View source for Glossary of algebraic geometry

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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.

{{Compact ToC|short1|sym=yes|x=[[#XYZ|XYZ]]|y=|z=|seealso=yes|refs=yes}}

==!$@==

{{glossary}}

{{term|<math>\eta</math>}}
{{defn|1=A [[generic point]]. For example, the point associated to the zero ideal for any integral affine scheme.}}

{{term|2=''F''(''n''), ''F''(''D'')}}
{{defn|no=1|1=If ''X'' is a projective scheme with [[Serre's twisting sheaf]] <math>\mathcal{O}_X(1)</math> and if ''F'' is an <math>\mathcal{O}_X</math>-module, then <math>F(n) = F \otimes_{\mathcal{O}_X} \mathcal{O}_X(n).</math>}}
{{defn|no=2|1=If ''D'' is a Cartier divisor and ''F'' is an <math>\mathcal{O}_X</math>-module (''X'' arbitrary), then <math>F(D) = F \otimes_{\mathcal{O}_X} \mathcal{O}_X(D).</math> 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'').)}}

{{term|<nowiki>|</nowiki>''D''<nowiki>|</nowiki>}}
{{defn|1=The [[complete linear system]] of a [[Weil divisor]] ''D'' on a normal complete<!-- the completeness is just to ensure a linear system is finite-dimensional --> variety ''X'' over an algebraically closed field ''k''; that is, <math>|D| = \mathbf{P}(\Gamma(X, \mathcal{O}_X(D)))</math>. There is a bijection between the set of ''k''-rational points of <nowiki>|</nowiki>''D''<nowiki>|</nowiki> and the set of effective Weil divisors on ''X'' that are linearly equivalent to ''D''.<ref>Proof: Let ''D'' be a Weil divisor on ''X''. If ''D<nowiki>'</nowiki>'' ~ ''D'', then there is a nonzero rational function ''f'' on ''X'' such that ''D'' + (''f'') <nowiki>=</nowiki> ''D<nowiki>'</nowiki>'' and then ''f'' is a section of ''O''<sub>''X''</sub>(''D'') if ''D<nowiki>'</nowiki>'' is effective. The opposite direction is similar. □</ref> The same definition is used if ''D'' is a [[Cartier divisor]] on a complete variety over ''k''.}}

{{term|[X/G]}}
{{defn|1=The [[quotient stack]] of, say, an algebraic space ''X'' by an action of a group scheme ''G''.}}

{{term|<math>X/\!/G</math>}}
{{defn|1=The [[GIT quotient]] of a scheme ''X'' by an [[action of a group scheme]] ''G''.}}

{{term|content=''L''<sup>''n''</sup>}}
{{defn|1=An ambiguous notation. It usually means an ''n''-th tensor power of ''L'' but can also mean the self-intersection number of ''L''. If <math>L = \mathcal{O}_X</math>, the structure sheaf on ''X'', then it means the direct sum of ''n'' copies of <math>\mathcal{O}_X</math>.}}

{{term|content=<math>\mathcal{O}_X(-1)</math>}}
{{defn|1=The [[tautological line bundle]]. It is the dual of [[Serre's twisting sheaf]] <math>\mathcal{O}_X(1)</math>.}}

{{term|content=<math>\mathcal{O}_X(1)</math>}}
{{defn|1=[[Serre's twisting sheaf]]. It is the dual of the [[tautological line bundle]] <math>\mathcal{O}_X(-1)</math>. It is also called the hyperplane bundle.}}

{{term|content=<math>\mathcal{O}_X(D)</math>}}
{{defn|no=1|If ''D'' is an [[effective Cartier divisor]] on ''X'', then it is the inverse of the ideal sheaf of ''D''.}}
{{defn|no=2|Most of the times, <math>\mathcal{O}_X(D)</math> is the image of ''D'' under the natural group homomorphism from the group of Cartier divisors to the Picard group <math>\operatorname{Pic}(X)</math> of ''X'', the group of isomorphism classes of line bundles on ''X''.}}
{{defn|no=3|In general, <math>\mathcal{O}_X(D)</math> is the sheaf corresponding to a [[Weil divisor]] ''D'' (on a [[normal scheme]]). It need not be locally free, only [[reflexive sheaf|reflexive]].}}
{{defn|no=4|If ''D'' is a <math>\mathbb{Q}</math>-divisor, then <math>\mathcal{O}_X(D)</math> is <math>\mathcal{O}_X</math> of the integral part of ''D''.}}

{{term|content=<math>\Omega_X^p</math>}}
{{defn|no=1|<math>\Omega_X^1</math> is the sheaf of [[Kähler differentials]] on ''X''.}}
{{defn|no=2|<math>\Omega_X^p</math> is the ''p''-th exterior power of <math>\Omega_X^1</math>.}}

{{term|content=<math>\Omega_X^p(\log D)</math>}}
{{defn|no=1|If ''p'' is 1, this is the sheaf of [[logarithmic Kähler differential]]s on ''X'' along ''D'' (roughly differential forms with simple poles along a divisor ''D''.)}}
{{defn|no=2|<math>\Omega_X^p(\log D)</math> is the ''p''-th exterior power of <math>\Omega_X^1(\log D)</math>.}}

{{term|'''P'''(''V'')}}
{{defn|1=The notation is ambiguous. Its traditional meaning is the [[projectivization]] of a finite-dimensional ''k''-vector space ''V''; i.e.,
<math>\mathbf{P}(V) = \operatorname{Proj}(k[V]) = \operatorname{Proj}(\operatorname{Sym}(V^*))</math>
(the [[Proj construction|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''.}}

{{term|Q-factorial}}
{{defn|1=A normal variety is <math>\mathbb{Q}</math>-factorial if every <math>\mathbb{Q}</math>-Weil divisor is <math>\mathbb{Q}</math>-Cartier.}}

{{term|Spec(''R'')}}
{{defn|1=The set of all prime ideals in a ring ''R'' with Zariski topology; it is called the [[prime spectrum]] of ''R''.}}

{{term|content=Spec<sub>''X''</sub>(''F'')}}
{{defn|1=The [[relative Spec]] of the ''O''<sub>''X''</sub>-algebra ''F''. It is also denoted by '''Spec'''(''F'') or simply Spec(''F'').}}

{{term|content=Spec<sup>an</sup>(''R'')}}
{{defn|1=The set of all valuations for a ring ''R'' with a certain weak topology; it is called the [[Berkovich spectrum]] of ''R''.}}

{{glossary end}}

==A==

{{glossary}}
{{term|1=abelian}}
{{defn|no=1|An [[abelian variety]] is a complete group variety. For example, consider the complex variety <math>\mathbb{C}^n/\mathbb{Z}^{2n} </math> or an elliptic curve <math>E</math> over a finite field <math>\mathbb{F}_q</math>.}}
{{defn|no=2|An [[abelian scheme]] is a (flat) family of abelian varieties.}}

{{term|1=adjunction formula}}
{{defn|no=1|If ''D'' is an effective Cartier divisor on an algebraic variety ''X'', both admitting [[dualizing sheaf|dualizing sheaves]] <math>\omega_D, \omega_X</math><!-- meaning Cohen-Macaulay -->, then the [[adjunction formula]] says:
<math>\omega_D = (\omega_X \otimes \mathcal{O}_X(D))|_D</math>.}}
{{defn|no=2|If, in addition, ''X'' and ''D'' are smooth, then the formula is equivalent to saying:
<math>K_D = (K_X + D)|_D</math>
where <math>K_D, K_X</math> are [[canonical divisor]]s on ''D'' and ''X''.}}

{{term|1=affine}}
{{defn|no=1|[[Affine space]] is roughly a vector space where one has forgotten which point is the origin}} 
{{defn|no=2|An [[affine variety]] is a variety in affine space}}
{{defn|no=3|An [[affine scheme]] is a scheme that is the [[prime spectrum]] of some commutative ring.}}
{{defn|no=4|1=
A morphism is called '''[[affine morphism|affine]]''' if the preimage of any open affine subset is again affine. In more fancy terms, affine morphisms are defined by the [[global Spec|global '''Spec''']] construction for sheaves of ''O<sub>X</sub>''-Algebras, defined by analogy with the [[spectrum of a ring]]. Important affine morphisms are [[vector bundle]]s, and [[finite morphism]]s.
}}
{{defn|no=5|The [[Cone (algebraic geometry)|affine cone]] over a closed subvariety ''X'' of a projective space is the Spec of the homogeneous coordinate ring of ''X''.}}

{{quote box
|align = right
|width=33%
|quote=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.
|source=From the preface to I.R. Shafarevich, Basic Algebraic Geometry.}} 
{{term|1=algebraic geometry}}
{{defn|1=[[Algebraic geometry]] is a branch of mathematics that studies solutions to algebraic equations.}}

{{term|1=algebraic geometry over the field with one element}}
{{defn|1=One goal is to prove the [[Riemann hypothesis]].<ref>{{Cite arXiv|title = An essay on the Riemann Hypothesis|eprint = 1509.05576|date = 2015-09-18|first = Connes|last = Alain|class = math.NT}}</ref> See also the [[field with one element]] and {{Cite arXiv|title = Mapping F_1-land:An overview of geometries over the field with one element|eprint=0909.0069 |date = 2009-08-31|first1 = Javier López|last1 = Peña|first2 = Oliver|last2 = Lorscheid|class=math.AG }} as well as <ref>{{Cite arXiv|title = Remarks on zeta functions and K-theory over F1|eprint=math/0605429 |date = 2006-05-16|first = Anton|last = Deitmar}}</ref><ref>{{Cite arXiv|title = Homological Algebra for Commutative Monoids|eprint=1503.02309 |date = 2015-03-08|first = Jaret|last = Flores|class=math.KT }}</ref>
.}}

{{term|1=algebraic group}}
{{defn|1=An [[algebraic group]] is an algebraic variety that is also a [[group (mathematics)|group]] in such a way the group operations are morphisms of varieties.}}

{{term|1=algebraic scheme}}
{{defn|1=A separated scheme of finite type over a field. For example, an algebraic variety is a reduced irreducible algebraic scheme.}}

{{term|1=algebraic set}}
{{defn|1=An [[algebraic set]] over a field ''k'' is a reduced separated scheme of finite type over <math>\operatorname{Spec}(k)</math>. An irreducible algebraic set is called an algebraic variety.}}

{{term|1=algebraic space}}
{{defn|1=An [[algebraic space]] is a quotient of a scheme by the [[étale equivalence relation]].}}

{{term|1=algebraic variety}}
{{defn|1=An [[algebraic variety]] over a field ''k'' is an integral separated scheme of finite type over <math>\operatorname{Spec}(k)</math>. Note, not assuming ''k'' is algebraically closed causes some pathology; for example, <math>\operatorname{Spec} \mathbb{C} \times_{\mathbb{R}} \operatorname{Spec} \mathbb{C}</math> is not a variety since the coordinate ring <math>\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}</math> is not an [[integral domain]].}}

{{term|1=algebraic vector bundle}}
{{defn|1=A [[locally free sheaf]] of a finite rank.}}

{{term|1=ample}}
{{defn|1=A line bundle on a projective variety is [[ample line bundle|ample]] if some tensor power of it is very ample.}}

{{term|1=Arakelov geometry}}
{{defn|1=Algebraic geometry over the compactification of Spec of the [[ring of rational integers]] <math>\mathbb{Z}</math>. See [[Arakelov geometry]].<ref>{{Cite arXiv|title = New Approach to Arakelov Geometry|eprint=0704.2030 |date = 2007-04-16|first = Nikolai|last = Durov|class=math.AG }}</ref>}}

{{term|1=arithmetic genus}}
{{defn|1=The [[arithmetic genus]] of a projective variety ''X'' of dimension ''r'' is <math>(-1)^r (\chi(\mathcal{O}_X) - 1)</math>.}}
<!--{{term|1=arithmetic}}
{{defn|1=The [[arithmetic scheme]] is a scheme over <math>\operatorname{Spec}(\mathbf{Z})</math>, the prime spectrum of the ring of rational integers.}} but any scheme is a scheme over Spec('''Z''').-->

{{term|1=Artin stack}}
{{defn|1=Another term for an [[algebraic stack]].}}

{{term|1=artinian}}
{{defn|1=0-dimensional and Noetherian. The definition applies both to a scheme and a ring.}}

{{glossary end}}

==B==

{{glossary}}
{{term|1=Behrend function}}
{{defn|1=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''.}} 

{{term|1=Behrend's trace formula}}
{{defn|1=[[Behrend's trace formula]] generalizes [[Grothendieck's trace formula]]; both formulas compute the trace of the [[Arithmetic and geometric Frobenius|Frobenius]] on ''l''-adic cohomology.}}

{{term|1=big}}
{{defn|1=A [[big line bundle]] ''L'' on ''X'' of dimension ''n'' is a line bundle such that <math>\displaystyle \limsup_{l \to \infty} \operatorname{dim} \Gamma(X, L^l) / l^n > 0</math>.}}

{{term|1=birational morphism}}
{{defn|1=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.}}

{{term|1=blow-up}}
{{defn|1=A [[Blowing up|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 <math>Z \subset X</math>, the blow-up of ''X'' along ''Z'' is a proper morphism <math>\pi: \widetilde{X} \to X</math> such that (1) <math>\pi^{-1}(Z) \hookrightarrow \widetilde{X}</math> is an effective Cartier divisor, called the exceptional divisor and (2) <math>\pi</math> is universal with respect to (1). Concretely, it is constructed as the relative Proj of the Rees algebra of <math>O_X</math> with respect to the ideal sheaf determining ''Z''.}}

{{glossary end}}

==C==

{{glossary}}
{{term|1=Calabi–Yau}}
{{defn|1=The [[Calabi–Yau metric]] is a Kähler metric whose Ricci curvature is zero.}}

{{term|1=canonical}}
{{defn|no=1|1=The [[canonical sheaf]] on a normal variety ''X'' of dimension ''n'' is <math>\omega_X = i_* \Omega_U^n</math> where ''i'' is the inclusion of the [[smooth locus]] ''U'' and <math>\Omega_U^n</math> 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.<!-- why not normalization? -->}}
{{defn|no=2|1=The [[canonical class]] <math>K_X</math> on a normal variety ''X'' is the divisor class such that <math>\mathcal{O}_X(K_X) = \omega_X</math>.}}
{{defn|no=3|1=The [[canonical divisor]] is a representative of the canonical class <math>K_X</math> denoted by the same symbol (and not well-defined.)}}
{{defn|no=4|1=The [[canonical ring]] of a normal variety ''X'' is the section ring of the [[canonical sheaf]].}}

{{term|1=canonical model}}
{{defn|The [[canonical model (algebraic geometry)|canonical model]] is the [[Proj construction|Proj]] of a canonical ring (assuming the ring is finitely generated.)}}

{{term|1=Cartier}}
{{defn|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).}}

{{term|1=Castelnuovo–Mumford regularity}}
{{defn|The [[Castelnuovo–Mumford regularity]] of a coherent sheaf ''F'' on a projective space <math>f: \mathbf{P}^n_S \to S</math> over a scheme ''S'' is the smallest integer ''r'' such that
:<math>R^i f_*F(r-i) = 0</math>
for all ''i'' > 0.}}

{{term|1=catenary}}
{{defn|1=
A scheme is [[Catenary (ring theory)|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.
}}

{{term|1=central fiber}}
{{defn|A special fiber.}}

{{term|1=Chow group}}
{{defn|The ''k''-th [[Chow group]] <math>A_k(X)</math> of a smooth variety ''X'' is the free abelian group generated by closed subvarieties of dimension ''k'' (group of ''k''-[[algebraic cycle|cycles]]) modulo [[rational equivalence]]s.}}

{{term|1=classification}}
{{defn|no=1|1=[[Classification theorem|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 [[Complete set of invariants|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]].}}
{{defn|no=2|1=[[Complete variety|Complete]] [[Singular point of an algebraic variety|smooth]] [[Algebraic curve|curves]] over an [[algebraically closed field]] are classified up to [[Birational geometry#Birational equivalence and rationality|rational equivalence]] by their [[Genus (mathematics)#Algebraic geometry|genus]] <math>g</math>. (a) <math>g=0</math>. [[Rational variety|rational]] curves, i.e. the curve is [[birational]] to the projective line <math>\mathbb{P}^1</math>. (b) <math>g=1</math>. [[Elliptic curves]], i.e. the curve is a complete 1-dimensional [[group scheme]] after choosing any point on the curve as identity. (c) <math>g\geq2</math>. '''Hyperbolic curves''', also called '''curves of general type'''. See [[Algebraic curve#Examples of curves|algebraic curves for examples]]. The classification of smooth curves can be refined by the [[Degree of an algebraic variety|degree]] for [[projective embedding|projectively embedded]] curves, in particular when restricted to [[Algebraic curve#Projective plane curves|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 [[Fiber product of schemes#Interpretations and special cases|base changed]] to an algebraic closure. See [[Faltings's theorem#Background|Faltings's theorem]] for details on the arithmetic implications.}}
{{defn|no=3|1=Classification of complete smooth [[Algebraic surface|surfaces]] over an algebraically closed field up to rational equivalence. See an [[Algebraic surface#Classification by the Kodaira dimension|overview of the classification]] or [[Enriques–Kodaira classification]] for details.}}
{{defn|no=4|1=Classification of [[Singular point of an algebraic variety|singularities]] resp. associated [[Zariski topology#Zariski topology of varieties|Zariski neighboorhoods]] over algebraically closed fields up to isomorphism. (a) In characteristic 0 [[Resolution of singularities#Hironaka's method 2|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 [[Algebraic curve#Classification of singularities|here for curves]] or [[Resolution of singularities#Resolution for schemes and status of the problem|here for curves and surfaces]].}}
{{defn|no=5|1= Classification of [[Fano variety#Classification in small dimensions|Fano varieties]] in small dimension.}}
{{defn|no=6|1= 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, [[Canonical singularity|terminal singularites]] naturally appear and are part of a wider classification.}}
{{defn|no=7|1= Classification of [[Reductive group#Classification of split reductive groups|split reductive groups]] up to isomorphism over algebraically closed fields.}}

{{term|1=classifying stack}}
{{defn|1=An analog of a [[classifying space]] for [[torsor (algebraic geometry)|torsor]]s in algebraic geometry; see [[classifying stack]].}}

{{term|1=closed}}
{{defn|1=
'''Closed subschemes''' of a scheme ''X'' are defined to be those occurring in the following construction. Let ''J'' be a [[Quasi-coherent sheaf|quasi-coherent]] sheaf of <math>\mathcal{O}_X</math>-[[sheaf of ideals|ideals]]. The [[support of a sheaf|support]] of the [[quotient sheaf]] <math>\mathcal{O}_X/J</math> is a closed subset ''Z'' of ''X'' and <math>(Z,(\mathcal{O}_X/J)|_Z)</math> is a scheme called the '''closed subscheme defined by the [[Quasi-coherent sheaf|quasi-coherent]] [[sheaf of ideals]] ''J'''''.<ref>{{harvnb|Grothendieck|Dieudonné|1960|loc=4.1.2 and 4.1.3}}</ref> 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.
}}

{{term|Cohen–Macaulay}}
{{defn|1=
A scheme is called Cohen-Macaulay
if all local rings are [[Cohen-Macaulay ring|Cohen-Macaulay]].
For example,  regular schemes, and ''Spec k''[''x,y'']/(''xy'') are Cohen–Macaulay, but
[[Image:Non cohen macaulay scheme thumb.png|50px]] is not. 
}}

{{term|1=coherent sheaf}}
{{defn|1=A [[coherent sheaf]] on a Noetherian scheme ''X'' is a quasi-coherent sheaf that is finitely generated as ''O''<sub>''X''</sub>-module.}}

{{term|1=conic}}
{{defn|1=An [[algebraic curve]] of degree two.}}

{{term|1=connected}}
{{defn|1=
  The scheme is ''[[Connected space|connected]]'' as a topological space. Since the [[connected component (topology)|connected components]] refine the [[irreducible component]]s any irreducible scheme is connected but not vice versa. An [[affine scheme]] ''Spec(R)'' is connected [[iff]] the ring ''R'' possesses no [[idempotent]]s 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''])
}}

{{term|1=compactification}}
{{defn|1=See for example [[Nagata's compactification theorem]].}}

{{term|1=Cox ring}}
{{defn|1=A generalization of a homogeneous coordinate ring. See [[Cox ring]].}}

{{term|1=crepant}}
{{defn|1=A [[crepant morphism]] <math>f: X \to Y</math> between normal varieties is a morphism such that <math>f^* \omega_Y = \omega_X</math>.}}

{{term|1=curve}}
{{defn|1=An algebraic variety of dimension one.}}

{{glossary end}}

==D==

{{glossary}}
{{term|1=deformation}}
{{defn|1=Let <math>S \to S'</math> be a morphism of schemes and ''X'' an ''S''-scheme. Then a deformation ''X''<nowiki>'</nowiki> of ''X'' is an ''S''<nowiki>'</nowiki>-scheme together with a pullback square in which ''X'' is the pullback of ''X''<nowiki>'</nowiki> (typically ''X''<nowiki>'</nowiki> is assumed to be [[flat morphism|flat]]).}}

{{term|1=degeneracy locus}}
{{defn|1=Given a vector-bundle map <math>f: E \to F</math> 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
<math>X_k(f) = \{ x \in X | \operatorname{rk}(f(x)) \le k \}</math>.}}

{{term|1=degeneration}}
{{defn|no=1|1=A scheme ''X'' is said to [[degeneration (algebraic geometry)|degenerate]] to a scheme <math>X_0</math> (called the limit of ''X'') if there is a scheme <math>\pi: Y \to \mathbf{A}^1</math> with [[generic fiber]] ''X'' and [[special fiber]] <math>X_0</math>.}}
{{defn|no=2|A [[flat degeneration]] is a degeneration such that <math>\pi</math> is flat.}}
{{term|1=dimension}}
{{defn|1=
The [[dimension of an algebraic variety|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 scheme]]s in dimension 0: [[Artinian ring|Artinian]] schemes, 1: [[algebraic curves]], 2: [[algebraic surfaces]]. 
}}

{{term|1=degree}}
{{defn|no=1|The degree of a line bundle ''L'' on a complete variety is an integer ''d'' such that <math>\chi(L^{\otimes m}) = {d \over n!} m^n + O(m^{n-1})</math>.}}
{{defn|no=2|If ''x'' is a cycle on a complete variety <math>f: X \to \operatorname{Spec} k</math> over a field ''k'', then its degree is <math>f_*(x) \in A_0(\operatorname{Spec} k) = \mathbb{Z}</math>.}}
{{defn|no=3|For the degree of a finite morphism, see [[morphism of varieties#Degree of a finite morphism]].}}

{{term|1=derived algebraic geometry}}
{{defn|1=An approach to algebraic geometry using ([[commutative ring spectrum|commutative]]) [[ring spectrum|ring spectra]] instead of [[commutative ring]]s; see [[derived algebraic geometry]].}}

{{term|1=divisorial}}
{{defn|no=1|1=A [[divisorial sheaf]] on a normal variety is a reflexive sheaf of the form ''O''<sub>''X''</sub>(''D'') for some [[Weil divisor]] ''D''.}}
{{defn|no=2|1=A [[divisorial scheme]] is a scheme admitting an ample family of invertible sheaves. A scheme admitting an ample invertible sheaf is a basic example.}}

{{term|1=dominant}}
{{defn|1=
A morphism {{nowrap|''f'' : ''X'' → ''Y''}} is called ''[[dominant morphism|dominant]]'', if the image ''f''(''X'') is [[dense set|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''.
}}

{{term|1=dualizing complex}}
{{defn|1=See [[Coherent duality]].}}

{{term|1=dualizing sheaf}}
{{defn|1=On a projective [[Cohen–Macaulay scheme]] of pure dimension ''n'', the [[dualizing sheaf]] is a coherent sheaf <math>\omega</math> on ''X'' such that
<math>H^{n-i}(X, F^{\vee} \otimes \omega) \simeq H^i(X, F)^*</math>
holds for any locally free sheaf ''F'' on ''X''; for example, if ''X'' is a smooth projective variety, then it is a [[canonical sheaf]].}}

{{glossary end}}

==E==

{{glossary}}

{{term|1=''Éléments de géométrie algébrique''}}
{{defn|1=
The [[Éléments de géométrie algébrique|EGA]] was an incomplete attempt to lay a foundation of algebraic geometry based on the notion of [[scheme (mathematics)|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.}}

{{term|1=elliptic curve}}
{{defn|1=An [[elliptic curve]] is a smooth [[projective curve]] of genus one.}}

{{term|1=essentially of finite type}}
{{defn|1=Localization of a finite type scheme.}}

{{term|1=étale}}
{{defn|1=
A morphism {{nowrap|''f'' : ''Y'' → ''X''}} is [[étale morphism|étale]] if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties <math> X </math> and <math> Y </math> over an algebraically closed [[field (mathematics)|field]], étale morphisms are precisely those inducing an isomorphism of tangent spaces <math> df: T_{y} Y \rightarrow T_{f(y)} X</math>, 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.
}}

{{term|1=Euler sequence}}
{{defn|1=The exact sequence of sheaves:
:<math>0 \to \mathcal{O}_{\mathbf{P}^n} \to \mathcal{O}_{\mathbf{P}^n}(1)^{\oplus (n+1)} \to T \mathbf{P}^n \to 0,</math>
where '''P'''<sup>''n''</sup> is the projective space over a field and the last nonzero term is the [[tangent sheaf]], is called the [[Euler sequence]].
}}

{{term|1=equivariant intersection theory}}
{{defn|1=See Chapter II of http://www.math.ubc.ca/~behrend/cet.pdf}}

{{glossary end}}

==F==

{{glossary}}

{{term|1=''F''-regular}}
{{defn|1=Related to [[Frobenius morphism]].<ref>{{Cite arXiv|title = Frobenius Splitting in Commutative Algebra|eprint=1409.1169 |date = 2014-09-03|first1 = Karen E.|last1 = Smith|first2 = Wenliang|last2 = Zhang|class=math.AC }}</ref>}}

{{term|1=Fano}}
{{defn|1=A [[Fano variety]] is a smooth [[projective variety]] ''X'' whose anticanonical sheaf <math>\omega_X^{-1}</math> is ample.}}

{{term|1=fiber}}
{{defn|1=Given <math>f: X \to Y</math> between schemes, the fiber of ''f'' over ''y'' is, as a set, the pre-image <math>f^{-1}(y) = \{x \in X|f(x)=y\}</math>; it has the natural structure of a scheme over the [[residue field]] of ''y'' as the fiber product <math>X \times_Y \{y\}</math>, where <math>\{y\}</math> has the natural structure of a scheme over ''Y'' as Spec of the residue field of ''y''.}}

{{term|1=fiber product}}
{{defn|no=1|1=Another term for the "[[pullback (category theory)|pullback]]" in the category theory.}}
{{defn|no=2|A stack <math>F \times_G H</math> given for <math>f: F \to G, g:H \to G</math>: an object over ''B'' is a triple (''x'', ''y'', ψ), ''x'' in ''F''(''B''), ''y'' in ''H''(''B''), ψ an isomorphism <math>f(x) \overset{\sim}\to g(y)</math> in ''G''(''B''); an arrow from (''x'', ''y'', ψ) to (''x''', ''y''{{'}}, ψ') is a pair of morphisms <math>\alpha: x \to x', \beta: y \to y'</math> such that <math>\psi' \circ f(\alpha) = g(\beta) \circ \psi</math>. The resulting square with obvious projections ''does not'' commute; rather, it commutes up to natural isomorphism; i.e., it [[2-commute]]s.}}

{{term|1=final}}
{{defn|1=
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 <math> \mathbb{Z} </math> of integers; so that any scheme <math> S </math> is ''over'' <math> \textrm{Spec} (\mathbb{Z}) </math>, and in a unique way.
}}

{{term|1=finite}}
{{defn|1=
The morphism {{nowrap|''f'' : ''Y'' → ''X''}} is '''finite''' if <math>X</math> may be covered by affine open sets <math> \text{Spec }B </math> such that each <math> f^{-1}(\text{Spec }B) </math> is affine — say of the form <math> \text{Spec }A </math> — and furthermore <math> A </math> is finitely generated as a <math> B </math>-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.

}}

{{term|1=finite type (locally)}}
{{defn|1=
The morphism {{nowrap|''f'' : ''Y'' → ''X''}} is '''locally of finite type''' if <math> X </math> may be covered by affine open sets <math> \text{Spec }B </math> such that each inverse image <math>f^{-1}(\text{Spec }B)</math> is covered by affine open sets <math>\text{Spec }A</math> where each <math> A </math> is finitely generated as a <math>B</math>-algebra.

The morphism {{nowrap|''f'' : ''Y'' → ''X''}} is '''of finite type''' if <math> X </math> may be covered by affine open sets <math> \text{Spec }B </math> such that each inverse image <math>f^{-1}(\text{Spec }B)</math> is covered by finitely many affine open sets <math>\text{Spec }A</math> where each <math> A </math> is finitely generated as a <math>B</math>-algebra.
}}

{{term|1=finite fibers}}
{{defn|1=
The morphism {{nowrap|''f'' : ''Y'' → ''X''}} has '''finite fibers''' if the fiber over each point <math> x \in X </math> is a finite set. A morphism is '''[[quasi-finite morphism|quasi-finite]]''' if it is of finite type and has finite fibers.
}}

{{term|1=finite presentation}}
{{defn|1=
If ''y'' is a point of ''Y'', then the morphism ''f'' is '''of finite presentation at ''y''''' (or '''finitely presented at ''y''''') if there is an open affine neighborhood ''U'' of ''f(y)'' and an open affine neighbourhood ''V'' of ''y'' such that ''f''(''V'')&nbsp;⊆&nbsp;''U'' and <math>\mathcal{O}_Y(V)</math> is a [[finitely presented algebra]] over <math>\mathcal{O}_X(U)</math>. 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.<ref>{{harvnb|Grothendieck|Dieudonné|1964|loc=§1.4}}</ref>

The morphism {{nowrap|''f'' : ''Y'' → ''X''}} is '''of finite presentation''' (or '''''Y'' is finitely presented over ''X''''') if it is locally of finite presentation, quasi-compact, and quasi-separated. If ''X'' is locally Noetherian, then ''f'' is of finite presentation if, and only if, it is of finite type.<ref>{{harvnb|Grothendieck|Dieudonné|1964|loc=§1.6}}</ref>
}}

{{term|1=flag variety}}
{{defn|1=The [[flag variety]] parametrizes a [[Flag (linear algebra)|flag]] of vector spaces.}}

{{term|1=flat}}
{{defn|1=
A morphism <math> f </math> is [[flat morphism (ring theory)|flat]] if it gives rise to a [[Flat map (ring theory)|flat map]] on stalks. When viewing a morphism {{nowrap|''f'' : ''Y'' → ''X''}} as a family of schemes parametrized by the points of <math> X </math>, the geometric meaning of flatness could roughly be described by saying that the fibers <math>f^{-1}(x)</math> do not vary too wildly.
}}

{{term|1=formal}}
{{defn|1=See [[formal scheme]].}}

{{glossary end}}

==G==

{{glossary}}
{{term|content=g<sup>r</sup><sub>d</sub><!-- not sure how to put r and d in parallel -->}}
{{defn|1=Given a curve ''C'', a divisor ''D'' on it and a vector subspace <math>V \subset H^0(C, \mathcal{O}(D))</math>, one says the linear system <math>\mathbb{P}(V)</math> is a g<sup>r</sup><sub>d</sub> if ''V'' has dimension ''r''+1 and ''D'' has degree ''d''. One says ''C'' has a g<sup>r</sup><sub>d</sub> if there is such a linear system.}}

{{term|1=Gabriel–Rosenberg reconstruction theorem}}
{{defn|1=The [[Gabriel–Rosenberg reconstruction theorem]] states a scheme ''X'' can be recovered from the category of [[quasi-coherent sheaf|quasi-coherent sheaves]] on ''X''.<ref>{{Cite arXiv|title = Tensor categorical foundations of algebraic geometry|eprint=1410.1716 |date = 2014-10-07|first = Martin|last = Brandenburg|class=math.AG }}</ref> 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}}

{{term|1=G-bundle}}
{{defn|1=A principal G-bundle.}}

{{term|1=generic point}}
{{defn|1=A dense point.}}

{{term|1=genus}}
{{defn|1=See [[#arithmetic genus]], [[#geometric genus]].}}

{{term|1=genus formula}}
{{defn|1=The [[genus formula]] for a nodal curve in the projective plane says the genus of the curve is given as
<math>g = (d-1)(d-2)/2 - \delta</math>
where ''d'' is the degree of the curve and δ is the number of nodes (which is zero if the curve is smooth).}}

{{term|1=geometric genus}}
{{defn|1=The [[geometric genus]] of a smooth projective variety ''X'' of dimension ''n'' is
<math>\dim \Gamma(X, \Omega^n_X) = \dim \operatorname{H}^n(X, \mathcal{O}_X)</math>
(where the equality is [[Serre's duality theorem]].)}}

{{term|1=geometric point}}
{{defn|1=The prime spectrum of an algebraically closed field.}}

{{term|1=geometric property}}
{{defn|1=A property of a scheme ''X'' over a field ''k'' is "[[geometrically (algebraic geometry)|geometric]]" if it holds for <math>X_E = X \times_{\operatorname{Spec} k} {\operatorname{Spec} E}</math> for any field extension <math>E/k</math>.}}

{{term|1=geometric quotient}}
{{defn|1=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.}}

{{term|1=gerbe}}
{{defn|1=A [[gerbe]] is (roughly) a [[stack (mathematics)|stack]] that is locally nonempty and in which two objects are locally isomorphic.}}

{{term|1=GIT quotient}}
{{defn|1=The [[GIT quotient]] <math>X / \! / G</math> is <math>\operatorname{Spec}(A^G)</math> when <math>X = \operatorname{Spec} A</math> and <math>\operatorname{Proj}(A^G)</math> when <math>X = \operatorname{Proj} A</math>.}}

{{term|1=good quotient}}
{{defn|1=The [[good quotient]] of a scheme ''X'' with the action of a group scheme ''G'' is an invariant morphism <math>f: X \to Y</math> such that <math>(f_* \mathcal{O}_X)^G = \mathcal{O}_Y.</math>}}

{{term|1=Gorenstein}}
{{defn|no=1|1=A [[Gorenstein scheme]] is a locally Noetherian scheme whose local rings are [[Gorenstein ring]]s.}}
{{defn|no=2|1=A normal variety is said to be <math>\mathbb{Q}</math>-Gorenstein if the canonical divisor on it is <math>\mathbb{Q}</math>-Cartier (and need not be Cohen–Macaulay).}}
{{defn|no=3|1=Some authors call a normal variety Gorenstein if the canonical divisor is Cartier; note this usage is inconsistent with meaning 1.}}

{{term|1=Grauert–Riemenschneider vanishing theorem}}
{{defn|1=The [[Grauert–Riemenschneider vanishing theorem]] extends the [[Kodaira vanishing theorem]] to higher direct image sheaves; see also https://arxiv.org/abs/1404.1827}}

{{term|1=Grothendieck ring of varieties}}
{{defn|1=The [[Grothendieck ring of varieties]] is the free abelian group generated by isomorphism classes of varieties with the relation:
<math>[X] = [Z] + [X - Z]</math>
where ''Z'' is a closed subvariety of a variety ''X'' and equipped with the multiplication
<math>[X] \cdot [Y] = [X \times Y].</math>}}

{{term|1=Grothendieck's vanishing theorem}}
{{defn|[[Grothendieck's vanishing theorem]] concerns [[local cohomology]].}}

{{term|1=group scheme}}
{{defn|A [[group scheme]] is a scheme whose sets of points have the structures of a [[group (mathematics)|group]].}}

{{term|1=group variety}}
{{defn|An old term for a "smooth" algebraic group.}}
{{glossary end}}

==H==

{{glossary}}
{{term|1=Hilbert polynomial}}
{{defn|1=The [[Hilbert polynomial]] of a projective scheme ''X'' over a field is the Euler characteristic <math>\chi(\mathcal{O}_X(s))</math>.}}

{{term|1=Hodge bundle}}
{{defn|1=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 <math>\Gamma(C, \omega_C)</math>.}}

{{term|1=hyperelliptic}}
{{defn|1=A curve is [[hyperelliptic curve|hyperelliptic]] if it has a ''g''<sup>1</sup><sub>2</sub> (i.e., there is a linear system of dimension 1 and degree 2.)}}

{{term|1=hyperplane bundle}}
{{defn|1=Another term for [[Serre's twisting sheaf]] <math>\mathcal{O}_X(1)</math>. It is the dual of the [[tautological line bundle]] (whence the term).}}
{{glossary end}}

==I==

{{glossary}}

{{term|1=image}}
{{defn|1=
If {{nowrap|''f'' : ''Y'' → ''X''}} is any morphism of schemes, the '''scheme-theoretic image''' of ''f'' is the unique ''closed'' subscheme {{nowrap|''i'' : ''Z'' → ''X''}} which satisfies the following [[universal property]]:
#''f'' factors through ''i'',
#if {{nowrap|''j'' : ''Z''′ → ''X''}} is any closed subscheme of ''X'' such that ''f'' factors through ''j'', then ''i'' also factors through ''j''.<ref>{{harvnb|Hartshorne|1977|loc=Exercise II.3.11(d)}}</ref><ref>[http://www.math.columbia.edu/algebraic_geometry/stacks-git/morphisms.pdf The Stacks Project], Chapter 21, §4.</ref>
This notion is distinct from that of the usual set-theoretic image of ''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''.
}}

{{term|1=immersion}}
{{defn|1=
'''Immersions''' {{nowrap|''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.<ref>{{harvnb|Grothendieck|Dieudonné|1960|loc=4.2.1}}</ref> 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 <math>f^\sharp: \mathcal{O}_X \to f_* \mathcal{O}_Y</math> is surjective.<ref name="HartshorneII3"/> A composition of immersions is again an immersion.<ref>{{harvnb|Grothendieck|Dieudonné|1960|loc=4.2.5}}</ref>

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.<ref>Q. Liu, ''Algebraic Geometry and Arithmetic Curves, exercise 2.3</ref>

Note that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not: <math>\operatorname{Spec} A/I</math> and <math>\operatorname{Spec} A/J</math> 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.
}}

{{term|1=ind-scheme}}
{{defn|1=An [[ind-scheme]] is an inductive limit of closed immersions of schemes.}}

{{term|1=invertible sheaf}}
{{defn|1=A locally free sheaf of a rank one. Equivalently, it is a [[torsor (algebraic geometry)|torsor]] for the multiplicative group <math>\mathbb{G}_m</math> (i.e., line bundle).}}
{{term|1=integral}}
{{defn|1=
{{Anchor|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 domain]]s. (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. 
}}

{{term|1=irreducible|content=irreducible {{anchor|irreducible|Irreducible}} }}
{{defn|1=
A scheme ''X'' is said to be ''[[hyperconnected space|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<sub>i</sub> all have exactly one minimal [[prime ideal]]. (Rings possessing exactly one minimal prime ideal are therefore also called [[Irreducible ring|irreducible]].) Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called its [[irreducible component]]s. [[Affine space]] and [[projective space]] are irreducible, while 
''Spec'' ''k''[''x,y'']/(''xy'') = [[Image:Reducible scheme.png|50px]] is not.
}}
{{glossary end}}

==J==

{{glossary}}
{{term|1=Jacobian variety}}
{{defn|1=The [[Jacobian variety]] of a projective curve ''X'' is the degree zero part of the [[Picard variety]] <math>\operatorname{Pic}(X)</math>.}}

{{glossary end}}

==K==

{{glossary}}
{{term|1=Kempf vanishing theorem}}
{{defn|1=The [[Kempf vanishing theorem]] concerns the vanishing of higher cohomology of a flag variety.}}

{{term|klt}}
{{defn|Abbreviation for "[[kawamata log terminal]]"}}

{{term|1=Kodaira dimension}}
{{defn|no=1|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''.}}
{{defn|no=2|1=The Kodaira dimension of a normal variety ''X'' is the Kodaira dimension of its canonical sheaf.}}

{{term|1=Kodaira vanishing theorem}}
{{defn|1=See the [[Kodaira vanishing theorem]].}}

{{term|1=Kuranishi map}}
{{defn|1=See [[Kuranishi structure]].}}

{{glossary end}}

==L==

{{glossary}}
{{term|1=Lelong number}}
{{defn|1=See [[Lelong number]].}}

{{term|1=level structure}}
{{defn|1=see http://math.stanford.edu/~conrad/248BPage/handouts/level.pdf}}

{{term|1=linearization}}
{{defn|1=Another term for the structure of an [[equivariant sheaf]]/vector bundle.}}

{{term|1=local|content=local {{anchor|local}} }}
{{defn|1=
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<sub>i</sub>'', i.e. ''X''=<math>\cup</math> ''X<sub>i</sub>'', every ''X<sub>i</sub>'' has the property ''P''. It is usually the case that it 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<sub>i</sub>''. Using the dictionary between [[commutative ring|(commutative) rings]] and [[affine scheme]]s local properties are thus properties of the rings ''A<sub>i</sub>''. A property ''P'' is local in the above sense, iff the corresponding property of rings is stable under [[Localization of a ring|localization]].

For example, we can speak of ''[[noetherian scheme|locally Noetherian]]'' schemes, namely those which are covered by the spectra of [[Noetherian ring]]s. 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 ring|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'' = <math>\cup</math> ''Spec A<sub>i</sub>'' be a covering of a scheme by open affine subschemes. For definiteness, let ''k'' denote a [[field (mathematics)|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,
Gorenstein.
}}

{{term|1=local complete intersection}}
{{defn|1=The local rings are [[complete intersection ring]]s. See also: [[regular embedding]].}}

{{term|1=local uniformization}}
{{defn|1=The [[local uniformization]] is a method of constructing a weaker form of [[resolution of singularities]] by means of [[valuation ring]]s.}}

{{term|1=locally factorial}}
{{defn|1=The local rings are [[unique factorization domain]]s.}}

{{term|1=locally of finite presentation}}
{{defn|1=
Cf. [[finitely presented scheme|finite presentation]] above.
}}

{{term|1=locally of finite type}}
{{defn|1=
The morphism {{nowrap|''f'' : ''Y'' → ''X''}} is '''locally of finite type''' if <math> X </math> may be covered by affine open sets <math> \text{Spec }B </math> such that each inverse image <math>f^{-1}(\text{Spec }B)</math> is covered by affine open sets <math>\text{Spec }A</math> where each <math> A </math> is finitely generated as a <math>B</math>-algebra. 
}}

{{term|1=locally Noetherian}}
{{defn|1=
A scheme ''X'' covered by Spec ''A<sub>i</sub>'', where the ''A<sub>i</sub>'' are [[Noetherian ring|Noetherian]] rings. If in addition a finite number of such affine spectra covers ''X'', the scheme is called ''[[Noetherian scheme|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
<math>GL_\infty = \cup GL_n</math> is not. 
}}

{{term|1=logarithmic geometry}}

{{term|1=log structure}}
{{defn|1=See [[log structure]]. The notion is due to Fontaine-Illusie and Kato.}}

{{term|1=loop group}}
{{defn|1=See [[loop group]] (the linked article does not discuss a loop group in algebraic geometry; for now see also [[ind-scheme]]).}}

{{glossary end}}

==M==

{{glossary}}
{{term|1=moduli}}
{{defn|1=See for example [[moduli space]].{{quote box
|align = right
|width=33%
|quote=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.
|source=Kollár, János, [https://web.math.princeton.edu/~kollar/book/chap1.pdf Chapter 1], "Book on Moduli of Surfaces".}}}}

{{term|1=Mori's minimal model program}}
{{defn|1=The [[minimal model program]] is a [[research program]] aiming to do [[birational classification]] of algebraic varieties of dimension greater than 2.}}

{{term|morphism}}
{{defn|no=1|A [[morphism of algebraic varieties]] is given locally by polynomials.}}
{{defn|no=2|A [[morphism of schemes]] is a morphism of [[locally ringed space]]s.}}
{{defn|no=3|A morphism <math>f: F \to G</math> of stacks (over, say, the category of ''S''-schemes) is a functor such that <math>P_G \circ f = P_F</math> where <math>P_F, P_G</math> are structure maps to the base category.}}

{{glossary end}}

==N==

{{glossary}}
{{term|1=nef}}
{{defn|1=See [[nef line bundle]].}}

{{term|1=nonsingular}}
{{defn|1=An archaic term for "smooth" as in a [[smooth variety]].}}

{{term|1=normal}}
{{defn|no=1|1=An integral scheme is called ''[[normal scheme|normal]]'', if the local rings are [[integrally closed domain]]s. 
For example, all regular schemes are normal, while singular curves are not.}}
{{defn|no=2|1=A smooth curve <math>C \subset \mathbf{P}^r</math> is said to be ''k''-normal if the hypersurfaces of degree ''k'' cut out the complete linear series <math>|\mathcal{O}_C(k)|</math>. 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.}}
{{defn|no=3|1=A closed subvariety <math>X \subset \mathbf{P}^r</math> 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.}}

{{term|1=normal}}
{{defn|no=1|1=If ''X'' is a closed subscheme of a scheme ''Y'' with ideal sheaf ''I'', then the [[normal sheaf]] to ''X'' is <math>(I/I^2)^* = \mathcal{H}om_{\mathcal{O}_Y}(I/I^2, \mathcal{O}_Y)</math>. If the embedded of ''X'' into ''Y'' is [[regular embedding|regular]], it is locally free and is called the [[normal bundle]].}}
{{defn|no=2|1=The [[Normal bundle#Algebraic geometry|normal cone]] to ''X'' is <math>\operatorname{Spec}_X(\oplus_0^{\infty} I^n/I^{n+1})</math>. if ''X'' is regularly embedded into ''Y'', then the normal cone is isomorphic to <math>\operatorname{Spec}_X(\mathcal{S}ym(I/I^2))</math>, the total space of the normal bundle to ''X''.}}

{{term|1=normal crossings}}
{{defn|1=Abbreviations '''nc''' for normal crossing and '''snc''' for simple normal crossing. Refers to several closely related notions such as nc divisor, nc singularity, snc divisor, and snc singularity. See [[normal crossings]].}}

{{term|1=normally generated}}
{{defn|1=A line bundle ''L'' on a variety ''X'' is said to be [[normally generated]] if, for each integer ''n'' > 0, the natural map <math>\Gamma(X, L)^{\otimes n} \to \Gamma(X, L^{\otimes n})</math> is surjective.}}
{{glossary end}}

==O==
{{glossary}}
{{term|1=open}}
{{defn|no=1|1=A morphism {{nowrap|''f'' : ''Y'' → ''X''}} of schemes is called ''open'' (''closed''), if the underlying map of topological spaces is [[open map|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.
}}
{{defn|no=2|1=An '''open subscheme''' of a scheme ''X'' is an open subset ''U'' with structure sheaf <math>\mathcal{O}_X|_U</math>.<ref name="HartshorneII3">{{harvnb|Hartshorne|1977|loc=§II.3}}</ref>
}}

{{term|1=orbifold}}
{{defn|1=Nowadays an [[orbifold]] is often defined as a [[Deligne–Mumford stack]] over the category of differentiable manifolds.<ref>{{Cite arXiv|title = Global quotients among toric Deligne-Mumford stacks|eprint=1302.0385 |date = 2013-02-02|first1 = Megumi|last1 = Harada|first2 = Derek|last2 = Krepski|class=math.DG }}</ref>}}
{{glossary end}}

==P==

{{glossary}}
{{term|1=''p''-divisible group}}
{{defn|1=See [[p-divisible group|''p''-divisible group]] (roughly an analog of torsion points of an abelian variety).}}

{{term|1=pencil}}
{{defn|1=A linear system of dimension one.}}

{{term|1=Picard group}}
{{defn|1=The [[Picard group]] of ''X'' is the group of the isomorphism classes of line bundles on ''X'', the multiplication being the [[tensor product]].}}

{{term|1=Plücker embedding}}
{{defn|1=The [[Plücker embedding]] is the [[closed embedding]] of the [[Grassmannian variety]] into a projective space.}}

{{term|1=plurigenus}}
{{defn|1=The ''n''-th [[plurigenus]] of a smooth projective variety is <math>\dim \Gamma(X, \omega_X^{\otimes n})</math>. See also [[Hodge number]].}}

{{term|1=Poincaré residue map}}
{{defn|See [[Poincaré residue]].}}

{{term|1=point}}
{{defn|1=
A scheme <math> S </math> is a [[locally ringed space]], so ''a fortiori'' a [[topological space]], but the meanings of ''point of <math> S </math>'' are threefold:

#a point <math> P </math> of the underlying topological space;
#a <math> T </math>-valued point of <math>S</math> is a morphism from <math> T </math> to <math> S </math>, for any scheme <math> T </math>;
#a ''geometric point'', where <math> S </math> is defined over (is equipped with a morphism to) <math> \textrm{Spec}(K) </math>, where <math> K </math> is a [[field (mathematics)|field]], is a morphism from <math> \textrm{Spec} (\overline{K}) </math> to <math> S </math> where <math> \overline{K} </math> is an [[algebraic closure]] of <math>K</math>.

Geometric points are what in the most classical cases, for example [[algebraic varieties]] that are [[complex manifold]]s, would be the ordinary-sense points. The points <math> P </math> of the underlying space include analogues of the [[generic point]]s (in the sense of [[Zariski]], not that of [[André Weil]]), which specialise to ordinary-sense points. The <math> T </math>-valued points are thought of, via [[Yoneda's lemma]], as a way of identifying <math> S </math> with the [[representable functor]] <math> h_{S} </math> it sets up. Historically there was a process by which [[projective geometry]] added more points (''e.g.'' complex points, [[line at infinity]]) to simplify the geometry by refining the basic objects. The <math> T </math>-valued points were a massive further step.

As part of the predominating [[Grothendieck's relative point of view|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 product]]s of two morphisms. For example, a '''geometric fiber''' of a morphism <math> S^{\prime} \to S </math> is thought of as
<math> S^{\prime} \times_{S} \textrm{Spec}(\overline{K}) </math>.
This makes the extension from [[affine scheme]]s, where it is just the [[tensor product of R-algebras]], to all schemes of the fiber product operation a significant (if technically anodyne) result.
}}
{{term|1=polarization}}
{{defn|an embedding into a projective space}}

{{term|1=Proj}}
{{defn|See [[Proj construction]].}}

{{term|1=projection formula}}
{{defn|The [[projection formula]] says that, for a morphism <math>f:X \to Y</math> of schemes, an <math> \mathcal{O}_X</math>-module <math>\mathcal{F}</math> and a [[locally free sheaf|locally free]] <math>\mathcal{O}_Y</math>-module <math>\mathcal{E}</math> of finite rank, there is a natural isomorphism
<math>f_* (F \otimes f^* E) = (f_* F) \otimes E</math>
(in short, <math>f_*</math> is linear with respect to the action of locally free sheaves.)
}}

{{term|1=projective}}
{{defn|no=1|1=A [[projective variety]] is a closed subvariety of a projective space.}}
{{defn|no=2|1=A [[projective scheme]] over a scheme ''S'' is an ''S''-scheme that factors through some projective space <math>\mathbf{P}^N_S \to S</math> as a closed subscheme.}}
{{defn|no=3|1=Projective morphisms are defined similarly to affine morphisms: {{nowrap|''f'' : ''Y'' → ''X''}} is called '''[[projective variety|projective]]''' if it factors as a closed immersion followed by the projection of a [[projective space]] <math> \mathbb{P}^{n}_X := \mathbb{P}^n \times_{\mathrm{Spec}\mathbb Z} X</math> to <math> X </math>.<ref>{{harvnb|Hartshorne|1977|loc=II.4}}</ref> Note that this definition is more restrictive than that of [[Éléments de géométrie algébrique|EGA]], II.5.5.2. The latter defines <math> f </math> to be projective if it is given by the [[global Proj|global '''Proj''']] of a quasi-coherent graded ''O<sub>X</sub>''-Algebra <math>\mathcal S</math> such that <math>\mathcal S_1</math> is finitely generated and generates the algebra <math>\mathcal S</math>. Both definitions coincide when <math>X</math> is affine or more generally if it is quasi-compact, separated and admits an ample sheaf,<ref>[[Éléments de géométrie algébrique|EGA]], II.5.5.4(ii).</ref> e.g. if <math>X</math> is an open subscheme of a projective space <math>\mathbb P^n_A</math> over a ring <math>A</math>.
}}

{{term|1=projective bundle}}
{{defn|1=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'':
<math>\mathbf{P}(E) = \mathbf{Proj}(\operatorname{Sym}_{\mathcal{O}_X}(E^{\vee})).</math>
Note this definition is standard nowadays (e.g., Fulton's ''Intersection theory'') but differs from EGA and Hartshorne (they don't take a dual).}}

{{term|1=projectively normal}}
{{defn|1=See [[#normal]].}}

{{term|1=proper}}
{{defn|1=
A morphism is '''[[proper morphism|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.
}}

{{term|property P}}
{{defn|Let '''P''' be a property of a scheme that is stable under base change (finite-type, proper, smooth, étale, etc.). Then a representable morphism <math>f: F \to G</math> is said to have property '''P''' if, for any <math>B \to G</math> with ''B'' a scheme, the base change <math>F \times_G B \to B</math> has property '''P'''.}}

{{term|1=pseudo-reductive}}
{{defn|1=[[Pseudo-reductive group|Pseudoreductive]] generalizes [[Reductive group|reductive]] in the context of [[Connected space|connected]] [[Singular point of an algebraic variety|smooth]] [[linear algebraic group]].}}

{{term|1=pure dimension}}
{{defn|1=A scheme has pure dimension ''d'' if each irreducible component has dimension ''d''.}}

{{glossary end}}

==Q==
{{glossary}}
{{term|1=quasi-coherent}}
{{defn|1=A quasi-coherent sheaf on a Noetherian scheme ''X'' is a [[sheaf of modules|sheaf of ''O''<sub>''X''</sub>-modules]] that is locally given by modules.
}}

{{term|1=quasi-compact}}
{{defn|1=
A morphism {{nowrap|''f'' : ''Y'' → ''X''}} is called ''[[quasi-compact morphism|quasi-compact]]'', if for some (equivalently: every) open affine cover of ''X'' by some ''U<sub>i</sub>'' = ''Spec B<sub>i</sub>'', the preimages ''f''<sup>−1</sup>(''U<sub>i</sub>'') are [[quasi-compact]].
}}

{{term|1=quasi-finite}}
{{defn|1=
The morphism {{nowrap|''f'' : ''Y'' → ''X''}} has '''finite fibers''' if the fiber over each point <math> x \in X </math> is a finite set. A morphism is '''[[quasi-finite morphism|quasi-finite]]''' if it is of finite type and has finite fibers.
}}

{{term|1=quasi-projective}}
{{defn|1=A [[quasi-projective variety]] is a [[locally closed]] subvariety of a projective space.}}

{{term|1=quasi-separated}} 
{{defn|1=
A morphism {{nowrap|''f'' : ''Y'' → ''X''}} is called '''[[quasi-separated]]''' or ('''''Y'' is quasi-separated over ''X''''') if the diagonal morphism {{nowrap|''Y'' → ''Y'' ×<sub>''X''</sub>''Y''}} is quasi-compact. A scheme ''Y'' is called '''quasi-separated''' if ''Y'' is quasi-separated over Spec('''Z''').<ref>{{harvnb|Grothendieck|Dieudonné|1964|loc=1.2.1}}</ref>
}}

{{term|1=quasi-split}} 
{{defn|1=A [[reductive group]] <math>G</math> defined over a field <math>k</math> is [[Quasi-split group|quasi-split]] if and only if it admits a Borel subgroup <math>B\subseteq G</math> defined over <math>k</math>. Any quasi-split reductive group is a split-reductive reductive group, but there are quasi-split reductive groups that are not split-reductive.}}

{{term|1=Quot scheme}}
{{defn|1=A [[Quot scheme]] parametrizes quotients of locally free sheaves on a projective scheme.}}

{{term|1=quotient stack}}
{{defn|Usually denoted by [''X''/''G''], a [[quotient stack]] generalizes a quotient of a scheme or variety.}}

{{glossary end}}

==R==

{{glossary}}
{{term|1=rational}}
{{defn|no=1|1=Over an algebraically closed field, a variety is [[rational variety|rational]] if it is birational to a projective space. For example, [[rational curve]]s and [[rational surface]]s are those birational to <math>\mathbb{P}^1, \mathbb{P}^2</math>.}}
{{defn|no=2|Given a field ''k'' and a relative scheme ''X'' → ''S'', a [[rational point|''k''-rational point]] of ''X'' is an ''S''-morphism <math>\operatorname{Spec}(k) \to X</math>.}}

{{term|1=rational function}}
{{defn|1=An element in the [[function field of an algebraic variety|function field]] <math>k(X) = \varinjlim k[U]</math> where the limit runs over all coordinates rings of open subsets ''U'' of an (irreducible) algebraic variety ''X''. See also [[function field (scheme theory)]].}}

{{term|1=rational normal curve}}
{{defn|1=A [[rational normal curve]] is the image of
<math>\mathbf{P}^1 \to \mathbf{P}^d, \, (s:t) \mapsto (s^d : s^{d-1} t : \cdots : t^d)</math>.
If ''d'' = 3, it is also called the [[twisted cubic]].}}

{{term|1=rational singularities}}
{{defn|1=A variety ''X'' over a field of characteristic zero has [[rational singularities]] if there is a resolution of singularities <math>f:X' \to X</math> such that <math>f_*(\mathcal{O}_{X'}) = \mathcal{O}_X</math> and <math>R^i f_*(\mathcal{O}_{X'}) = 0, \, i \ge 1</math>.}}

{{term|1=reduced|content=reduced {{anchor|reduced|Reduced}} }}
{{defn|no=1|1=A commutative ring <math>R</math> is [[reduced ring|reduced]] if it has no nonzero nilpotent elements, i.e., its nilradical is the zero ideal, <math>\sqrt{(0)} = (0)</math>. Equivalently, <math>R</math> is reduced if <math>\operatorname{Spec}(R)</math> is a reduced scheme.}}
{{defn|no=2|1=A scheme X is reduced if its stalks <math>\mathcal{O}_{X,x}</math> are reduced rings. Equivalently X is reduced if, for each open subset <math>U \subset X</math>, <math>\mathcal{O}_X(U)</math> is a reduced ring, i.e., <math>X</math> has no nonzero nilpotent sections.}}

{{term|1=reductive}}
{{defn|1=A [[Connected space|connected]] [[linear algebraic group]] <math>G</math> over a field <math>k</math> is a [[reductive group]] if and only if the unipotent radical <math>R_u(G_{\overline{k}})</math>  of the base change <math>G_{\overline{k}}</math> of <math>G</math> to an algebraic closure <math>\overline{k}</math> is trivial.}}

{{term|1=reflexive sheaf}}
{{defn|1=A coherent sheaf is [[reflexive sheaf|reflexive]] if the canonical map to the second dual is an isomorphism.}}

{{term|1=regular}}
{{defn|1=A [[regular scheme]] is a scheme where the local rings are [[regular local ring]]s. For example,  [[smooth scheme|smooth varieties]] over a field are regular, while
''Spec k''[''x,y'']/(''x''<sup>2</sup>+''x''<sup>3</sup>-''y''<sup>2</sup>)=[[Image:Non regular scheme thumb.png|50px]] is not.
}}

{{term|1=regular embedding}}
{{defn|1=A [[closed immersion]] <math>i: X \hookrightarrow Y</math> 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, <math>\mathcal{I}/\mathcal{I}^2</math> when <math>\mathcal{I}</math> is the ideal sheaf of ''X'', is locally free.}}

{{term|1=regular function}}
{{defn|1=A [[morphism of varieties|morphism]] from an algebraic variety to the [[affine line]].}}

{{term|representable morphism}}
{{defn|A morphism <math>F \to G</math> of stacks such that, for any morphism <math>B \to G</math> from a scheme ''B'', the base change <math>F \times_G B</math> is an algebraic space. If "algebraic space" is replaced by "scheme", then it is said to be strongly representable.}}

{{term|1=resolution of singularities}}
{{defn|1=A [[resolution of singularities]] of a scheme ''X'' is a proper [[birational morphism]] <math>\pi: Z \to X</math> such that ''Z'' is [[smooth scheme|smooth]].}}

{{term|1=Riemann–Hurwitz formula}}
{{defn|1=Given a finite separable morphism <math>\pi:X \to Y</math> between smooth projective curves, if <math>\pi</math> 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 index|ramification indices]]:
<math>2g(X) - 2 = \operatorname{deg}(\pi) (2g(Y) - 2) + \sum_{y \in Y} (e_y - 1)</math>.
Nowadays, the formula is viewed as a consequence of the more general formula (which is valid even if π is not tame):
<math>K_X \sim \pi^*K_Y + R</math>
where <math>\sim</math> means a [[linear equivalence]] and <math>R = \sum_{P \in X} \operatorname{length}_{\mathcal{O}_P} (\Omega_{X/Y})P</math> is the divisor of the relative cotangent sheaf <math>\Omega_{X/Y}</math> (called the [[different (mathematics)|different]]).}}

{{term|1=Riemann–Roch formula}}
{{defn|no=1|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'':
<math>\chi(L) = d - g + 1</math>.
For example, the formula implies the degree of the canonical divisor ''K'' is 2''g'' - 2.}}
{{defn|no=2|1=The general version is due to Grothendieck and called the [[Grothendieck–Riemann–Roch formula]]. It says: if <math>\pi:X \to S</math> is a proper morphism with smooth ''X'', ''S'' and if ''E'' is a vector bundle on ''X'', then as equality in the rational [[Chow group]]
<math>\operatorname{ch}(\pi_! E) \cdot \operatorname{td}(S) = \pi_*(\operatorname{ch}(E) \cdot \operatorname{td}(X))</math>
where <math>\pi_! = \sum_i (-1)^i R^i \pi_*</math>, <math>\operatorname{ch}</math> means a [[Chern character]] and <math>\operatorname{td}</math> a [[Todd class]] of the tangent bundle of a space, and, over the complex numbers, <math>\pi_*</math> is an [[integration along fibers]]. For example, if the base ''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 <math>\pi_*(e^{c_1(L)}(1-c_1(T^*X)/2)) = \operatorname{deg}(L) - g + 1.</math>}}

{{term|1=rigid}}
{{defn|1=Every [[infinitesimal deformation]] is trivial. For example, the [[projective space]] is rigid since <math>\operatorname{H}^1(\mathbf{P}^n, T_{\mathbf{P}^n}) = 0</math> (and using the [[Kodaira–Spencer map]]).}}

{{term|1=rigidify}}
{{defn|1=A heuristic term, roughly equivalent to "killing automorphisms". For example, one might say "we introduce [[Level structure (algebraic geometry)|level structure]]s resp. [[Moduli of algebraic curves#Moduli of marked curves|marked points]] to rigidify the geometric situation."}}

{{glossary end}}

==S==

{{glossary}}
{{quote box
|align = right
|width=33%
|quote=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]]''?
|source=[http://www.landsburg.com/grothendieck/mclarty1.pdf]
}}{{term|1=scheme}}
{{defn|1=A [[scheme (mathematics)|scheme]] is a [[locally ringed space]] that is locally a [[prime spectrum]] of a [[commutative ring]].}}

{{term|1=Schubert}}
{{defn|no=1|1=A [[Schubert cell]] is a ''B''-orbit on the Grassmannian <math>\operatorname{Gr}(d, n)</math> where ''B'' is the standard Borel; i.e., the group of upper triangular matrices.}}
{{defn|no=2|1=A [[Schubert variety]] is the closure of a Schubert cell.}}

{{term|1=scroll}}
{{defn|1=A [[rational normal scroll]] is a [[Ruled surface#Ruled surfaces in algebraic geometry|ruled surface]] which is of [[Degree of an algebraic variety|degree]] <math>n</math> in a [[Projective space#Algebraic geometry|projective space]] <math>\mathbb{P}^{n+1}</math> for some <math>n\in\mathbb{N}_{>1}</math>.}}

{{term|1=secant variety}}
{{defn|1=The [[secant variety]] to a projective variety <math>V \subset \mathbb{P}^r</math> is the closure of the union of all secant lines to ''V'' in <math>\mathbb{P}^r</math>.}}

{{term|1=section ring}}
{{defn|1=The [[section ring]] or the ring of sections of a line bundle ''L'' on a scheme ''X'' is the graded ring <math>\oplus_0^\infty \Gamma(X, L^n)</math>.}}

{{term|content=Serre's conditions ''S''<sub>''n''</sub>}}
{{defn|1=See [[Serre's conditions on normality]]. See also https://mathoverflow.net/q/22228 }}

{{term|1=Serre duality}}
{{defn|1=See [[#dualizing sheaf]]}}

{{term|1=separated}}
{{defn|1=A [[separated morphism]] is a morphism <math> f </math> such that the [[fiber product]] of <math> f </math> with itself along <math> f </math> has its [[diagonal]] as a closed subscheme &mdash; in other words, the [[diagonal morphism (algebraic geometry)|diagonal morphism]] is a ''[[closed immersion]]''.}}

{{term|1=sheaf generated by global sections}}
{{defn|1=A sheaf with a set of global sections that span the stalk of the sheaf at every point. See [[Sheaf generated by global sections]].}}

{{term|1=simple}}
{{defn|no=1|1=The term "simple point" is an old term for a "smooth point".}}
{{defn|no=2|1=A [[Normal crossing singularity|simple 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 in [[Resolution of singularities#Definitions|strong desingularization]] as well as in stabilization for compactifying moduli problems.}}
{{defn|no=3|1=In the context of [[linear algebraic group]]s there are [[Linear algebraic group#Semisimple and reductive groups|semisimple groups]] and [[Reductive group#Simple reductive groups|simple 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.}}

{{term|1=smooth}}
{{defn|no=1|1=
{{main|smooth morphism}}
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 {{nowrap|''f'' : ''Y'' → ''X''}}:
# for any ''y'' &isin; ''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 space|affine ''n''-space]] over ''U''.
# ''f'' is flat, locally of finite presentation, and for every geometric point <math>\bar{y}</math> of ''Y'' (a morphism from the spectrum of an algebraically closed field <math>k(\bar{y})</math> to ''Y''), the geometric fiber <math>X_{\bar{y}}:=X\times_Y \mathrm{Spec} (k(\bar{y}))</math> is a smooth ''n''-dimensional variety over <math>k(\bar{y})</math> in the sense of classical algebraic geometry.
}}
{{defn|no=2|1=A [[smooth scheme]] over a perfect field ''k'' is a scheme ''X'' that is locally of finite type and [[regular scheme|regular]] over ''k''.}}
{{defn|no=3|1=A smooth scheme over a field ''k'' is a scheme ''X'' that is geometrically smooth: <math>X \times_k \overline{k}</math> is smooth.}}

{{term|1=special}}
{{defn|1=A divisor ''D'' on a smooth curve ''C'' is [[special divsor|special]] if <math>h^0(\mathcal{O}(K - D))</math>, which is called the index of speciality, is positive.}}

{{term|1=spherical variety}}
{{defn|1=A [[spherical variety]] is a normal ''G''-variety (''G'' connected reductive) with an open dense orbit by a Borel subgroup of ''G''.}}

{{term|1=split}}
{{defn|no=1|1=In the context of an [[algebraic group]] <math>G</math> for certain properties <math>P</math> there is the derived property split-<math>P</math>. Usually <math>P</math> is a property that is automatic or more common over algebraically closed fields <math>\overline{k}</math>. If this property holds already for <math>G</math> defined over a not necessarily algebraically closed field <math>k</math> then <math>G</math> is said to satisfy split-<math>P</math>.}}
{{defn|no=2|1=A linear algebraic group <math>G</math> defined over a field <math>k</math> is a [[Linear algebraic group#Tori|torus]] if only if its base change <math>G_{\overline{k}}</math> to an algebraic closure <math>\overline{k}</math> is isomorphic to a product of multiplicative groups <math>G_{m,\overline{k}}^{n}</math>. <math>G</math> is a '''split torus''' if and only if it is isomorphic to <math>G_{m,k}^{n}</math> without any base change. <math>G</math> is said to '''split''' over an intermediate field <math>k\subseteq L\subseteq\overline{k}</math> if and only if its base change <math>G_{L}</math> to <math>L</math> is isomorphic to <math>G_{m,L}^{n}</math>.}}
{{defn|no=3|1=A [[reductive group]] <math>G</math> defined over a field <math>k</math> is '''split-reductive''' if and only if a maximal torus <math>T\subseteq G</math> defined over <math>k</math> is a split torus. Since any [[Reductive group#Simple reductive groups|simple group]] is reductive a '''split simple group''' means a simple group that is split-reductive.}}
{{defn|no=4|1=A [[Connected space|connected]] [[Solvable group|solvable]] linear algebraic group <math>G</math> defined over a field <math>k</math> is '''split''' if and only if it has [[composition series]] <math>B=B_0\supset B_1\supset \ldots\supset B_t=\{1\}</math> defined over <math>k</math> such that each successive quotient <math>B_i/B_{i+1}</math> is isomorphic to either the multiplicative group <math>G_{m,k}</math> or the additive group <math>G_{m,a}</math> over <math>k</math>.}}
{{defn|no=5|1=A linear algebraic group <math>G</math> defined over a field <math>k</math> is '''split''' if and only if it has a [[Borel subgroup]] <math>B\subseteq G</math> defined over <math>k</math> that is split in the sense of connected solvable linear algebraic groups.}}
{{defn|no=6|1=In the classification of [[Real form (Lie theory)|real Lie algebras]] [[split Lie algebra]]s play an important role. There is a close connection between linear Lie groups, their associated Lie algebras and linear algebraic groups over <math>k=\mathbb{R}</math> resp. <math>\mathbb{C}</math>. The term '''split''' has similar meanings for Lie theory and linear algebraic groups.}}




{{term|1=stable}}
{{defn|no=1|A [[stable curve]] is a curve with some "mild" singularity, used to construct a good-behaving [[moduli space of curves]].}}
{{defn|no=2|A [[stable vector bundle]] is used to construct the [[moduli space of vector bundles]].}}

{{term|1=stack}}
{{defn|1=A [[stack (mathematics)|stack]] parametrizes sets of points together with automorphisms.}}

{{term|1=strict transform}}
{{defn|1=Given a blow-up <math>\pi: \widetilde{X} \to X</math> along a closed subscheme ''Z'' and a morphism <math>f: Y \to X</math>, the [[strict transform]] of ''Y'' (also called proper transform) is the [[Blowing up|blow-up]] <math>\widetilde{Y} \to Y</math> of ''Y'' along the closed subscheme <math>f^{-1} Z</math>. If ''f'' is a closed immersion, then the induced map <math>\widetilde{Y} \hookrightarrow \widetilde{X}</math> is also a closed immersion.}}

{{term|1=subscheme}}
{{defn|1=
A '''subscheme''', without qualifier, of ''X'' is a closed subscheme of an open subscheme of ''X''.
}}

{{term|1=surface}}
{{defn|1=An algebraic variety of dimension two.}}

{{term|1=symmetric variety}}
{{defn|1=An analog of a [[symmetric space]]. See [[symmetric variety]].}}

{{glossary end}}

==T==

{{glossary}}
{{term|1=tangent space}}
{{defn|1=See [[Zariski tangent space]].}}

{{term|1=tautological line bundle}}
{{defn|1=The [[tautological line bundle]] of a projective scheme ''X'' is the dual of [[Serre's twisting sheaf]] <math>\mathcal{O}_X(1)</math>; that is, <math>\mathcal{O}_X(-1)</math>.}}

{{term|1=theorem}}
{{defn|1=See [[Zariski's main theorem]], [[theorem on formal functions]], [[cohomology base change theorem]], [[:Category:Theorems in algebraic geometry]].}}

{{term|1=torus embedding}}
{{defn|1=An old term for a [[toric variety]]}}

{{term|1=toric variety}}
{{defn|1=A [[toric variety]] is a normal variety with the action of a torus such that the torus has an open dense orbit.}}

{{term|1=tropical geometry}}
{{defn|1=A kind of a piecewise-linear algebraic geometry. See [[tropical geometry]].}}

{{term|1=torus}}
{{defn|1=A [[algebraic torus|split torus]] is a product of finitely many [[multiplicative group]]s <math>\mathbb{G}_m</math>.}}

{{glossary end}}

==U==

{{glossary}}
{{term|1=universal}}
{{defn|no=1|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 [[Moduli of curves|universal curve]]. A [[tautological bundle]] would be another example of a universal object.}}
{{defn|no=2|Let <math>\mathcal{M}_{g}</math> be the moduli of smooth projective curves of genus ''g'' and <math>\mathcal{C}_g = \mathcal{M}_{g, 1}</math> that of smooth projective curves of genus ''g'' with single marked points. In literature, the forgetful map
<math>\pi: \mathcal{C}_{g} \to \mathcal{M}_{g}</math>
is often called a universal curve.}}

{{term|1=universally}}
{{defn|1=
A morphism has some property universally if all base changes of the morphism have this property. Examples include [[universally catenary]], [[universally injective]].
}}

{{term|1=unramified}}
{{defn|1=
For a point <math> y </math> in <math> Y </math>, consider the corresponding morphism of local rings
<math>f^\# \colon \mathcal{O}_{X, f(y)} \to \mathcal{O}_{Y, y}</math>.

Let <math> \mathfrak{m} </math> be the maximal ideal of <math> \mathcal{O}_{X,f(y)} </math>, and let
<math>\mathfrak{n} = f^\#(\mathfrak{m}) \mathcal{O}_{Y,y}</math>

be the ideal generated by the image of <math> \mathfrak{m} </math> in <math>\mathcal{O}_{Y,y} </math>. The morphism <math> f </math> is '''[[unramified morphism|unramified]]''' (resp. '''G-unramified''') if it is locally of finite type (resp. locally of finite presentation) and if for all <math> y </math> in <math> Y </math>, <math> \mathfrak{n} </math> is the maximal ideal of <math> \mathcal{O}_{Y,y} </math> and the induced map
<math>\mathcal{O}_{X,f(y)}/\mathfrak{m} \to \mathcal{O}_{Y,y}/\mathfrak{n} </math>

is a [[finite field extension|finite]] [[separable field extension]].<ref>The notion G-unramified is what is called "unramified" in EGA, but we follow Raynaud's definition of "unramified", so that [[closed immersion]]s are unramified.  See [http://stacks.math.columbia.edu/tag/02G4 Tag 02G4 in the Stacks Project] for more details.</ref> This is the geometric version (and generalization) of an [[Ramification (mathematics)|unramified field extension]] in [[algebraic number theory]].
}}
{{glossary end}}

==V==

{{glossary}}
{{term|1=variety}}
{{defn|1=a synonym with "algebraic variety".}}

{{term|1=very ample}}
{{defn|1=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.}}

{{glossary end}}

==W==

{{glossary}}
{{term|1=weakly normal}}
{{defn|1=a scheme is weakly normal if any finite birational morphism to it is an isomorphism.}}

{{term|1=Weil divisor}}
{{defn|1=Another but more standard term for a "codimension-one cycle"; see [[divisor (algebraic geometry)|divisor]].}}

{{term|1=Weil reciprocity}}
{{defn|1=See [[Weil reciprocity]].}}

{{glossary end}}

==Z==

{{glossary}}
{{term|1=Zariski–Riemann space}}
{{defn|1=A [[Zariski–Riemann space]] is a locally ringed space whose points are valuation rings.}}

{{glossary end}}

== Notes ==
{{Reflist|30em}}

== References ==
*{{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Intersection theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=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] | isbn=978-3-540-62046-4|mr=1644323 | year=1998 | volume=2 | doi=10.1007/978-1-4612-1700-8| doi-access=free }}
*{{EGA | book=I}}
*{{EGA | book=II}}
*{{EGA | book=III-1}}
*{{EGA | book=III-2}}
*{{EGA | book=IV-1}}
*{{EGA | book=IV-2}}
*{{EGA | book=IV-3}}
*{{EGA | book=IV-4}}
*{{Hartshorne AG}}
*[[János Kollár|Kollár, János]], "Book on Moduli of Surfaces" available at his website [https://web.math.princeton.edu/~kollar/]
*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 [http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1 book] worked out by many authors.

== See also ==
*[[Glossary of arithmetic and Diophantine geometry]]
*[[Glossary of classical algebraic geometry]]
*[[Glossary of differential geometry and topology]]
*[[Glossary of Riemannian and metric geometry]]
*[[List of complex and algebraic surfaces]]
*[[List of surfaces]]
*[[List of curves]]

[[Category:Glossaries of mathematics|Algebraic geometry]]
[[Category:Algebraic geometry]]
[[Category:Scheme theory]]
[[Category:Wikipedia glossaries using description lists]]