immersion, then the induced map is also a closed immersion.

**subscheme**, without qualifier, of

*X*is a closed subscheme of an open subscheme of

*X*.

## TEdit

- tangent space
- See Zariski tangent space.
- tautological line bundle
- The tautological line bundle of a projective scheme
*X*is the dual of Serre's twisting sheaf ; that is, . - theorem
- See Zariski's main theorem, theorem on formal functions, cohomology base change theorem, Category:Theorems in algebraic geometry.
- torus embedding
- An old term for a toric variety
- toric variety
- A toric variety is a normal variety with the action of a torus such that the torus has an open dense orbit.
- tropical geometry
- A kind of a piecewise-linear algebraic geometry. See tropical geometry.
- torus
- A split torus is a product of finitely many multiplicative groups .

## UEdit

- universal
- 1. If a moduli functor
*F*is represented by some scheme or algebraic space*M*, then a universal object is an element of*F*(*M*) that corresponds to the identity morphism*M*→*M*(which is an*M*-point of*M*). If the values of*F*are isomorphism classes of curves with extra structure, say, then a universal object is called a universal curve. A tautological bundle would be another example of a universal object. - 2. Let be the moduli of smooth projective curves of genus
*g*and that of smooth projective curves of genus*g*with single marked points. In literature, the forgetful map - universally
- A morphism has some property universally if all base changes of the morphism have this property. Examples include universally catenary, universally injective.
- unramified
- For a point in , consider the corresponding morphism of local rings
- .

**unramified**(resp.**G-unramified**) if it is locally of finite type (resp. locally of finite presentation) and if for all in , is the maximal ideal of and the induced map^{[21]}This is the geometric version (and generalization) of an unramified field extension in algebraic number theory.

## VEdit

- variety
- a synonym with "algebraic variety".
- very ample
- A line bundle
*L*on a variety*X*is very ample if*X*can be embedded into a projective space so that*L*is the restriction of Serre's twisting sheaf*O*(1) on the projective space.

## WEdit

- weakly normal
- a scheme is weakly normal if any finite birational morphism to it is an isomorphism.
- Weil divisor
- Another but more standard term for a "codimension-one cycle"; see divisor.
- Weil reciprocity
- See Weil reciprocity.

## ZEdit

- Zariski–Riemann space
- A Zariski–Riemann space is a locally ringed space whose points are valuation rings.

## NotesEdit

**^**Proof: Let*D*be a Weil divisor on*X*. If*D'*~*D*, then there is a nonzero rational function*f*on*X*such that*D*+ (*f*) =*D'*and then*f*is a section of*O*_{X}(*D*) if*D'*is effective. The opposite direction is similar. □**^**Alain, Connes (2015-09-18). "An essay on the Riemann Hypothesis". arXiv:1509.05576.**^**Deitmar, Anton (2006-05-16). "Remarks on zeta functions and K-theory over F1". arXiv:math/0605429.**^**Flores, Jaret (2015-03-08). "Homological Algebra for Commutative Monoids". arXiv:1503.02309.**^**Durov, Nikolai (2007-04-16). "New Approach to Arakelov Geometry". arXiv:0704.2030.**^**Grothendieck & Dieudonné 1960, 4.1.2 and 4.1.3**^**Smith, Karen E.; Zhang, Wenliang (2014-09-03). "Frobenius Splitting in Commutative Algebra". arXiv:1409.1169.**^**Grothendieck & Dieudonné 1964, §1.4**^**Grothendieck & Dieudonné 1964, §1.6**^**Brandenburg, Martin (2014-10-07). "Tensor categorical foundations of algebraic geometry". arXiv:1410.1716.**^**Hartshorne 1977, Exercise II.3.11(d)**^**The Stacks Project, Chapter 21, §4.**^**Grothendieck & Dieudonné 1960, 4.2.1- ^
^{a}^{b}Hartshorne 1977, §II.3 **^**Grothendieck & Dieudonné 1960, 4.2.5**^**Q. Liu,*Algebraic Geometry and Arithmetic Curves, exercise 2.3***^**Harada, Megumi; Krepski, Derek (2013-02-02). "Global quotients among toric Deligne-Mumford stacks". arXiv:1302.0385.**^**Hartshorne 1977, II.4**^**EGA, II.5.5.4(ii).**^**Grothendieck & Dieudonné 1964, 1.2.1**^**The notion G-unramified is what is called "unramified" in EGA, but we follow Raynaud's definition of "unramified", so that closed immersions are unramified. See Tag 02G4 in the Stacks Project for more details.

## ReferencesEdit

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*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics],**2**, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1700-8, ISBN 978-3-540-62046-4, MR 1644323 - Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas".
*Publications Mathématiques de l'IHÉS*.**4**. doi:10.1007/bf02684778. MR 0217083. - Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes".
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*Publications Mathématiques de l'IHÉS*.**11**. doi:10.1007/bf02684274. MR 0217085. - Grothendieck, Alexandre; Dieudonné, Jean (1963). "Éléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Seconde partie".
*Publications Mathématiques de l'IHÉS*.**17**. doi:10.1007/bf02684890. MR 0163911. - Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie".
*Publications Mathématiques de l'IHÉS*.**20**. doi:10.1007/bf02684747. MR 0173675. - Grothendieck, Alexandre; Dieudonné, Jean (1965). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie".
*Publications Mathématiques de l'IHÉS*.**24**. doi:10.1007/bf02684322. MR 0199181. - Grothendieck, Alexandre; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie".
*Publications Mathématiques de l'IHÉS*.**28**. doi:10.1007/bf02684343. MR 0217086. - Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie".
*Publications Mathématiques de l'IHÉS*.**32**. doi:10.1007/bf02732123. MR 0238860. - Hartshorne, Robin (1977),
*Algebraic Geometry*, Graduate Texts in Mathematics,**52**, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 - Kollár, János, "Book on Moduli of Surfaces" available at his website [2]
- Martin's Olsson's course notes written by Anton, https://web.archive.org/web/20121108104319/http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf
- A book worked out by many authors.