Glossary of algebraic geometry

This is a glossary of algebraic geometry.

See also glossary of commutative algebra, glossary of classical algebraic geometry, glossary of scheme theory and glossary of ring theory.

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

Contents :

!$@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
XYZ
See also
References

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$\eta$

A generic point.

|D|

The complete linear system of a Cartier divisor D on a scheme X; that is,

|D| = \mathbf{P}(\Gamma(X, \mathcal{O}_X(D)))

[X/G]

The quotient stack of, say, an algebraic space X by the action of a group scheme G.

$X/\!/G$

The GIT quotient of a scheme X by the 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 = \mathcal{O}_X

, the structure sheaf on X, then it means the direct sum of n copies of

\mathcal{O}_X

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

The tautological line bundle. It is the dual of Serre's twisting sheaf

\mathcal{O}_X(1)

$\mathcal{O}_X(1)$

Serre's twisting sheaf. It is the dual of the tautological line bundle

\mathcal{O}_X(-1)

. It is also called the hyperplane bundle.

$\mathcal{O}_X(D)$

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,

\mathcal{O}_X(D)

is the image of D under the natural group homomorphism from the group of Cartier divisors to the Picard group

\operatorname{Pic}(X)

of X, the group of isomorphism classes of line bundles on X.

3. In general,

\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.

$\Omega_X^p$

\Omega_X^1

is the sheaf of Kähler differentials on X.

\Omega_X^p

is the p-th exterior power of

\Omega_X^1

$\Omega_X^p(\log D)$

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.)

\Omega_X^p(\log D)

is the p-th exterior power of

\Omega_X^1(\log D)

P(V)

Unfortunately, the notation is ambiguous. Its traditional meaning is the projectivization of a finite-dimensional k-vector space V; i.e.,

\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.

Q-factorial

A normal variety is

\mathbb{Q}

-factorial if every

\mathbb{Q}

-Weil divisor is

\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.

Spec^an(R)

The set of all valuations for a ring R with a certain weak topology; it is called the Berkovich spectrum of R.

A

abelian: 1. An abelian variety is a complete group variety.; 2. An abelian scheme is a (flat) family of abelian varieties.
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. 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 O_X-Algebras, defined by analogy with the spectrum of a ring. Important affine morphisms are vector bundles, and finite morphisms.
algebraic set: An algebraic set over a field k is a reduced separated scheme of finite type over $\operatorname{Spec}(k)$ . 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 $\operatorname{Spec}(k)$ . Note, not assuming k is algebraically closed causes some pathology; for example, $\mathbb{C} \times_{\mathbb{R}} \mathbb{C}$ is not a variety since the coordinate ring is not an integral domain.
algebraic vector bundle: A locally free sheaf of a finite rank.
arithmetic genus: The arithmetic genus of a projective variety X of dimension r is $(-1)^r (\chi(\mathcal{O}_X) - 1)$ .
arithmetic: The arithmetic scheme is a scheme over $\operatorname{Spec}(\mathbf{Z})$ , the prime spectrum of the ring of rational integers.
artinian: 0-dimensional and Noetherian. The definition applies both to a scheme and a ring.

B

big: A big line bundle L on X of dimension n is a line bundle such that $\displaystyle \limsup_{l \to \infty} \operatorname{dim} \Gamma(X, L^l) / l^n > 0$ .
birational morphism: A birational morphism between schemes is a morphism that becomes an isomorphism after restricted to some open dense subset.

C

Calabi–Yau: 1. The Calabi–Yau metric is a Kähler metric whose Ricci curvature is zero.
canonical: 1. The canonical sheaf on a normal variety X of dimension n is $\omega_X = i_* \Omega_U^n$ where i is the inclusion of the smooth locus U and $\Omega_U^n$ is the sheaf of differential forms on U of degree n. If the base field has characteristic zero instead of normality, then one replaces i by a resolution of singularities.; 2. The canonical class $K_X$ on a normal variety X is the divisor class such that $\mathcal{O}_X(K_X) = \omega_X$ .; 3. The canonical divisor is a representative of the canonical class $K_X$ 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: 1. The canonical model is the Proj of a canonical ring (assuming the ring is finitely generated.)
Cartier: 1. A 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).
catenary: A scheme is catenary, if all chains between two irreducible closed subschemes have the same length. Examples include virtually everything, e.g. varieties over a field, and it is hard to construct examples that are not catenary.
central fiber: 1. A special fiber.
Chow group: The k-th Chow group $A_k(X)$ of a smooth variety X is the free abelian group generated by closed subvarieties of dimension k (cycles) modulo principal cycles.
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 $\mathcal{O}_X$ -ideals. The support of the quotient sheaf $\mathcal{O}_X/J$ is a closed subset Z of X and $(Z,(\mathcal{O}_X/J)|_Z)$ is a scheme called the closed subscheme defined by the quasi-coherent sheaf of ideals J.^[1] 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.
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])
crepant: A crepant morphism $f: X \to Y$ between normal varieties is a morphism such that $f^* \omega_Y = \omega_X$ .
curve: An algebraic variety of dimension one.

D

deformation: An older term (a la Kodaira) for #degeneration
degeneration: 1. A scheme X is said to degenerate to a scheme $X_0$ (called the limit of X) if there is a scheme $\pi: Y \to \mathbf{A}^1$ with generic fiber X and special fiber $X_0$ .; 2. A flat degeneration is a degeneration such that that $\pi$ is flat; e.g., toric degeneration.
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.
dominant: A morphism f : Y → X is called dominant, if the image f(Y) 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
dualizing sheaf: An invertible sheaf $\omega_X$ on X such that Serre duality $H^{n-i}(X, L^{\vee} \otimes \omega) \simeq H^i(X, L)^*$ holds for any line bundle L on X.

E

elliptic curve

An elliptic curve is a smooth projective curve of genus one.

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

and

Y

over an algebraically closed field, étale morphisms are precisely those inducing an isomorphism of tangent spaces

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 \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.

F

Fano

A Fano variety is a smooth projective variety X whose anticanonical sheaf

\omega_X^{-1}

is ample.

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

\mathbb{Z}

of integers; so that any scheme

S

is over

\textrm{Spec} (\mathbb{Z})

, and in a unique way.

finite

The morphism f : Y → X is finite if

X

may be covered by affine open sets

\text{Spec }B

such that each

f^{-1}(\text{Spec }B)

is affine — say of the form

\text{Spec }A

— and furthermore

A

is finitely generated as a

B

-module. See finite morphism.

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

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

The morphism f : Y → X has finite fibers if the fiber over each point $x \in X$ is a finite set. A morphism is quasi-finite if it is of finite type and has finite fibers.

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.

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 subset U of f(y) and an open affine neighbourhood V of y such that f(V) ⊆ U and $\mathcal{O}_Y(V)$ is a finitely presented algebra over $\mathcal{O}_X(U)$ . 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.^[2]

The morphism 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.^[3]

flat

A morphism

f

is flat if it gives rise to a flat map on stalks. When viewing a morphism f : Y → X as a family of schemes parametrized by the points of

X

, the geometric meaning of flatness could roughly be described by saying that the fibers

f^{-1}(x)

do not vary too wildly.

G

G-bundle: A principal G-bundle.
generic point: A dense point.
geometric point: The prime spectrum of an algebraically closed field.
geometric property: A property of a scheme X over a field k is "geometric" if it holds for $X_E = X \times_{\operatorname{Spec} k} {\operatorname{Spec} E}$ for any field extension $E/k$ .
geometric quotient: The geometric quotient of a scheme X with the action of a group scheme G is a good quotient such that the fibers are orbits.
GIT quotient: The GIT quotient $X / \! / G$ is $\operatorname{Spec}(A^G)$ when $X = \operatorname{Spec} A$ and $\operatorname{Proj}(A^G)$ when $X = \operatorname{Proj} A$ .
good quotient: The good quotient of a scheme X with the action of a group scheme G is an invariant morphism $f: X \to Y$ such that $(f_* \mathcal{O}_X)^G = \mathcal{O}_Y.$

Gorenstein

Cohen-Macaulay and the dualizing sheaf is invertible.

Grothendieck's vanishing theorem

Grothendieck's vanishing theorem concerns local cohomology.

H

Hilbert polynomial: The Hilbert polynomial of a projective scheme X over a field is the Euler characteristic $\chi(\mathcal{O}_X(s))$ .
hyperplane bundle: Another term for Serre's twisting sheaf $\mathcal{O}_X(1)$ . It is the dual of the tautological line bundle (whence the term).

I

image

If f : Y → X is any morphism of schemes, the scheme-theoretic image of f is the unique closed subscheme i : Z → X which satisfies the following universal property:

f factors through i,
if j : Z′ → X is any closed subscheme of X such that f factors through j, then i also factors through j.^[4]^[5]

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.

immersion

Immersions f : Y → X are maps that factor through isomorphisms with subschemes. Specifically, an open immersion factors through an isomorphism with an open subscheme and a closed immersion factors through an isomorphism with a closed subscheme.^[6] 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

f^\sharp: \mathcal{O}_X \to f_* \mathcal{O}_Y

is surjective.^[7] A composition of immersions is again an immersion.^[8]

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.^[9]

Note that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not:

\operatorname{Spec} A/I

and

\operatorname{Spec} A/J

may be homeomorphic but not isomorphic. This happens, for example, if I is the radical of J but J is not a radical ideal. When specifying a closed subset of a scheme without mentioning the scheme structure, usually the so-called reduced scheme structure is meant, that is, the scheme structure corresponding to the unique radical ideal consisting of all functions vanishing on that closed subset.

ind-scheme

An ind-scheme is an inductive limit of closed immersions of schemes.

invertible sheaf

A locally free sheaf of a rank one. Equivalently, it is a torsor for the multiplicative group

\mathbb{G}_m

(i.e., line bundle).

integral

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

irreducible

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

is not.

J

Jacobian variety: The Jacobian variety of a projective curve X is the degree zero part of the Picard variety $\operatorname{Pic}(X)$ .

K

Kodaira dimension: 1. The Kodaira dimension (also called the Iitaka dimension) of a semi-ample line bundle L is the dimension of Proj of the section ring of L.; 2. The Kodaira dimension of a normal variety X is the Kodaira dimension of its canonical sheaf.

L

linearlization

Another term for the structure of an equivariant sheaf/vector bundle.

local

Most important properties of schemes are local in nature, i.e. a scheme X has a certain property P if and only if for any cover of X by open subschemes X_i, i.e. X=

\cup

X_i, every X_i has the property P. It is usually the case that is enough to check one cover, not all possible ones. One also says that a certain property is Zariski-local, if one needs to distinguish between the Zariski topology and other possible topologies, like the étale topology.

Consider a scheme X and a cover by affine open subschemes Spec A_i. Using the dictionary between (commutative) rings and affine schemes local properties are thus properties of the rings A_i. A property P is local in the above sense, iff the corresponding property of rings is stable under localization.

For example, we can speak of locally Noetherian schemes, namely those which are covered by the spectra of Noetherian rings. The fact that localizations of a Noetherian ring are still noetherian then means that the property of a scheme of being locally Noetherian is local in the above sense (whence the name). Another example: if a ring is reduced (i.e., has no non-zero nilpotent elements), then so are its localizations.

An example for a non-local property is separatedness (see below for the definition). Any affine scheme is separated, therefore any scheme is locally separated. However, the affine pieces may glue together pathologically to yield a non-separated scheme.

The following is a (non-exhaustive) list of local properties of rings, which are applied to schemes. Let X = $\cup$ Spec A_i be a covering of a scheme by open affine subschemes. For definiteness, let k denote a field in the following. Most of the examples also work with the integers Z as a base, though, or even more general bases. Connected, irreducible, reduced, integral, normal, regular, Cohen-Macaulay, locally noetherian, dimension,

catenary,

local uniformization

The local uniformization is a method of constructing a weaker form of resolution of singularities by means of valuation rings.

locally factorial

The local rings are unique factorization domains.

locally of finite type

The morphism f : Y → X is locally of finite type if

X

may be covered by affine open sets

\text{Spec }B

such that each inverse image

f^{-1}(\text{Spec }B)

is covered by affine open sets

\text{Spec }A

where each

A

is finitely generated as a

B

-algebra.

locally Noetherian

The A_i are Noetherian rings. If in addition a finite number of such affine spectra covers X, the scheme is called noetherian. While it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false.

For example, most schemes in finite-dimensional algebraic geometry are locally Noetherian, but

GL_\infty = \cup GL_n

is not.

logarithmic geometry

log structure

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

M

Mori's minimal model program: The minimal model program is a research program aiming to do birational classification of algebraic varieties of dimension greater than 2.
multicone: The multicone of line bundles $L_1, \dots, L_m$ are the Spec of the section ring of $L_1 \otimes \cdots \otimes L_m$ .

N

nonsingular: An archaic term for "smooth" as in a smooth variety.
normal: An integral scheme is called normal, if the A_i are integrally closed domains. For example, all regular schemes are normal, while singular curves are not.
normal crossings: See normal crossings.

O

open: A morphism f : Y → X of schemes is called open (closed), if the underlying map of topological spaces is open (closed, respectively), i.e. if open subschemes of Y are mapped to open subschemes of X (and similarly for closed). For example, finitely presented flat morphisms are open and proper maps are closed.; An open subscheme of a scheme X is an open subset U with structure sheaf $\mathcal{O}_X|_U$ .^[7]. {{{1}}}

P

p-divisible group

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

Plücker embedding

The Plücker embedding is the closed embedding of the Grassmannian variety into a projective space.

Poincaré residue map

See Poincaré residue.

point

A scheme

S

is a locally ringed space, so a fortiori a topological space, but the meanings of point of $S$ are threefold:

a point $P$ of the underlying topological space;
a $T$ -valued point of $S$ is a morphism from $T$ to $S$ , for any scheme $T$ ;
a geometric point, where $S$ is defined over (is equipped with a morphism to) $\textrm{Spec}(K)$ , where $K$ is a field, is a morphism from $\textrm{Spec} (\overline{K})$ to $S$ where $\overline{K}$ is an algebraic closure of $K$ .

Geometric points are what in the most classical cases, for example algebraic varieties that are complex manifolds, would be the ordinary-sense points. The points $P$ of the underlying space include analogues of the generic points (in the sense of Zariski, not that of André Weil), which specialise to ordinary-sense points. The $T$ -valued points are thought of, via Yoneda's lemma, as a way of identifying $S$ with the representable functor $h_{S}$ 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 $T$ -valued points were a massive further step.

As part of the predominating Grothendieck approach, there are three corresponding notions of fiber of a morphism: the first being the simple inverse image of a point. The other two are formed by creating fiber products of two morphisms. For example, a geometric fiber of a morphism $S^{\prime} \to S$ is thought of as

S^{\prime} \times_{S} \textrm{Spec}(\overline{K})

This makes the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of the fiber product operation a significant (if technically anodyne) result.

polarization

an embedding into a projective space

Proj

See Proj construction.

projective

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

2. A projective scheme over a scheme S is an S-scheme that factors through some projective space

\mathbf{P}^N_S \to S

as a closed subscheme.

{{{1}}}

projectively normal

A projective variety is projectively normal if, with respect to some fixed embedding into a projective space, the homogeneous coordinate ring is an integrally closed domain.

proper

A morphism is proper if it is separated, universally closed (i.e. such that fiber products with it are closed maps), and of finite type. Projective morphisms are proper; but the converse is not in general true. See also complete variety. A deep property of proper morphisms is the existence of a Stein factorization, namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism.

Q

quasi-compact: A morphism f : Y → X is called quasi-compact, if for some (equivalently: every) open affine cover of X by some U_i = Spec B_i, the preimages f⁻¹(U_i) are quasi-compact.
quasi-finite: The morphism f : Y → X has finite fibers if the fiber over each point $x \in X$ is a finite set. A morphism is quasi-finite if it is of finite type and has finite fibers.
quasi-separated: A morphism f : Y → X is called quasi-separated or (Y is quasi-separated over X) if the diagonal morphism Y → Y ×_XY is quasi-compact. A scheme Y is called quasi-separated if Y is quasi-separated over Spec(Z).^[10]
Quot scheme: A Quot scheme parametrizes quotients of locally free sheaves on a projective scheme.

R

rational function: An element in the field $k(X) = \varinjlim k[U]$ where the limit runs over all coordinates rings of open subsets U of an (irreducible) algebraic variety X. See also function field (scheme theory).
rational singularities: A variety X has rational singularities if there is a resolution of singularities $f:X' \to X$ such that $f_*(\mathcal{O}_{X'}) = \mathcal{O}_X$ and $R^i f_*(\mathcal{O}_{X'}) = 0, \, i \ge 1$ .
reduced: The A_i are reduced rings. Equivalently, none of its rings of sections $\mathcal O_X(U)$ (U any open subset of X) has any nonzero nilpotent element. Allowing non-reduced schemes is one of the major generalizations from varieties to schemes.
Any variety is reduced (by definition) while
Spec k[x]/(x²) is not.
reflexive sheaf: A coherent sheaf is reflexive if the canonical map to the second dual is an isomorphism.
regular: Main article: regular scheme

The A_i are regular rings. For example, smooth varieties over a field are regular, while
Spec k[x,y]/(x²+x³-y²)= is not.
regular function: A morphism from an algebraic variety to the affine line.
resolution of singularities: A resolution of singularities of a scheme X is a proper birational morphism $\pi: Z \to X$ such that Z is smooth.

S

Schubert

1. A Schubert cell is a B-orbit on the Grassmannian

\operatorname{Gr}(d, n)

where B is the standard Borel; i.e., the group of upper triangular matrices.

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

section ring

The section ring or the ring of sections of a line bundle L on a scheme X is the graded ring

\oplus_0^\infty \Gamma(X, L^n)

Serre's conditions S_n

See Serre's conditions on normality. See also http://mathoverflow.net/questions/22228/what-is-serres-condition-s-n-for-sheaves

Serre duality

See #dualizing sheaf

semi-ample

A semi-ample line bundle is a line bundle such that some tensor power of it is generated by global sections of the power.

separated

A separated morphism is a morphism

f

such that the fiber product of

f

with itself along

f

has its diagonal as a closed subscheme — in other words, the diagonal map is a closed immersion.

As a consequence, a scheme $X$ is separated when the diagonal of $X$ within the scheme product of $X$ with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism $X \rightarrow \textrm{Spec} (\mathbb{Z})$ is separated.

Notice that a topological space Y is Hausdorff iff the diagonal embedding

Y \stackrel{\Delta}{\longrightarrow} Y \times Y

is closed. In algebraic geometry, the above formulation is used because a scheme which is a Hausdorff space is necessarily empty or zero-dimensional. The difference between the topological and algebro-geometric context comes from the topological structure of the fiber product (in the category of schemes) $X \times_{\textrm{Spec} (\mathbb{Z})} X$ , which is different from the product of topological spaces.

Any affine scheme Spec A is separated, because the diagonal corresponds to the surjective map of rings (hence is a closed immersion of schemes):

$A \otimes_{\mathbb Z} A \rightarrow A, a \otimes a' \mapsto a \cdot a'$ .

sheaf generated by global sections

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

smooth

Main article: 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 f : Y → X:

1) for any y ∈ Y, there are open affine neighborhoods V and U of y, x=f(y), respectively, such that the restriction of f to V factors as an étale morphism followed by the projection of affine n-space over U.

2) f is flat, locally of finite presentation, and for every geometric point

\bar{y}

of Y (a morphism from the spectrum of an algebraically closed field

k(\bar{y})

to Y), the geometric fiber

X_{\bar{y}}:=X\times_Y \mathrm{Spec} (k(\bar{y}))

is a smooth n-dimensional variety over

k(\bar{y})

in the sense of classical algebraic geometry.

2. A smooth scheme over a perfect field k is a scheme X that is of locally of finite type and regular over k.

3. A smooth scheme over a field k is a scheme X that is geometrically smooth:

X \times_k \overline{k}

is smooth.

spherical variety

A spherical variety is a normal G-variety (G connected reductive) with an open dense orbit by a Borel subgroup of G.

stable

1. A stable curve is a curve with some "mild" singularity, used to construct a good-behaving moduli space of curves.

2. A stable vector bundle is used to construct the moduli space of vector bundles.

subscheme

A subscheme, without qualifier, of X is a closed subscheme of an open subscheme of X.

surface

An algebraic variety of dimension two.

T

tautological line bundle: The tautological line bundle of a projective scheme X is the dual of Serre's twisting sheaf $\mathcal{O}_X(1)$ ; that is, $\mathcal{O}_X(-1)$ .
torus embedding: An old term for a toric variety
toric variety: A toric variety is a normal variety with the action of a torus such that the torus has an open dense orbit.
tropical geometry: A kind of a piecewise-linear algebraic geometry. See tropical geometry.
torus: A torus is a product of finitely many multiplicative groups $\mathbb{G}_m$ .

U

universally

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

unramified

For a point

y

Y

, consider the corresponding morphism of local rings

f^\# \colon \mathcal{O}_{X, f(y)} \to \mathcal{O}_{Y, y}

Let $\mathfrak{m}$ be the maximal ideal of $\mathcal{O}_{X,f(y)}$ , and let

\mathfrak{n} = f^\#(\mathfrak{m}) \mathcal{O}_{Y,y}

be the ideal generated by the image of $\mathfrak{m}$ in $\mathcal{O}_{Y,y}$ . The morphism $f$ is unramified (resp. G-unramified) if it is locally of finite type (resp. locally of finite presentation) and if for all $y$ in $Y$ , $\mathfrak{n}$ is the maximal ideal of $\mathcal{O}_{Y,y}$ and the induced map

\mathcal{O}_{X,f(y)}/\mathfrak{m} \to \mathcal{O}_{Y,y}/\mathfrak{n}

is a finite separable field extension.^[11] This is the geometric version (and generalization) of an unramified field extension in algebraic number theory.

V

variety: a synonym with "algebraic variety".

W

weakly normal: a scheme is weakly normal if any finite birational morphism to it is an isomorphism.
Weil reciprocity: See Weil reciprocity.

Z

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

References

↑ Grothendieck & Dieudonné 1960, 4.1.2 and 4.1.3
↑ Grothendieck & Dieudonné 1964, §1.4
↑ Grothendieck & Dieudonné 1964, §1.6
↑ Hartshorne 1977, Exercise II.3.11(d)
↑ The Stacks Project, Chapter 21, §4.
↑ Grothendieck & Dieudonné 1960, 4.2.1
↑ 7.0 7.1 Hartshorne 1977, §II.3
↑ Grothendieck & Dieudonné 1960, 4.2.5
↑ Q. Liu, Algebraic Geometry and Arithmetic Curves, exercise 2.3
↑ Grothendieck & Dieudonné 1964, 1.2.1
↑ The notion G-unramified is what is called "unramified" in EGA, but we follow Raynaud's definition of "unramified", so that closed immersions are unramified. See Tag 02G4 in the Stacks Project for more details.

Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Kollár, János, "Book on Moduli of Surfaces" available at his website