This is aglossary of algebraic geometry.

See alsoglossary of commutative algebra,glossary of classical algebraic geometry, andglossary of ring theory. For the number-theoretic applications, seeglossary 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 schemeS and a morphism anS-morphism.

${\displaystyle \eta }$

Ageneric point. For example, the point associated to the zero ideal for any integral affine scheme.

F(n),F(D)

1.  IfX is a projective scheme withSerre's twisting sheaf${\displaystyle {\mathcal{O}}_X(1)}$ and ifF is an${\displaystyle {\mathcal{O}}_X}$-module, then${\displaystyle F(n)=F{\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(n).}$

2.  IfD is a Cartier divisor andF is an${\displaystyle {\mathcal{O}}_X}$-module (X arbitrary), then${\displaystyle F(D)=F{\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(D).}$ IfD is a Weil divisor andF is reflexive, then one replacesF(D) by its reflexive hull (and calls the result stillF(D)).

|D|

Thecomplete linear system of aWeil divisorD on a normal complete varietyX over an algebraically closed fieldk; that is,${\displaystyle |D|=\mathbf{P}(\mathrm{\Gamma }(X,{\mathcal{O}}_X(D)))}$. There is a bijection between the set ofk-rational points of |D| and the set of effective Weil divisors onX that are linearly equivalent toD.^[1] The same definition is used ifD is aCartier divisor on a complete variety overk.

[X/G]

Thequotient stack of, say, an algebraic spaceX by an action of a group schemeG.

${\displaystyle X//G}$

TheGIT quotient of a schemeX by anaction of a group schemeG.

Lⁿ

An ambiguous notation. It usually means ann-th tensor power ofL but can also mean the self-intersection number ofL. If${\displaystyle L={\mathcal{O}}_X}$, the structure sheaf onX, then it means the direct sum ofn copies of${\displaystyle {\mathcal{O}}_X}$.

${\displaystyle {\mathcal{O}}_X(-1)}$

Thetautological line bundle. It is the dual ofSerre's twisting sheaf${\displaystyle {\mathcal{O}}_X(1)}$.

${\displaystyle {\mathcal{O}}_X(1)}$

Serre's twisting sheaf. It is the dual of thetautological line bundle${\displaystyle {\mathcal{O}}_X(-1)}$. It is also called the hyperplane bundle.

${\displaystyle {\mathcal{O}}_X(D)}$

1.  IfD is aneffective Cartier divisor onX, then it is the inverse of the ideal sheaf ofD.

2.  Most of the times,${\displaystyle {\mathcal{O}}_X(D)}$ is the image ofD under the natural group homomorphism from the group of Cartier divisors to the Picard group${\displaystyle \mathrm{Pic}(X)}$ ofX, the group of isomorphism classes of line bundles onX.

3.  In general,${\displaystyle {\mathcal{O}}_X(D)}$ is the sheaf corresponding to aWeil divisorD (on anormal scheme). It need not be locally free, onlyreflexive.

4.  IfD is a${\displaystyle \mathbb{Q}}$-divisor, then${\displaystyle {\mathcal{O}}_X(D)}$ is${\displaystyle {\mathcal{O}}_X}$ of the integral part ofD.

${\displaystyle {\mathrm{\Omega }}_X^p}$

1.  ${\displaystyle {\mathrm{\Omega }}_X^1}$ is the sheaf ofKähler differentials onX.

2.  ${\displaystyle {\mathrm{\Omega }}_X^p}$ is thep-thexterior power of${\displaystyle {\mathrm{\Omega }}_X^1}$.

${\displaystyle {\mathrm{\Omega }}_X^p(\log D)}$

1.  Ifp is 1, this is the sheaf oflogarithmic Kähler differentials onX alongD (roughly differential forms with simple poles along a divisorD.)

2.  ${\displaystyle {\mathrm{\Omega }}_X^p(\log D)}$ is thep-th exterior power of${\displaystyle {\mathrm{\Omega }}_X^1(\log D)}$.

P(V)

The notation is ambiguous. Its traditional meaning is theprojectivization of a finite-dimensionalk-vector spaceV; i.e.,${\displaystyle \mathbf{P}(V)=\mathrm{Proj}(k[V])=\mathrm{Proj}(\mathrm{Sym}(V^{\ast }))}$(theProj of thering of polynomial functionsk[V]) and itsk-points correspond to lines inV. In contrast, Hartshorne and EGA writeP(V) for the Proj of the symmetric algebra ofV.

Q-factorial

A normal variety is${\displaystyle \mathbb{Q}}$-factorial if every${\displaystyle \mathbb{Q}}$-Weil divisor is${\displaystyle \mathbb{Q}}$-Cartier.

Spec(R)

The set of all prime ideals in a ringR with Zariski topology; it is called theprime spectrum ofR.

Spec_X(F)

Therelative Spec of theO_X-algebraF. It is also denoted bySpec(F) or simply Spec(F).

Spec^an(R)

The set of all valuations for a ringR with a certain weak topology; it is called theBerkovich spectrum ofR.

abelian

1.  Anabelian variety is a complete group variety. For example, consider the complex variety${\displaystyle {\mathbb{C}}^n/{\mathbb{Z}}^{2n}}$ or an elliptic curve${\displaystyle E}$ over a finite field${\displaystyle {\mathbb{F}}_q}$.

2.  Anabelian scheme is a (flat) family of abelian varieties.

adjunction formula

1.  IfD is an effective Cartier divisor on an algebraic varietyX, both admittingdualizing sheaves${\displaystyle {\omega }_D,{\omega }_X}$, then theadjunction formula says:${\displaystyle {\omega }_D=({\omega }_X\otimes {\mathcal{O}}_X(D)){|}_D}$.

2.  If, in addition,X andD are smooth, then the formula is equivalent to saying:${\displaystyle K_D=(K_X+D){|}_D}$where${\displaystyle K_D,K_X}$ arecanonical divisors onD andX.

affine

1.  Affine space is roughly a vector space where one has forgotten which point is the origin

2.  Anaffine variety is a variety in affine space

3.  Anaffine scheme is a scheme that is theprime spectrum of some commutative ring.

4.  A morphism is calledaffine if the preimage of any open affine subset is again affine. In more fancy terms, affine morphisms are defined by theglobalSpec construction for sheaves ofO_X-Algebras, defined by analogy with thespectrum of a ring. Important affine morphisms arevector bundles, andfinite morphisms.

5.  Theaffine cone over a closed subvarietyX of a projective space is the Spec of the homogeneous coordinate ring ofX.

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.

algebraic geometry

Algebraic geometry is a branch of mathematics that studies solutions to algebraic equations.

algebraic geometry over the field with one element

One goal is to prove theRiemann hypothesis.^[2] See also thefield with one element andPeña, Javier López; Lorscheid, Oliver (2009-08-31). "Mapping F_1-land:An overview of geometries over the field with one element".arXiv:0909.0069 [math.AG]. as well as^[3][4].

algebraic group

Analgebraic group is an algebraic variety that is also agroup in such a way the group operations are morphisms of varieties.

algebraic scheme

A separated scheme of finite type over a field. For example, an algebraic variety is a reduced irreducible algebraic scheme.

algebraic set

Analgebraic set over a fieldk is a reduced separated scheme of finite type over${\displaystyle \mathrm{Spec}(k)}$. An irreducible algebraic set is called an algebraic variety.

algebraic space

Analgebraic space is a quotient of a scheme by theétale equivalence relation.

algebraic variety

Analgebraic variety over a fieldk is an integral separated scheme of finite type over${\displaystyle \mathrm{Spec}(k)}$. Note, not assumingk is algebraically closed causes some pathology; for example,${\displaystyle \mathrm{Spec}\mathbb{C}{\times }_{\mathbb{R}}\mathrm{Spec}\mathbb{C}}$ is not a variety since the coordinate ring${\displaystyle \mathbb{C}{\otimes }_{\mathbb{R}}\mathbb{C}}$ is not anintegral domain.

algebraic vector bundle

Alocally free sheaf of a finite rank.

ample

A line bundle on a projective variety isample if some tensor power of it is very ample.

Arakelov geometry

Algebraic geometry over the compactification of Spec of thering of rational integers${\displaystyle \mathbb{Z}}$. SeeArakelov geometry.^[5]

arithmetic genus

Thearithmetic genus of a projective varietyX of dimensionr is${\displaystyle (-1{)}^r(\chi ({\mathcal{O}}_X)-1)}$.

Artin stack

Another term for analgebraic stack.

artinian

0-dimensional and Noetherian. The definition applies both to a scheme and a ring.

Behrend function

Theweighted Euler characteristic of a (nice) stackX with respect to theBehrend function is the degree of thevirtual fundamental class ofX.

Behrend's trace formula

Behrend's trace formula generalizesGrothendieck's trace formula; both formulas compute the trace of theFrobenius onl-adic cohomology.

big

Abig line bundleL onX of dimensionn is a line bundle such that${\displaystyle {\displaystyle \underset{l\to \mathrm{\infty }}{lim sup}\dim \mathrm{\Gamma }(X,L^l)/l^n>0}}$.

birational morphism

Abirational 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

Ablow-up is a birational transformation that replaces a closed subscheme with an effective Cartier divisor. Precisely, given a noetherian schemeX and a closed subscheme${\displaystyle Z\subset X}$, the blow-up ofX alongZ is a proper morphism${\displaystyle \pi :\tilde{X}\to X}$ such that (1)${\displaystyle {\pi }^{-1}(Z)\hookrightarrow \tilde{X}}$ is an effective Cartier divisor, called theexceptional divisor, and (2)${\displaystyle \pi }$ is universal with respect to (1). Concretely, it is constructed as the relative Proj of the Rees algebra of${\displaystyle O_X}$ with respect to the ideal sheaf determiningZ.

Calabi–Yau

TheCalabi–Yau metric is a Kähler metric whose Ricci curvature is zero.

canonical

1.  Thecanonical sheaf on a normal varietyX of dimensionn is${\displaystyle {\omega }_X=i_{\ast }{\mathrm{\Omega }}_U^n}$ wherei is the inclusion of thesmooth locusU and${\displaystyle {\mathrm{\Omega }}_U^n}$ is the sheaf of differential forms onU of degreen. If the base field has characteristic zero instead of normality, then one may replacei by a resolution of singularities.

2.  Thecanonical class${\displaystyle K_X}$ on a normal varietyX is the divisor class such that${\displaystyle {\mathcal{O}}_X(K_X)={\omega }_X}$.

3.  Thecanonical divisor is a representative of the canonical class${\displaystyle K_X}$ denoted by the same symbol (and not well-defined.)

4.  Thecanonical ring of a normal varietyX is the section ring of thecanonical sheaf.

canonical model

Thecanonical model is theProj of a canonical ring (assuming the ring is finitely generated.)

Cartier

An effectiveCartier divisorD on a schemeX overS is a closed subscheme ofX that is flat overS and whose ideal sheaf is invertible (locally free of rank one).

Castelnuovo–Mumford regularity

TheCastelnuovo–Mumford regularity of a coherent sheafF on a projective space${\displaystyle f:{\mathbf{P}}_S^n\to S}$ over a schemeS is the smallest integerr such that

${\displaystyle R^i{f}_{\ast }F(r-i)=0}$

for alli > 0.

catenary

A scheme iscatenary, 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

A special fiber.

Chow group

Thek-thChow group${\displaystyle A_k(X)}$ of a smooth varietyX is the free abelian group generated by closed subvarieties of dimensionk (group ofk-cycles) modulorational equivalences.

classification

1.  Classification is a guiding principle in all of mathematics where one tries to describe all objects satisfying certain properties up to given equivalences by more accessible data such asinvariants or even some constructive process. In algebraic geometry one distinguishes between discrete and continuous invariants. For continuous classifying invariants one additionally attempts to provide some geometric structure which leads tomoduli spaces.

2.  Completesmoothcurves over analgebraically closed field are classified up torational equivalence by theirgenus${\displaystyle g}$. (a)${\displaystyle g=0}$.rational curves, i.e. the curve isbirational to the projective line${\displaystyle {\mathbb{P}}^1}$. (b)${\displaystyle g=1}$.Elliptic curves, i.e. the curve is a complete 1-dimensionalgroup scheme after choosing any point on the curve as identity. (c)${\displaystyle g\ge 2}$.Hyperbolic curves, also calledcurves of general type. Seealgebraic curves for examples. The classification of smooth curves can be refined by thedegree forprojectively embedded curves, in particular when restricted toplane curves. Note that all complete smooth curves are projective in the sense that they admit embeddings into projective space, but for the degree to be well-defined a choice of such an embedding has to be explicitly specified. The arithmetic of a complete smooth curve over anumber field (in particular number and structure of its rational points) is governed by the classification of the associated curvebase changed to an algebraic closure. SeeFaltings's theorem for details on the arithmetic implications.

3.  Classification of complete smoothsurfaces over an algebraically closed field up to rational equivalence. See anoverview of the classification orEnriques–Kodaira classification for details.

4.  Classification ofsingularities resp. associatedZariski neighboorhoods over algebraically closed fields up to isomorphism. (a) In characteristic 0Hironaka's resolution result attaches invariants to a singularity which classify them. (b) For curves and surfaces resolution is known in any characteristic which also yields a classification. Seehere for curves orhere for curves and surfaces.

5.  Classification ofFano varieties in small dimension.

6.  Theminimal model program is an approach to birational classification of complete smooth varieties in higher dimension (at least 2). While the original goal is about smooth varieties,terminal singularites naturally appear and are part of a wider classification.

7.  Classification ofsplit reductive groups up to isomorphism over algebraically closed fields.

classifying stack

An analog of aclassifying space fortorsors in algebraic geometry; seeclassifying stack.

closed

Closed subschemes of a schemeX are defined to be those occurring in the following construction. LetJ be aquasi-coherent sheaf of${\displaystyle {\mathcal{O}}_X}$-ideals. Thesupport of thequotient sheaf${\displaystyle {\mathcal{O}}_X/J}$ is a closed subsetZ ofX and${\displaystyle (Z,({\mathcal{O}}_X/J){|}_Z)}$ is a scheme called theclosed subscheme defined by the quasi-coherent sheaf of idealsJ.^[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-Macaulayif all local rings areCohen-Macaulay.For example, regular schemes, andSpec k[x,y]/(xy) are Cohen–Macaulay, but is not.

coherent sheaf

Acoherent sheaf on a Noetherian schemeX is a quasi-coherent sheaf that is finitely generated asO_X-module.

conic

Analgebraic curve of degree two.

connected

The scheme isconnected as a topological space. Since theconnected components refine theirreducible components any irreducible scheme is connected but not vice versa. Anaffine schemeSpec(R) is connectediff the ringR possesses noidempotents other than 0 and 1; such a ring is also called aconnected ring.Examples of connected schemes includeaffine space,projective space, and an example of a scheme that is not connected isSpec(k[x]×k[x])

compactification

See for exampleNagata's compactification theorem.

Cox ring

A generalization of a homogeneous coordinate ring. SeeCox ring.

crepant

Acrepant morphism${\displaystyle f:X\to Y}$ between normal varieties is a morphism such that${\displaystyle f^{\ast }{\omega }_Y={\omega }_X}$.

curve

An algebraic variety of dimension one.

deformation

Let${\displaystyle S\to S^{\prime }}$ be a morphism of schemes andX anS-scheme. Then a deformationX' ofX is anS'-scheme together with a pullback square in whichX is the pullback ofX' (typicallyX' is assumed to beflat).

degeneracy locus

Given a vector-bundle map${\displaystyle f:E\to F}$ over a varietyX (that is, a schemeX-morphism between the total spaces of the bundles), thedegeneracy locus is the (scheme-theoretic) locus${\displaystyle X_k(f)=\{x\in X|\mathrm{rk}(f(x))\le k\}}$.

degeneration

1.  A schemeX is said todegenerate to a scheme${\displaystyle X_0}$ (called the limit ofX) if there is a scheme${\displaystyle \pi :Y\to {\mathbf{A}}^1}$ withgeneric fiberX andspecial fiber${\displaystyle X_0}$.

2.  Aflat degeneration is a degeneration such that${\displaystyle \pi }$ is flat.

dimension

Thedimension, by definition the maximal length of a chain of irreducible closed subschemes, is a global property. It can be seen locally if a scheme is irreducible. It depends only on the topology, not on the structure sheaf. See alsoGlobal dimension.Examples:equidimensional schemes in dimension 0:Artinian schemes, 1:algebraic curves, 2:algebraic surfaces.

degree

1.  The degree of a line bundleL on a complete variety is an integerd such that${\displaystyle \chi (L^{\otimes m})=\frac{d}{n!}{m}^n+O(m^{n-1})}$.

2.  Ifx is a cycle on a complete variety${\displaystyle f:X\to \mathrm{Spec}k}$ over a fieldk, then its degree is${\displaystyle f_{\ast }(x)\in A_0(\mathrm{Spec}k)=\mathbb{Z}}$.

3.  For the degree of a finite morphism, seemorphism of varieties#Degree of a finite morphism.

derived algebraic geometry

An approach to algebraic geometry using (commutative)ring spectra instead ofcommutative rings; seederived algebraic geometry.

divisorial

1.  Adivisorial sheaf on a normal variety is a reflexive sheaf of the formO_X(D) for someWeil divisorD.

2.  Adivisorial scheme is a scheme admitting an ample family of invertible sheaves. A scheme admitting an ample invertible sheaf is a basic example.

dominant

A morphismf :X →Y is calleddominant, if the imagef(X) isdense. A morphism of affine schemesSpec A →Spec B is dense if and only if the kernel of the corresponding mapB →A is contained in the nilradical ofB.

dualizing complex

SeeCoherent duality.

dualizing sheaf

On a projectiveCohen–Macaulay scheme of pure dimensionn, thedualizing sheaf is a coherent sheaf${\displaystyle \omega }$ onX such that${\displaystyle H^{n-i}(X,F^{\vee }\otimes \omega )\simeq H^i(X,F{)}^{\ast }}$holds for any locally free sheafF onX; for example, ifX is a smooth projective variety, then it is acanonical sheaf.

Éléments de géométrie algébrique

TheEGA was an incomplete attempt to lay a foundation of algebraic geometry based on the notion ofscheme, 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

Anelliptic curve is a smoothprojective curve of genus one.

essentially of finite type

Localization of a finite type scheme.

étale

A morphismf :Y →X isétale if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties${\displaystyle X}$ and${\displaystyle Y}$ over an algebraically closedfield, étale morphisms are precisely those inducing an isomorphism of tangent spaces${\displaystyle df:T_{y}Y\to T_{f(y)}X}$, 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-calledétale topology and consequently theétale cohomology, which is nowadays one of the cornerstones of algebraic geometry.

Euler sequence

The exact sequence of sheaves:

${\displaystyle 0\to {\mathcal{O}}_{{\mathbf{P}}^n}\to {\mathcal{O}}_{{\mathbf{P}}^n}(1{)}^{\oplus (n+1)}\to T{\mathbf{P}}^n\to 0,}$

wherePⁿ is the projective space over a field and the last nonzero term is thetangent sheaf, is called theEuler sequence.

equivariant intersection theory

See Chapter II ofhttp://www.math.ubc.ca/~behrend/cet.pdf

F-regular

Related toFrobenius morphism.^[7]

Fano

AFano variety is a smoothprojective varietyX whose anticanonical sheaf${\displaystyle {\omega }_X^{-1}}$ is ample.

fiber

Given${\displaystyle f:X\to Y}$ between schemes, the fiber off overy is, as a set, the pre-image${\displaystyle f^{-1}(y)=\{x\in X|f(x)=y\}}$; it has the natural structure of a scheme over theresidue field ofy as the fiber product${\displaystyle X{\times }_Y\{y\}}$, where${\displaystyle \{y\}}$ has the natural structure of a scheme overY as Spec of the residue field ofy.

fiber product

1.  Another term for the "pullback" in the category theory.

2.  A stack${\displaystyle F{\times }_{G}H}$ given for${\displaystyle f:F\to G,g:H\to G}$: an object overB is a triple (x,y, ψ),x inF(B),y inH(B), ψ an isomorphism${\displaystyle f(x)\stackrel{\sim }{\to }g(y)}$ inG(B); an arrow from (x,y, ψ) to (x',y', ψ') is a pair of morphisms${\displaystyle \alpha :x\to x^{\prime },\beta :y\to y^{\prime }}$ such that${\displaystyle {\psi }^{\prime }\circ f(\alpha )=g(\beta )\circ \psi }$. The resulting square with obvious projectionsdoes not commute; rather, it commutes up to natural isomorphism; i.e., it2-commutes.

final

One of Grothendieck's fundamental ideas is to emphasizerelative notions, i.e. conditions on morphisms rather than conditions on schemes themselves. The category of schemes has afinal object, the spectrum of the ring${\displaystyle \mathbb{Z}}$ of integers; so that any scheme${\displaystyle S}$ isover${\displaystyle \text{Spec}(\mathbb{Z})}$, and in a unique way.

finite

The morphismf :Y →X isfinite if${\displaystyle X}$ may be covered by affine open sets${\displaystyle \text{Spec }B}$ such that each${\displaystyle f^{-1}(\text{Spec }B)}$ is affine — say of the form${\displaystyle \text{Spec }A}$ — and furthermore${\displaystyle A}$ is finitely generated as a${\displaystyle B}$-module. Seefinite 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 morphismf :Y →X islocally of finite type if${\displaystyle X}$ may be covered by affine open sets${\displaystyle \text{Spec }B}$ such that each inverse image${\displaystyle f^{-1}(\text{Spec }B)}$ is covered by affine open sets${\displaystyle \text{Spec }A}$ where each${\displaystyle A}$ is finitely generated as a${\displaystyle B}$-algebra.The morphismf :Y →X isof finite type if${\displaystyle X}$ may be covered by affine open sets${\displaystyle \text{Spec }B}$ such that each inverse image${\displaystyle f^{-1}(\text{Spec }B)}$ is covered by finitely many affine open sets${\displaystyle \text{Spec }A}$ where each${\displaystyle A}$ is finitely generated as a${\displaystyle B}$-algebra.

finite fibers

The morphismf :Y →X hasfinite fibers if the fiber over each point${\displaystyle x\in X}$ is a finite set. A morphism isquasi-finite if it is of finite type and has finite fibers.

finite presentation

Ify is a point ofY, then the morphismf isof finite presentation aty (orfinitely presented aty) if there is an open affine neighborhoodU off(y) and an open affine neighbourhoodV ofy such thatf(V) ⊆ U and${\displaystyle {\mathcal{O}}_Y(V)}$ is afinitely presented algebra over${\displaystyle {\mathcal{O}}_X(U)}$. The morphismf islocally of finite presentation if it is finitely presented at all points ofY. IfX is locally Noetherian, thenf is locally of finite presentation if, and only if, it is locally of finite type.^[8]The morphismf :Y →X isof finite presentation (orY is finitely presented overX) if it is locally of finite presentation, quasi-compact, and quasi-separated. IfX is locally Noetherian, thenf is of finite presentation if, and only if, it is of finite type.^[9]

flag variety

Theflag variety parametrizes aflag of vector spaces.

flat

A morphism${\displaystyle f}$ isflat if it gives rise to aflat map on stalks. When viewing a morphismf :Y →X as a family of schemes parametrized by the points of${\displaystyle X}$, the geometric meaning of flatness could roughly be described by saying that the fibers${\displaystyle f^{-1}(x)}$ do not vary too wildly.

formal

Seeformal scheme.

g^r_d

Given a curveC, a divisorD on it and a vector subspace${\displaystyle V\subset H^0(C,\mathcal{O}(D))}$, one says the linear system${\displaystyle \mathbb{P}(V)}$ is a g^r_d ifV has dimensionr+1 andD has degreed. One saysC has a g^r_d if there is such a linear system.

Gabriel–Rosenberg reconstruction theorem

TheGabriel–Rosenberg reconstruction theorem states a schemeX can be recovered from the category ofquasi-coherent sheaves onX.^[10] The theorem is a starting point fornoncommutative algebraic geometry since, taking the theorem as an axiom, defining anoncommutative scheme amounts to defining the category of quasi-coherent sheaves on it. See alsohttps://mathoverflow.net/q/16257

G-bundle

A principal G-bundle.

generic point

A dense point.

genus

See#arithmetic genus,#geometric genus.

genus formula

Thegenus formula for a nodal curve in the projective plane says the genus of the curve is given as${\displaystyle g=(d-1)(d-2)/2-\delta }$whered is the degree of the curve and δ is the number of nodes (which is zero if the curve is smooth).

geometric genus

Thegeometric genus of a smooth projective varietyX of dimensionn is${\displaystyle \dim \mathrm{\Gamma }(X,{\mathrm{\Omega }}_X^n)=\dim {\mathrm{H}}^n(X,{\mathcal{O}}_X)}$(where the equality isSerre's duality theorem.)

geometric point

The prime spectrum of an algebraically closed field.

geometric property

A property of a schemeX over a fieldk is "geometric" if it holds for${\displaystyle X_E=X{\times }_{\mathrm{Spec}k}\mathrm{Spec}E}$ for any field extension${\displaystyle E/k}$.

geometric quotient

Thegeometric quotient of a schemeX with the action of a group schemeG is a good quotient such that the fibers are orbits.

gerbe

Agerbe is (roughly) astack that is locally nonempty and in which two objects are locally isomorphic.

GIT quotient

TheGIT quotient${\displaystyle X//G}$ is${\displaystyle \mathrm{Spec}(A^G)}$ when${\displaystyle X=\mathrm{Spec}A}$ and${\displaystyle \mathrm{Proj}(A^G)}$ when${\displaystyle X=\mathrm{Proj}A}$.

good quotient

Thegood quotient of a schemeX with the action of a group schemeG is an invariant morphism${\displaystyle f:X\to Y}$ such that${\displaystyle (f_{\ast }{\mathcal{O}}_X{)}^G={\mathcal{O}}_Y.}$

Gorenstein

1.  AGorenstein scheme is a locally Noetherian scheme whose local rings areGorenstein rings.

2.  A normal variety is said to be${\displaystyle \mathbb{Q}}$-Gorenstein if the canonical divisor on it is${\displaystyle \mathbb{Q}}$-Cartier (and need not be Cohen–Macaulay).

3.  Some authors call a normal variety Gorenstein if the canonical divisor is Cartier; note this usage is inconsistent with meaning 1.

Grauert–Riemenschneider vanishing theorem

TheGrauert–Riemenschneider vanishing theorem extends theKodaira vanishing theorem to higher direct image sheaves; see alsohttps://arxiv.org/abs/1404.1827

Grothendieck ring of varieties

TheGrothendieck ring of varieties is the free abelian group generated by isomorphism classes of varieties with the relation:${\displaystyle [X]=[Z]+[X-Z]}$whereZ is a closed subvariety of a varietyX and equipped with the multiplication${\displaystyle [X]\cdot [Y]=[X\times Y].}$

Grothendieck's vanishing theorem

Grothendieck's vanishing theorem concernslocal cohomology.

group scheme

Agroup scheme is a scheme whose sets of points have the structures of agroup.

group variety

An old term for a "smooth" algebraic group.

Hilbert polynomial

TheHilbert polynomial of a projective schemeX over a field is the Euler characteristic${\displaystyle \chi ({\mathcal{O}}_X(s))}$.

Hodge bundle

TheHodge bundle on themoduli space of curves (of fixed genus) is roughly a vector bundle whose fiber over a curveC is the vector space${\displaystyle \mathrm{\Gamma }(C,{\omega }_C)}$.

hyperelliptic

A curve ishyperelliptic if it has ag¹₂ (i.e., there is a linear system of dimension 1 and degree 2.)

hyperplane bundle

Another term forSerre's twisting sheaf${\displaystyle {\mathcal{O}}_X(1)}$. It is the dual of thetautological line bundle (whence the term).

image

Iff :Y →X is any morphism of schemes, thescheme-theoretic image off is the uniqueclosed subschemei :Z →X which satisfies the followinguniversal property:

1. f factors throughi,
2. ifj :Z′ →X is any closed subscheme ofX such thatf factors throughj, theni also factors throughj.^[11][12]

This notion is distinct from that of the usual set-theoretic image off,f(Y). For example, the underlying space ofZ always contains (but is not necessarily equal to) the Zariski closure off(Y) inX, so ifY is any open (and not closed) subscheme ofX andf is the inclusion map, thenZ is different fromf(Y). WhenY is reduced, thenZ is the Zariski closure off(Y) endowed with the structure of reduced closed subscheme. But in general, unlessf is quasi-compact, the construction ofZ is not local onX.

immersion

Immersionsf :Y →X are maps that factor through isomorphisms with subschemes. Specifically, anopen immersion factors through an isomorphism with an open subscheme and aclosed 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 ofY to a closed subset of the underlying topological space ofX, and if the morphism${\displaystyle f^{\mathrm{♯}}:{\mathcal{O}}_X\to f_{\ast }{\mathcal{O}}_Y}$ is surjective.^[14] A composition of immersions is again an immersion.^[15]Some authors, such as Hartshorne in his bookAlgebraic Geometry and Q. Liu in his bookAlgebraic 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 whenf 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:${\displaystyle \mathrm{Spec}A/I}$ and${\displaystyle \mathrm{Spec}A/J}$ may be homeomorphic but not isomorphic. This happens, for example, ifI is the radical ofJ butJ is not a radical ideal. When specifying a closed subset of a scheme without mentioning the scheme structure, usually the so-calledreduced 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

Anind-scheme is aninductive limit of closed immersions of schemes.

invertible sheaf

A locally free sheaf of a rank one. Equivalently, it is atorsor for the multiplicative group${\displaystyle {\mathbb{G}}_m}$ (i.e., line bundle).

integral

A scheme that is both reduced and irreducible is calledintegral. For locally Noetherian schemes, to be integral is equivalent to being a connected scheme that is covered by the spectra ofintegral domains. (Strictly speaking, this is not a local property, because thedisjoint union of two integral schemes is not integral. However, for irreducible schemes, it is a local property.)For example, the schemeSpec k[t]/f,firreducible polynomial is integral, whileSpec A×B (A,B ≠ 0) is not.

irreducible

A schemeX is said to beirreducible when (as a topological space) it is not the union of two closed subsets except if one is equal toX. Using the correspondence of prime ideals and points in an affine scheme, this meansX is irreducibleiffX is connected and the rings A_i all have exactly one minimalprime ideal. (Rings possessing exactly one minimal prime ideal are therefore also calledirreducible.) Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called itsirreducible components.Affine space andprojective space are irreducible, whileSpeck[x,y]/(xy) = is not.

Jacobian variety

TheJacobian variety of a projective curveX is the degree zero part of thePicard variety${\displaystyle \mathrm{Pic}(X)}$.

Kempf vanishing theorem

TheKempf vanishing theorem concerns the vanishing of higher cohomology of a flag variety.

klt

Abbreviation for "kawamata log terminal"

Kodaira dimension

1.  TheKodaira dimension (also called theIitaka dimension) of a semi-ample line bundleL is the dimension of Proj of the section ring ofL.

2.  The Kodaira dimension of a normal varietyX is the Kodaira dimension of its canonical sheaf.

Kodaira vanishing theorem

See theKodaira vanishing theorem.

Kuranishi map

SeeKuranishi structure.

Lelong number

SeeLelong number.

level structure

seehttp://math.stanford.edu/~conrad/248BPage/handouts/level.pdf

linearization

Another term for the structure of anequivariant sheaf/vector bundle.

local

Most important properties of schemes arelocal in nature, i.e. a schemeX has a certain propertyP if and only if for any cover ofX by open subschemesX_i, i.e.X=${\displaystyle \cup }$X_i, everyX_i has the propertyP. It is usually the case that it is enough to check one cover, not all possible ones. One also says that a certain property isZariski-local, if one needs to distinguish between theZariski topology and other possible topologies, like theétale topology.Consider a schemeX and a cover by affine open subschemesSpec A_i. Using the dictionary between(commutative) rings andaffine schemes local properties are thus properties of the ringsA_i. A propertyP is local in the above sense, iff the corresponding property of rings is stable underlocalization.For example, we can speak oflocally Noetherian schemes, namely those which are covered by the spectra ofNoetherian 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 isreduced (i.e., has no non-zeronilpotent elements), then so are its localizations.An example for a non-local property isseparatedness (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. LetX =${\displaystyle \cup }$Spec A_i be a covering of a scheme by open affine subschemes. For definiteness, letk denote afield in the following. Most of the examples also work with the integersZ as a base, though, or even more general bases.Connected,irreducible,reduced,integral,normal,regular,Cohen-Macaulay,locally noetherian,dimension, catenary,Gorenstein.

local complete intersection

The local rings arecomplete intersection rings. See also:regular embedding.

local uniformization

Thelocal uniformization is a method of constructing a weaker form ofresolution of singularities by means ofvaluation rings.

locally factorial

The local rings areunique factorization domains.

locally of finite presentation

Cf.finite presentation above.

locally of finite type

The morphismf :Y →X islocally of finite type if${\displaystyle X}$ may be covered by affine open sets${\displaystyle \text{Spec }B}$ such that each inverse image${\displaystyle f^{-1}(\text{Spec }B)}$ is covered by affine open sets${\displaystyle \text{Spec }A}$ where each${\displaystyle A}$ is finitely generated as a${\displaystyle B}$-algebra.

locally Noetherian

A schemeX covered by SpecA_i, where theA_i areNoetherian rings. If in addition a finite number of such affine spectra coversX, the scheme is callednoetherian. While it is true that the spectrum of a noetherian ring is anoetherian topological space, the converse is false. For example, most schemes in finite-dimensional algebraic geometry are locally Noetherian, but${\displaystyle G{L}_{\mathrm{\infty }}=\cup G{L}_n}$ is not.

logarithmic geometry

log structure

Seelog structure. The notion is due to Fontaine-Illusie and Kato.

loop group

Seeloop group (the linked article does not discuss a loop group in algebraic geometry; for now see alsoind-scheme).

moduli

See for examplemoduli 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

Theminimal model program is aresearch program aiming to dobirational classification of algebraic varieties of dimension greater than 2.

morphism

1.  Amorphism of algebraic varieties is given locally by polynomials.

2.  Amorphism of schemes is a morphism oflocally ringed spaces.

3.  A morphism${\displaystyle f:F\to G}$ of stacks (over, say, the category ofS-schemes) is a functor such that${\displaystyle P_G\circ f=P_F}$ where${\displaystyle P_F,P_G}$ are structure maps to the base category.

nef

Seenef line bundle.

nonsingular

An archaic term for "smooth" as in asmooth variety.

normal

1.  An integral scheme is callednormal, if the local rings areintegrally closed domains. For example, all regular schemes are normal, while singular curves are not.

2.  A smooth curve${\displaystyle C\subset {\mathbf{P}}^r}$ is said to bek-normal if the hypersurfaces of degreek cut out the complete linear series${\displaystyle |{\mathcal{O}}_C(k)|}$. It isprojectively normal if it isk-normal for allk > 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${\displaystyle X\subset {\mathbf{P}}^r}$ is said to be projectively normal if theaffine cover overX is anormal scheme; i.e., the homogeneous coordinate ring ofX is an integrally closed domain. This meaning is consistent with that of 2.

normal

1.  IfX is a closed subscheme of a schemeY with ideal sheafI, then thenormal sheaf toX is${\displaystyle (I/I^2{)}^{\ast }=\mathcal{H}o{m}_{{\mathcal{O}}_Y}(I/I^2,{\mathcal{O}}_Y)}$. If the embedded ofX intoY isregular, it is locally free and is called thenormal bundle.

2.  Thenormal cone toX is${\displaystyle {\mathrm{Spec}}_X({\oplus }_0^{\mathrm{\infty }}{I}^n/I^{n+1})}$. ifX is regularly embedded intoY, then the normal cone is isomorphic to${\displaystyle {\mathrm{Spec}}_X(\mathcal{S}ym(I/I^2))}$, the total space of the normal bundle toX.

normal crossings

Abbreviationsnc for normal crossing andsnc for simple normal crossing. Refers to several closely related notions such as nc divisor, nc singularity, snc divisor, and snc singularity. Seenormal crossings.

normally generated

A line bundleL on a varietyX is said to benormally generated if, for each integern > 0, the natural map${\displaystyle \mathrm{\Gamma }(X,L{)}^{\otimes n}\to \mathrm{\Gamma }(X,L^{\otimes n})}$ is surjective.

open

1.  A morphismf :Y →X of schemes is calledopen (closed), if the underlying map of topological spaces isopen (closed, respectively), i.e. if open subschemes ofY are mapped to open subschemes ofX (and similarly for closed). For example, finitely presented flat morphisms are open and proper maps are closed.

2.  Anopen subscheme of a schemeX is an open subsetU with structure sheaf${\displaystyle {\mathcal{O}}_X{|}_U}$.^[14]

orbifold

Nowadays anorbifold is often defined as aDeligne–Mumford stack over the category of differentiable manifolds.^[17]

p-divisible group

Seep-divisible group (roughly an analog of torsion points of an abelian variety).

pencil

A linear system of dimension one.

Picard group

ThePicard group ofX is the group of the isomorphism classes of line bundles onX, the multiplication being thetensor product.

Plücker embedding

ThePlücker embedding is theclosed embedding of theGrassmannian variety into a projective space.

plurigenus

Then-thplurigenus of a smooth projective variety is${\displaystyle \dim \mathrm{\Gamma }(X,{\omega }_X^{\otimes n})}$. See alsoHodge number.

Poincaré residue map

SeePoincaré residue.

point

A scheme${\displaystyle S}$ is alocally ringed space, soa fortiori atopological space, but the meanings ofpoint of${\displaystyle S}$ are threefold:

1. a point${\displaystyle P}$ of the underlying topological space;
2. a${\displaystyle T}$-valued point of${\displaystyle S}$ is a morphism from${\displaystyle T}$ to${\displaystyle S}$, for any scheme${\displaystyle T}$;
3. ageometric point, where${\displaystyle S}$ is defined over (is equipped with a morphism to)${\displaystyle \text{Spec}(K)}$, where${\displaystyle K}$ is afield, is a morphism from${\displaystyle \text{Spec}(\overline{K})}$ to${\displaystyle S}$ where${\displaystyle \overline{K}}$ is analgebraic closure of${\displaystyle K}$.

Geometric points are what in the most classical cases, for examplealgebraic varieties that arecomplex manifolds, would be the ordinary-sense points. The points${\displaystyle P}$ of the underlying space include analogues of thegeneric points (in the sense ofZariski, not that ofAndré Weil), which specialise to ordinary-sense points. The${\displaystyle T}$-valued points are thought of, viaYoneda's lemma, as a way of identifying${\displaystyle S}$ with therepresentable functor${\displaystyle h_S}$ it sets up. Historically there was a process by whichprojective geometry added more points (e.g. complex points,line at infinity) to simplify the geometry by refining the basic objects. The${\displaystyle T}$-valued points were a massive further step.As part of the predominatingGrothendieck approach, there are three corresponding notions offiber of a morphism: the first being the simpleinverse image of a point. The other two are formed by creatingfiber products of two morphisms. For example, ageometric fiber of a morphism${\displaystyle S^{\mathrm{\prime }}\to S}$ is thought of as${\displaystyle S^{\mathrm{\prime }}{\times }_S\text{Spec}(\overline{K})}$.This makes the extension fromaffine schemes, where it is just thetensor product of R-algebras, to all schemes of the fiber product operation a significant (if technically anodyne) result.

polarization

an embedding into a projective space

Proj

SeeProj construction.

projection formula

Theprojection formula says that, for a morphism${\displaystyle f:X\to Y}$ of schemes, an${\displaystyle {\mathcal{O}}_X}$-module${\displaystyle \mathcal{F}}$ and alocally free${\displaystyle {\mathcal{O}}_Y}$-module${\displaystyle \mathcal{E}}$ of finite rank, there is a natural isomorphism${\displaystyle f_{\ast }(F\otimes f^{\ast }E)=(f_{\ast }F)\otimes E}$(in short,${\displaystyle f_{\ast }}$ is linear with respect to the action of locally free sheaves.)

projective

1.  Aprojective variety is a closed subvariety of a projective space.

2.  Aprojective scheme over a schemeS is anS-scheme that factors through some projective space${\displaystyle {\mathbf{P}}_S^N\to S}$ as a closed subscheme.

3.  Projective morphisms are defined similarly to affine morphisms:f :Y →X is calledprojective if it factors as a closed immersion followed by the projection of aprojective space${\displaystyle {\mathbb{P}}_X^n:={\mathbb{P}}^n{\times }_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathbb{Z}}X}$ to${\displaystyle X}$.^[18] Note that this definition is more restrictive than that ofEGA, II.5.5.2. The latter defines${\displaystyle f}$ to be projective if it is given by theglobalProj of a quasi-coherentgradedO_X-algebra${\displaystyle \mathcal{S}}$ such that${\displaystyle {\mathcal{S}}_1}$ is finitely generated and generates the algebra${\displaystyle \mathcal{S}}$. Both definitions coincide when${\displaystyle X}$ is affine or more generally if it is quasi-compact, separated and admits an ample sheaf,^[19] e.g. if${\displaystyle X}$ is an open subscheme of a projective space${\displaystyle {\mathbb{P}}_A^n}$ over a ring${\displaystyle A}$.

projective bundle

IfE is a locally free sheaf on a schemeX, theprojective bundleP(E) ofE is theglobal Proj of the symmetric algebra of the dual ofE:${\displaystyle \mathbf{P}(E)=\mathbf{P}\mathbf{r}\mathbf{o}\mathbf{j}({\mathrm{Sym}}_{{\mathcal{O}}_X}(E^{\vee })).}$Note this definition is standard nowadays (e.g., Fulton'sIntersection theory) but differs from EGA and Hartshorne (they don't take a dual).

projectively normal

See#normal.

proper

A morphism isproper 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 alsocomplete variety. A deep property of proper morphisms is the existence of aStein 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

LetP be a property of a scheme that is stable under base change (finite-type, proper, smooth, étale, etc.). Then a representable morphism${\displaystyle f:F\to G}$ is said to have propertyP if, for any${\displaystyle B\to G}$ withB a scheme, the base change${\displaystyle F{\times }_{G}B\to B}$ has propertyP.

pseudo-reductive

Pseudoreductive generalizesreductive in the context ofconnectedsmoothlinear algebraic group.

pure dimension

A scheme has pure dimensiond if each irreducible component has dimensiond.

quasi-coherent

A quasi-coherent sheaf on a Noetherian schemeX is asheaf ofO_X-modules that is locally given by modules.

quasi-compact

A morphismf :Y →X is calledquasi-compact, if for some (equivalently: every) open affine cover ofX by someU_i =Spec B_i, the preimagesf⁻¹(U_i) arequasi-compact.

quasi-finite

The morphismf :Y →X hasfinite fibers if the fiber over each point${\displaystyle x\in X}$ is a finite set. A morphism isquasi-finite if it is of finite type and has finite fibers.

quasi-projective

Aquasi-projective variety is alocally closed subvariety of a projective space.

quasi-separated

A morphismf :Y →X is calledquasi-separated or (Y is quasi-separated overX) if the diagonal morphismY →Y ×_XY is quasi-compact. A schemeY is calledquasi-separated ifY is quasi-separated over Spec(Z).^[20]

quasi-split

Areductive group${\displaystyle G}$ defined over a field${\displaystyle k}$ isquasi-split if and only if it admits a Borel subgroup${\displaystyle B\subseteq G}$ defined over${\displaystyle k}$. Any quasi-split reductive group is a split-reductive reductive group, but there are quasi-split reductive groups that are not split-reductive.

Quot scheme

AQuot scheme parametrizes quotients of locally free sheaves on a projective scheme.

quotient stack

Usually denoted by [X/G], aquotient stack generalizes a quotient of a scheme or variety.

rational

1.  Over an algebraically closed field, a variety isrational if it is birational to a projective space. For example,rational curves andrational surfaces are those birational to${\displaystyle {\mathbb{P}}^1,{\mathbb{P}}^2}$.

2.  Given a fieldk and a relative schemeX →S, ak-rational point ofX is anS-morphism${\displaystyle \mathrm{Spec}(k)\to X}$.

rational function

An element in thefunction field${\displaystyle k(X)=\underrightarrow{\lim }k[U]}$ where the limit runs over all coordinates rings of open subsetsU of an (irreducible) algebraic varietyX. See alsofunction field (scheme theory).

rational normal curve

Arational normal curve is the image of${\displaystyle {\mathbf{P}}^1\to {\mathbf{P}}^d,(s:t)\mapsto (s^d:s^{d-1}t:\cdots :t^d)}$.Ifd = 3, it is also called thetwisted cubic.

rational singularities

A varietyX over a field of characteristic zero hasrational singularities if there is a resolution of singularities${\displaystyle f:X^{\prime }\to X}$ such that${\displaystyle f_{\ast }({\mathcal{O}}_{X^{\prime }})={\mathcal{O}}_X}$ and${\displaystyle R^i{f}_{\ast }({\mathcal{O}}_{X^{\prime }})=0,i\ge 1}$.

reduced

1.  A commutative ring${\displaystyle R}$ isreduced if it has no nonzero nilpotent elements, i.e., its nilradical is the zero ideal,${\displaystyle \sqrt{(0)}=(0)}$. Equivalently,${\displaystyle R}$ is reduced if${\displaystyle \mathrm{Spec}(R)}$ is a reduced scheme.

2.  A scheme X is reduced if its stalks${\displaystyle {\mathcal{O}}_{X,x}}$ are reduced rings. Equivalently X is reduced if, for each open subset${\displaystyle U\subset X}$,${\displaystyle {\mathcal{O}}_X(U)}$ is a reduced ring, i.e.,${\displaystyle X}$ has no nonzero nilpotent sections.

reductive

Aconnectedlinear algebraic group${\displaystyle G}$ over a field${\displaystyle k}$ is areductive group if and only if the unipotent radical${\displaystyle R_u(G_{\overline{k}})}$ of the base change${\displaystyle G_{\overline{k}}}$ of${\displaystyle G}$ to an algebraic closure${\displaystyle \overline{k}}$ is trivial.

reflexive sheaf

A coherent sheaf isreflexive if the canonical map to the second dual is an isomorphism.

regular

Aregular scheme is a scheme where the local rings areregular local rings. For example,smooth varieties over a field are regular, whileSpec k[x,y]/(x²+x³-y²)= is not.

regular embedding

Aclosed immersion${\displaystyle i:X\hookrightarrow Y}$ is aregular embedding if each point ofX has an affine neighborhood inY so that the ideal ofX there is generated by aregular sequence. Ifi is a regular embedding, then theconormal sheaf ofi, that is,${\displaystyle \mathcal{I}/{\mathcal{I}}^2}$ when${\displaystyle \mathcal{I}}$ is the ideal sheaf ofX, is locally free.

regular function

Amorphism from an algebraic variety to theaffine line.

representable morphism

A morphism${\displaystyle F\to G}$ of stacks such that, for any morphism${\displaystyle B\to G}$ from a schemeB, the base change${\displaystyle F{\times }_{G}B}$ is an algebraic space. If "algebraic space" is replaced by "scheme", then it is said to be strongly representable.

resolution of singularities

Aresolution of singularities of a schemeX is a properbirational morphism${\displaystyle \pi :Z\to X}$ such thatZ issmooth.

Riemann–Hurwitz formula

Given a finite separable morphism${\displaystyle \pi :X\to Y}$ between smooth projective curves, if${\displaystyle \pi }$ istamely ramified (no wild ramification), for example, over a field of characteristic zero, then theRiemann–Hurwitz formula relates the degree of π, the genera ofX,Y and theramification indices:${\displaystyle 2g(X)-2=\mathrm{deg}(\pi )(2g(Y)-2)+\sum _{y\in Y}(e_y-1)}$.Nowadays, the formula is viewed as a consequence of the more general formula (which is valid even if π is not tame):${\displaystyle K_X\sim {\pi }^{\ast }{K}_Y+R}$where${\displaystyle \sim }$ means alinear equivalence and${\displaystyle R=\sum _{P\in X}{\mathrm{length}}_{{\mathcal{O}}_P}({\mathrm{\Omega }}_{X/Y})P}$ is the divisor of the relative cotangent sheaf${\displaystyle {\mathrm{\Omega }}_{X/Y}}$ (called thedifferent).

Riemann–Roch formula

1.  IfL is a line bundle of degreed on a smooth projective curve of genusg, then theRiemann–Roch formula computes theEuler characteristic ofL:${\displaystyle \chi (L)=d-g+1}$.For example, the formula implies the degree of the canonical divisorK is 2g - 2.

2.  The general version is due to Grothendieck and called theGrothendieck–Riemann–Roch formula. It says: if${\displaystyle \pi :X\to S}$ is a proper morphism with smoothX,S and ifE is a vector bundle onX, then as equality in the rationalChow group${\displaystyle \mathrm{ch}({\pi }_{!}E)\cdot \mathrm{td}(S)={\pi }_{\ast }(\mathrm{ch}(E)\cdot \mathrm{td}(X))}$where${\displaystyle {\pi }_{!}=\sum _i(-1{)}^i{R}^i{\pi }_{\ast }}$,${\displaystyle \mathrm{ch}}$ means aChern character and${\displaystyle \mathrm{td}}$ aTodd class of the tangent bundle of a space, and, over the complex numbers,${\displaystyle {\pi }_{\ast }}$ is anintegration along fibers. For example, if the baseS is a point,X is a smooth curve of genusg andE is a line bundleL, then the left-hand side reduces to the Euler characteristic while the right-hand side is${\displaystyle {\pi }_{\ast }(e^{c_1(L)}(1-c_1(T^{\ast }X)/2))=\mathrm{deg}(L)-g+1.}$

rigid

Everyinfinitesimal deformation is trivial. For example, theprojective space is rigid since${\displaystyle {\mathrm{H}}^1({\mathbf{P}}^n,T_{{\mathbf{P}}^n})=0}$ (and using theKodaira–Spencer map).

rigidify

A heuristic term, roughly equivalent to "killing automorphisms". For example, one might say "we introducelevel structures resp.marked points 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 ofÉléments de géométrie algébrique?

[1]

scheme

Ascheme is alocally ringed space that is locally aprime spectrum of acommutative ring.

Schubert

1.  ASchubert cell is aB-orbit on the Grassmannian${\displaystyle \mathrm{Gr}(d,n)}$ whereB is the standard Borel; i.e., the group of upper triangular matrices.

2.  ASchubert variety is the closure of a Schubert cell.

scroll

Arational normal scroll is aruled surface which is ofdegree${\displaystyle n}$ in aprojective space${\displaystyle {\mathbb{P}}^{n+1}}$ for some${\displaystyle n\in {\mathbb{N}}_{>1}}$.

secant variety

Thesecant variety to a projective variety${\displaystyle V\subset {\mathbb{P}}^r}$ is the closure of the union of all secant lines toV in${\displaystyle {\mathbb{P}}^r}$.

section ring

Thesection ring or the ring of sections of a line bundleL on a schemeX is the graded ring${\displaystyle {\oplus }_0^{\mathrm{\infty }}\mathrm{\Gamma }(X,L^n)}$.

Serre's conditionsS_n

SeeSerre's conditions on normality. See alsohttps://mathoverflow.net/q/22228

Serre duality

See#dualizing sheaf

separated

Aseparated morphism is a morphism${\displaystyle f}$ such that thefiber product of${\displaystyle f}$ with itself along${\displaystyle f}$ has itsdiagonal as a closed subscheme — in other words, thediagonal morphism is aclosed immersion.

sheaf generated by global sections

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

simple

1.  The term "simple point" is an old term for a "smooth point".

2.  Asimple normal crossing (snc) divisor is another name for a smooth normal crossing divisor, i.e. a divisor that has only smooth normal crossing singularities. They appear instrong desingularization as well as in stabilization for compactifying moduli problems.

3.  In the context oflinear algebraic groups there aresemisimple groups andsimple groups which are themselves semisimple groups with additional properties. Since all simple groups are reductive, a split simple group is a simple group that is split-reductive.

smooth

1.  

Main article:smooth morphism

The higher-dimensional analog of étale morphisms aresmooth morphisms. There are many different characterisations of smoothness. The following are equivalent definitions of smoothness of the morphismf :Y →X:

1. for anyy ∈Y, there are open affine neighborhoodsV andU ofy,x=f(y), respectively, such that the restriction off toV factors as an étale morphism followed by the projection ofaffinen-space overU.
2. f is flat, locally of finite presentation, and for every geometric point${\displaystyle \overline{y}}$ ofY (a morphism from the spectrum of an algebraically closed field${\displaystyle k(\overline{y})}$ toY), the geometric fiber${\displaystyle X_{\overline{y}}:=X{\times }_Y\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(k(\overline{y}))}$ is a smoothn-dimensional variety over${\displaystyle k(\overline{y})}$ in the sense of classical algebraic geometry.

2.  Asmooth scheme over a perfect fieldk is a schemeX that is locally of finite type andregular overk.

3.  A smooth scheme over a fieldk is a schemeX that is geometrically smooth:${\displaystyle X{\times }_k\overline{k}}$ is smooth.

special

A divisorD on a smooth curveC isspecial if${\displaystyle h^0(\mathcal{O}(K-D))}$, which is called the index of speciality, is positive.

spherical variety

Aspherical variety is a normalG-variety (G connected reductive) with an open dense orbit by a Borel subgroup ofG.

split

1.  In the context of analgebraic group${\displaystyle G}$ for certain properties${\displaystyle P}$ there is the derived property split-${\displaystyle P}$. Usually${\displaystyle P}$ is a property that is automatic or more common over algebraically closed fields${\displaystyle \overline{k}}$. If this property holds already for${\displaystyle G}$ defined over a not necessarily algebraically closed field${\displaystyle k}$ then${\displaystyle G}$ is said to satisfy split-${\displaystyle P}$.

2.  A linear algebraic group${\displaystyle G}$ defined over a field${\displaystyle k}$ is atorus if only if its base change${\displaystyle G_{\overline{k}}}$ to an algebraic closure${\displaystyle \overline{k}}$ is isomorphic to a product of multiplicative groups${\displaystyle G_{m,\overline{k}}^n}$.${\displaystyle G}$ is asplit torus if and only if it is isomorphic to${\displaystyle G_{m,k}^n}$ without any base change.${\displaystyle G}$ is said tosplit over an intermediate field${\displaystyle k\subseteq L\subseteq \overline{k}}$ if and only if its base change${\displaystyle G_L}$ to${\displaystyle L}$ is isomorphic to${\displaystyle G_{m,L}^n}$.

3.  Areductive group${\displaystyle G}$ defined over a field${\displaystyle k}$ issplit-reductive if and only if a maximal torus${\displaystyle T\subseteq G}$ defined over${\displaystyle k}$ is a split torus. Since anysimple group is reductive asplit simple group means a simple group that is split-reductive.

4.  Aconnectedsolvable linear algebraic group${\displaystyle G}$ defined over a field${\displaystyle k}$ issplit if and only if it hascomposition series${\displaystyle B=B_0\supset B_1\supset \dots \supset B_t=\{1\}}$ defined over${\displaystyle k}$ such that each successive quotient${\displaystyle B_i/B_{i+1}}$ is isomorphic to either the multiplicative group${\displaystyle G_{m,k}}$ or the additive group${\displaystyle G_{m,a}}$ over${\displaystyle k}$.

5.  A linear algebraic group${\displaystyle G}$ defined over a field${\displaystyle k}$ issplit if and only if it has aBorel subgroup${\displaystyle B\subseteq G}$ defined over${\displaystyle k}$ that is split in the sense of connected solvable linear algebraic groups.

6.  In the classification ofreal Lie algebrassplit Lie algebras play an important role. There is a close connection between linear Lie groups, their associated Lie algebras and linear algebraic groups over${\displaystyle k=\mathbb{R}}$ resp.${\displaystyle \mathbb{C}}$. The termsplit has similar meanings for Lie theory and linear algebraic groups.

stable

1.  Astable curve is a curve with some "mild" singularity, used to construct a good-behavingmoduli space of curves.

2.  Astable vector bundle is used to construct themoduli space of vector bundles.

stack

Astack parametrizes sets of points together with automorphisms.

strict transform

Given a blow-up${\displaystyle \pi :\tilde{X}\to X}$ along a closed subschemeZ and a morphism${\displaystyle f:Y\to X}$, thestrict transform ofY (also called proper transform) is theblow-up${\displaystyle \tilde{Y}\to Y}$ ofY along the closed subscheme${\displaystyle f^{-1}Z}$. Iff is a closed immersion, then the induced map${\displaystyle \tilde{Y}\hookrightarrow \tilde{X}}$ is also a closed immersion.

subscheme

Asubscheme, without qualifier, ofX is a closed subscheme of an open subscheme ofX.

surface

An algebraic variety of dimension two.

symmetric variety

An analog of asymmetric space. Seesymmetric variety.

tangent space

SeeZariski tangent space.

tautological line bundle

Thetautological line bundle of a projective schemeX is the dual ofSerre's twisting sheaf${\displaystyle {\mathcal{O}}_X(1)}$; that is,${\displaystyle {\mathcal{O}}_X(-1)}$.

theorem

SeeZariski's main theorem,theorem on formal functions,cohomology base change theorem,Category:Theorems in algebraic geometry.

torus embedding

An old term for atoric variety

toric variety

Atoric 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. Seetropical geometry.

torus

Asplit torus is a product of finitely manymultiplicative groups${\displaystyle {\mathbb{G}}_m}$.

universal

1.  If amoduli functorF is represented by some scheme or algebraic spaceM, then auniversal object is an element ofF(M) that corresponds to the identity morphismM →M (which is anM-point ofM). If the values ofF are isomorphism classes of curves with extra structure, say, then a universal object is called auniversal curve. Atautological bundle would be another example of a universal object.

2.  Let${\displaystyle {\mathcal{M}}_g}$ be the moduli of smooth projective curves of genusg and${\displaystyle {\mathcal{C}}_g={\mathcal{M}}_{g,1}}$ that of smooth projective curves of genusg with single marked points. In literature, the forgetful map${\displaystyle \pi :{\mathcal{C}}_g\to {\mathcal{M}}_g}$is often called a universal curve.

universally

A morphism has some property universally if all base changes of the morphism have this property. Examples includeuniversally catenary,universally injective.

unramified

For a point${\displaystyle y}$ in${\displaystyle Y}$, consider the corresponding morphism of local rings${\displaystyle f^{\mathrm{\#}}:{\mathcal{O}}_{X,f(y)}\to {\mathcal{O}}_{Y,y}}$.Let${\displaystyle \mathfrak{m}}$ be the maximal ideal of${\displaystyle {\mathcal{O}}_{X,f(y)}}$, and let${\displaystyle \mathfrak{n}=f^{\mathrm{\#}}(\mathfrak{m}){\mathcal{O}}_{Y,y}}$be the ideal generated by the image of${\displaystyle \mathfrak{m}}$ in${\displaystyle {\mathcal{O}}_{Y,y}}$. The morphism${\displaystyle f}$ isunramified (resp.G-unramified) if it is locally of finite type (resp. locally of finite presentation) and if for all${\displaystyle y}$ in${\displaystyle Y}$,${\displaystyle \mathfrak{n}}$ is the maximal ideal of${\displaystyle {\mathcal{O}}_{Y,y}}$ and the induced map${\displaystyle {\mathcal{O}}_{X,f(y)}/\mathfrak{m}\to {\mathcal{O}}_{Y,y}/\mathfrak{n}}$is afiniteseparable field extension.^[21] This is the geometric version (and generalization) of anunramified field extension inalgebraic number theory.

variety

a synonym with "algebraic variety".

very ample

A line bundleL on a varietyX isvery ample ifX can be embedded into a projective space so thatL is the restriction of Serre's twisting sheafO(1) on the projective space.

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"; seedivisor.

Weil reciprocity

SeeWeil reciprocity.

Zariski–Riemann space

AZariski–Riemann space is a locally ringed space whose points are valuation rings.

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