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.
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η {\displaystyle \eta }
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 O X ( 1 ) {\displaystyle {\mathcal {O}}_{X}(1)} and if F is an O X {\displaystyle {\mathcal {O}}_{X}} -module, then F ( n ) = F ⊗ O X O X ( n ) . {\displaystyle F(n)=F\otimes _{{\mathcal {O}}_{X}}{\mathcal {O}}_{X}(n).}
2. If D is a Cartier divisor and F is an O X {\displaystyle {\mathcal {O}}_{X}} -module (X arbitrary), then F ( D ) = F ⊗ O X O X ( D ) . {\displaystyle F(D)=F\otimes _{{\mathcal {O}}_{X}}{\mathcal {O}}_{X}(D).} If D is a Weil divisor and F is reflexive, then one replaces F(D) by its reflexive hull (and calls 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, | D | = P ( Γ ( X , O X ( D ) ) ) {\displaystyle |D|=\mathbf {P} (\Gamma (X,{\mathcal {O}}_{X}(D)))} . 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. 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.
X / / G {\displaystyle X/\!/G}
The GIT quotient of a scheme X by an action of a group scheme G.
Ln An ambiguous notation. It usually means an n-th tensor power of L but can also mean the self-intersection number of L. If L = O X {\displaystyle L={\mathcal {O}}_{X}} , the structure sheaf on X, then it means the direct sum of n copies of O X {\displaystyle {\mathcal {O}}_{X}} .
O X ( − 1 ) {\displaystyle {\mathcal {O}}_{X}(-1)}
The tautological line bundle. It is the dual of Serre's twisting sheaf O X ( 1 ) {\displaystyle {\mathcal {O}}_{X}(1)} .
O X ( 1 ) {\displaystyle {\mathcal {O}}_{X}(1)}
Serre's twisting sheaf. It is the dual of the tautological line bundle O X ( − 1 ) {\displaystyle {\mathcal {O}}_{X}(-1)} . It is also called the hyperplane bundle.
O X ( D ) {\displaystyle {\mathcal {O}}_{X}(D)}
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, O X ( D ) {\displaystyle {\mathcal {O}}_{X}(D)} is the image of D under the natural group homomorphism from the group of Cartier divisors to the Picard group Pic ( X ) {\displaystyle \operatorname {Pic} (X)} of X, the group of isomorphism classes of line bundles on X. 3. In general, O X ( D ) {\displaystyle {\mathcal {O}}_{X}(D)} 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 {\displaystyle \mathbb {Q} } -divisor, then O X ( D ) {\displaystyle {\mathcal {O}}_{X}(D)} is O X {\displaystyle {\mathcal {O}}_{X}} of the integral part of D.
Ω X p {\displaystyle \Omega _{X}^{p}}
1. Ω X 1 {\displaystyle \Omega _{X}^{1}} is the sheaf of Kähler differentials on X. 2. Ω X p {\displaystyle \Omega _{X}^{p}} is the p-th exterior power of Ω X 1 {\displaystyle \Omega _{X}^{1}} .
Ω X p ( log D ) {\displaystyle \Omega _{X}^{p}(\log D)}
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. Ω X p ( log D ) {\displaystyle \Omega _{X}^{p}(\log D)} is the p-th exterior power of Ω X 1 ( log D ) {\displaystyle \Omega _{X}^{1}(\log D)} .
P(V) The notation is ambiguous. Its traditional meaning is the projectivization of a finite-dimensional k-vector space V; i.e.,
P ( V ) = Proj ( k [ V ] ) = Proj ( Sym ( V ∗ ) ) {\displaystyle \mathbf {P} (V)=\operatorname {Proj} (k[V])=\operatorname {Proj} (\operatorname {Sym} (V^{*}))}
(the Proj of the ring of polynomial functions 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 Q {\displaystyle \mathbb {Q} } -factorial if every Q {\displaystyle \mathbb {Q} } -Weil divisor is Q {\displaystyle \mathbb {Q} } -Cartier.
Spec(R) The set of all prime ideals in a ring R with Zariski topology; it is called the prime spectrum of R.
SpecX(F) The relative Spec of the OX-algebra F. It is also denoted by Spec(F) or simply Spec(F).
Specan(R) The set of all valuations for a ring R with a certain weak topology; it is called the Berkovich spectrum of R.
A
abelian 1. An abelian variety is a complete group variety. For example, consider the complex variety C n / Z 2 n {\displaystyle \mathbb {C} ^{n}/\mathbb {Z} ^{2n}} or an elliptic curve E {\displaystyle E} over a finite field F q {\displaystyle \mathbb {F} _{q}} . 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 ω D , ω X {\displaystyle \omega _{D},\omega _{X}} , then the adjunction formula says:
ω D = ( ω X ⊗ O X ( D ) ) | D {\displaystyle \omega _{D}=(\omega _{X}\otimes {\mathcal {O}}_{X}(D))|_{D}} . 2. If, in addition, X and D are smooth, then the formula is equivalent to saying:
K D = ( K X + D ) | D {\displaystyle K_{D}=(K_{X}+D)|_{D}}
where K D , K X {\displaystyle K_{D},K_{X}} are canonical divisors on D and X.
affine 1. Affine space is roughly a vector space where one has forgotten which point is the origin 2. An affine variety is a variety in affine space 3. An affine scheme is a scheme that is the prime spectrum of some commutative ring. 4. A morphism is called 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 OX-algebras, defined by analogy with the spectrum of a ring. Important affine morphisms are vector bundles, and finite morphisms. 5. The affine cone over a closed subvariety X of a projective space is the Spec of the homogeneous coordinate ring of X.
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 the Riemann hypothesis. 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 [math.AG]. as well as .
algebraic group An algebraic group is an algebraic variety that is also a group 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 An algebraic set over a field k is a reduced separated scheme of finite type over Spec ( k ) {\displaystyle \operatorname {Spec} (k)} . An irreducible algebraic set is called an algebraic variety.
algebraic space An algebraic space is a quotient of a scheme by the étale equivalence relation.
algebraic variety An algebraic variety over a field k is an integral separated scheme of finite type over Spec ( k ) {\displaystyle \operatorname {Spec} (k)} . Note, not assuming k is algebraically closed causes some pathology; for example, Spec C × R Spec C {\displaystyle \operatorname {Spec} \mathbb {C} \times _{\mathbb {R} }\operatorname {Spec} \mathbb {C} } is not a variety since the coordinate ring C ⊗ R C {\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} } is not an integral domain.
algebraic vector bundle A locally free sheaf of a finite rank.
ample A line bundle on a projective variety is ample if some tensor power of it is very ample.
Arakelov geometry Algebraic geometry over the compactification of Spec of the ring of rational integers Z {\displaystyle \mathbb {Z} } . See Arakelov geometry.
arithmetic genus The arithmetic genus of a projective variety X of dimension r is ( − 1 ) r ( χ ( O X ) − 1 ) {\displaystyle (-1)^{r}(\chi ({\mathcal {O}}_{X})-1)} .
Artin stack Another term for an algebraic stack.
artinian 0-dimensional and Noetherian. The definition applies both to a scheme and a ring.
B
base change A fiber product of schemes, in particular one of a k-scheme X with Spec E for a field extension E/k.
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 lim sup l → ∞ dim Γ ( X , L l ) / l n > 0 {\displaystyle \displaystyle \limsup _{l\to \infty }\operatorname {dim} \Gamma (X,L^{l})/l^{n}>0} .
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 Z ⊂ X {\displaystyle Z\subset X} , the blow-up of X along Z is a proper morphism π : X ~ → X {\displaystyle \pi :{\widetilde {X}}\to X} such that (1) π − 1 ( Z ) ↪ X ~ {\displaystyle \pi ^{-1}(Z)\hookrightarrow {\widetilde {X}}} is an effective Cartier divisor, called the exceptional divisor, and (2) π {\displaystyle \pi } is universal with respect to (1). Concretely, it is constructed as the relative Proj of the Rees algebra of O X {\displaystyle O_{X}} with respect to the ideal sheaf determining Z.
C
Calabi–Yau 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 ω X = i ∗ Ω U n {\displaystyle \omega _{X}=i_{*}\Omega _{U}^{n}} where i is the inclusion of the smooth locus U and Ω U n {\displaystyle \Omega _{U}^{n}} 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 K X {\displaystyle K_{X}} on a normal variety X is the divisor class such that O X ( K X ) = ω X {\displaystyle {\mathcal {O}}_{X}(K_{X})=\omega _{X}} . 3. The canonical divisor is a representative of the canonical class K X {\displaystyle K_{X}} 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 The canonical model is the Proj of a canonical ring (assuming the ring is finitely generated.)
Cartier 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 f : P S n → S {\displaystyle f:\mathbf {P} _{S}^{n}\to S} over a scheme S is the smallest integer r such that
R i f ∗ F ( r − i ) = 0 {\displaystyle R^{i}f_{*}F(r-i)=0}
for all 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 A special fiber.
Chow group The k-th Chow group A k ( X ) {\displaystyle A_{k}(X)} of a smooth variety X is the free abelian group generated by closed subvarieties of dimension k (group of k-cycles) modulo rational 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 as invariants 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 to moduli spaces. 2. Complete smooth curves over an algebraically closed field are classified up to rational equivalence by their genus g {\displaystyle g} . (a) g = 0 {\displaystyle g=0} . rational curves, i.e. the curve is birational to the projective line P 1 {\displaystyle \mathbb {P} ^{1}} . (b) g = 1 {\displaystyle g=1} . Elliptic curves, i.e. the curve is a complete 1-dimensional group scheme after choosing any point on the curve as identity. (c) g ≥ 2 {\displaystyle g\geq 2} . Hyperbolic curves, also called curves of general type. See algebraic curves for examples. The classification of smooth curves can be refined by the degree for projectively embedded curves, in particular when restricted to plane 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 a number field (in particular number and structure of its rational points) is governed by the classification of the associated curve base changed to an algebraic closure. See Faltings' theorem for details on the arithmetic implications. 3. Classification of complete smooth surfaces over an algebraically closed field up to rational equivalence. See an overview of the classification or Enriques–Kodaira classification for details. 4. Classification of singularities resp. associated Zariski neighboorhoods over algebraically closed fields up to isomorphism. (a) In characteristic 0 Hironaka'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. See here for curves or here for curves and surfaces. 5. Classification of Fano varieties in small dimension. 6. The minimal 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 of split reductive groups up to isomorphism over algebraically closed fields.
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 O X {\displaystyle {\mathcal {O}}_{X}} -ideals. The support of the quotient sheaf O X / J {\displaystyle {\mathcal {O}}_{X}/J} is a closed subset Z of X and ( Z , ( O X / J ) | Z ) {\displaystyle (Z,({\mathcal {O}}_{X}/J)|_{Z})} is a scheme called the closed subscheme defined by the quasi-coherent sheaf of ideals J. 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 OX-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 f : X → Y {\displaystyle f:X\to Y} between normal varieties is a morphism such that f ∗ ω Y = ω X {\displaystyle f^{*}\omega _{Y}=\omega _{X}} .
curve An algebraic variety of dimension one.
D
deformation Let S → S ′ {\displaystyle S\to S'} 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 f : E → F {\displaystyle f:E\to F} 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 k ( f ) = { x ∈ X | rk ( f ( x ) ) ≤ k } {\displaystyle X_{k}(f)=\{x\in X|\operatorname {rk} (f(x))\leq k\}} .
degeneration 1. A scheme X is said to degenerate to a scheme X 0 {\displaystyle X_{0}} (called the limit of X) if there is a scheme π : Y → A 1 {\displaystyle \pi :Y\to \mathbf {A} ^{1}} with generic fiber X and special fiber X 0 {\displaystyle X_{0}} . 2. A flat degeneration is a degeneration such that π {\displaystyle \pi } is flat. dimension The dimension, 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 also Global dimension.
Examples: equidimensional schemes in dimension 0: Artinian schemes, 1: algebraic curves, 2: algebraic surfaces.
degree 1. The degree of a line bundle L on a complete variety is an integer d such that χ ( L ⊗ m ) = d n ! m n + O ( m n − 1 ) {\displaystyle \chi (L^{\otimes m})={d \over n!}m^{n}+O(m^{n-1})} . 2. If x is a cycle on a complete variety f : X → Spec k {\displaystyle f:X\to \operatorname {Spec} k} over a field k, then its degree is f ∗ ( x ) ∈ A 0 ( Spec k ) = Z {\displaystyle f_{*}(x)\in A_{0}(\operatorname {Spec} k)=\mathbb {Z} } . 3. For the degree of a finite morphism, see morphism of varieties#Degree of a finite morphism.
derived algebraic geometry An approach to algebraic geometry using (commutative) ring spectra instead of commutative rings; see derived algebraic geometry.
divisorial 1. A divisorial sheaf on a normal variety is a reflexive sheaf of the form OX(D) for some Weil divisor D. 2. A divisorial scheme is a scheme admitting an ample family of invertible sheaves. A scheme admitting an ample invertible sheaf is a basic example.
dominant A morphism 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.
dualizing complex See Coherent duality.
dualizing sheaf On a projective Cohen–Macaulay scheme of pure dimension n, the dualizing sheaf is a coherent sheaf ω {\displaystyle \omega } on X such that
H n − i ( X , F ∨ ⊗ ω ) ≃ H i ( X , F ) ∗ {\displaystyle H^{n-i}(X,F^{\vee }\otimes \omega )\simeq H^{i}(X,F)^{*}}
holds for any locally free sheaf F on X; for example, if X is a smooth projective variety, then it is a canonical sheaf.
E
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.
equidimensional A scheme whose irreducible components have the same dimension. See also § pure dimension.
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 X {\displaystyle X} and Y {\displaystyle Y} over an algebraically closed field, étale morphisms are precisely those inducing an isomorphism of tangent spaces d f : T y Y → T f ( y ) X {\displaystyle df:T_{y}Y\rightarrow 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-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:
0 → O P n → O P n ( 1 ) ⊕ ( n + 1 ) → T P n → 0 , {\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,}
where Pn 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
F
F-regular Related to Frobenius morphism.
Fano A Fano variety is a smooth projective variety X whose anticanonical sheaf ω X − 1 {\displaystyle \omega _{X}^{-1}} is ample.
fiber Given f : X → Y {\displaystyle f:X\to Y} between schemes, the fiber of f over y is, as a set, the pre-image f − 1 ( y ) = { x ∈ X | f ( x ) = y } {\displaystyle f^{-1}(y)=\{x\in X|f(x)=y\}} ; it has the natural structure of a scheme over the residue field of y as the fiber product X × Y { y } {\displaystyle X\times _{Y}\{y\}} , where { y } {\displaystyle \{y\}} 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. In particular, fiber product of schemes. 2. A stack F × G H {\displaystyle F\times _{G}H} given for f : F → G , g : H → G {\displaystyle f:F\to G,g:H\to G} : an object over B is a triple (x, y, ψ), x in F(B), y in H(B), ψ an isomorphism f ( x ) → ∼ g ( y ) {\displaystyle f(x){\overset {\sim }{\to }}g(y)} in G(B); an arrow from (x, y, ψ) to (x', y', ψ') is a pair of morphisms α : x → x ′ , β : y → y ′ {\displaystyle \alpha :x\to x',\beta :y\to y'} such that ψ ′ ∘ f ( α ) = g ( β ) ∘ ψ {\displaystyle \psi '\circ f(\alpha )=g(\beta )\circ \psi } . 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 Z {\displaystyle \mathbb {Z} } of integers; so that any scheme S {\displaystyle S} is over Spec ( Z ) {\displaystyle {\textrm {Spec}}(\mathbb {Z} )} , and in a unique way.
finite The morphism f : Y → X is finite if X {\displaystyle X} may be covered by affine open sets Spec B {\displaystyle {\text{Spec }}B} such that each f − 1 ( Spec B ) {\displaystyle f^{-1}({\text{Spec }}B)} is affine — say of the form Spec A {\displaystyle {\text{Spe