Glossary of algebraic geometry

{{defn|no=1|1=If X is a projective scheme with Serre's twisting sheaf \mathcal{O}_X(1) and if F is an \mathcal{O}_X-module, then F(n) = F \otimes_{\mathcal{O}_X} \mathcal{O}_X(n).}} {{defn|no=2|1=If D is a Cartier divisor and F is an \mathcal{O}_X-module (X arbitrary), then F(D) = F \otimes_{\mathcal{O}_X} \mathcal{O}_X(D). If D is a Weil divisor and F is reflexive, then one replaces F(D) by its reflexive hull (and calls the result still F(D)).}} {{defn|1=The complete linear system of a Weil divisor D on a normal complete variety X over an algebraically closed field k; that is, |D| = \mathbf{P}(\Gamma(X, \mathcal{O}_X(D))). There is a bijection between the set of k-rational points of |D| and the set of effective Weil divisors on X that are linearly equivalent to D. The same definition is used if D is a Cartier divisor on a complete variety over k.}} {{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 L = \mathcal{O}_X, the structure sheaf on X, then it means the direct sum of n copies of \mathcal{O}_X.}} {{term|content=\mathcal{O}_X(-1)}} {{defn|1=The tautological line bundle. It is the dual of Serre's twisting sheaf \mathcal{O}_X(1).}} {{term|content=\mathcal{O}_X(1)}} {{defn|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.}} {{term|content=\mathcal{O}_X(D)}} {{defn|no=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.}} {{defn|no=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.}} {{defn|no=4|If D is a \mathbb{Q}-divisor, then \mathcal{O}_X(D) is \mathcal{O}_X of the integral part of D.}} {{defn|1=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. In contrast, Hartshorne and EGA write P(V) for the Proj of the symmetric algebra of V.}} {{defn|1=A normal variety is \mathbb{Q}-factorial if every \mathbb{Q}-Weil divisor is \mathbb{Q}-Cartier.}} ==A==

{{defn|no=1|An abelian variety is a complete group variety. For example, consider the complex variety \mathbb{C}^n/\mathbb{Z}^{2n} or an elliptic curve E over a finite field \mathbb{F}_q.}} {{defn|no=1|If D is an effective Cartier divisor on an algebraic variety X, both admitting dualizing sheaves \omega_D, \omega_X, then the adjunction formula says: \omega_D = (\omega_X \otimes \mathcal{O}_X(D))|_D.}} {{quote box {{defn|1=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.}} {{defn|1=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, \operatorname{Spec} \mathbb{C} \times_{\mathbb{R}} \operatorname{Spec} \mathbb{C} is not a variety since the coordinate ring \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} is not an integral domain.}} {{defn|1=Algebraic geometry over the compactification of Spec of the ring of rational integers \mathbb{Z}. See Arakelov geometry.}} {{defn|1=The arithmetic genus of a projective variety X of dimension r is (-1)^r (\chi(\mathcal{O}_X) - 1).}} \operatorname{Spec}(\mathbf{Z}), the prime spectrum of the ring of rational integers.}} but any scheme is a scheme over Spec(Z).--> ==B==

{{defn|1=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.}} {{defn|1=A blow-up is a birational transformation that replaces a closed subscheme with an effective Cartier divisor. Precisely, given a noetherian scheme X and a closed subscheme Z \subset X, the blow-up of X along Z is a proper morphism \pi: \widetilde{X} \to X such that (1) \pi^{-1}(Z) \hookrightarrow \widetilde{X} is an effective Cartier divisor, called the exceptional divisor, and (2) \pi is universal with respect to (1). Concretely, it is constructed as the relative Proj of the Rees algebra of O_X with respect to the ideal sheaf determining Z.}} ==C==

{{defn|no=2|1=The canonical class K_X on a normal variety X is the divisor class such that \mathcal{O}_X(K_X) = \omega_X.}} {{defn|The Castelnuovo–Mumford regularity of a coherent sheaf F on a projective space f: \mathbf{P}^n_S \to S over a scheme S is the smallest integer r such that :R^i f_*F(r-i) = 0 for all i > 0.}} {{defn|no=2|1=Complete smooth curves over an algebraically closed field are classified up to rational equivalence by their genus g. (a) g=0. rational curves, i.e. the curve is birational to the projective line \mathbb{P}^1. (b) g=1. Elliptic curves, i.e. the curve is a complete 1-dimensional group scheme after choosing any point on the curve as identity. (c) g\geq2. Hyperbolic curves, also called curves of general type. See algebraic curves for examples. The classification of smooth curves can be refined by the degree for projectively embedded curves, in particular when restricted to plane curves. Note that all complete smooth curves are projective in the sense that they admit embeddings into projective space, but for the degree to be well-defined a choice of such an embedding has to be explicitly specified. The arithmetic of a complete smooth curve over a number field (in particular number and structure of its rational points) is governed by the classification of the associated curve base changed to an algebraic closure. See Faltings' theorem for details on the arithmetic implications.}} {{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 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'''''. 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. }} ==D==

{{defn|1=Given a vector-bundle map f: E \to F over a variety X (that is, a scheme X-morphism between the total spaces of the bundles), the degeneracy locus is the (scheme-theoretic) locus X_k(f) = \{ x \in X | \operatorname{rk}(f(x)) \le k \}.}} {{defn|no=1|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.}} {{defn|no=1|The degree of a line bundle L on a complete variety is an integer d such that \chi(L^{\otimes m}) = {d \over n!} m^n + O(m^{n-1}).}} {{defn|no=2|If x is a cycle on a complete variety f: X \to \operatorname{Spec} k over a field k, then its degree is f_*(x) \in A_0(\operatorname{Spec} k) = \mathbb{Z}.}} {{defn|1=On a projective Cohen–Macaulay scheme of pure dimension n, the dualizing sheaf is a coherent sheaf \omega on X such that H^{n-i}(X, F^{\vee} \otimes \omega) \simeq H^i(X, F)^* holds for any locally free sheaf F on X; for example, if X is a smooth projective variety, then it is a canonical sheaf.}} ==E==

{{defn|1= A morphism 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. }} {{defn|1=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==

{{defn|1=A Fano variety is a smooth projective variety X whose anticanonical sheaf \omega_X^{-1} is ample.}} {{defn|1=Given f: X \to Y between schemes, the fiber of f over y is, as a set, the pre-image f^{-1}(y) = \{x \in X|f(x)=y\}; it has the natural structure of a scheme over the residue field of y as the fiber product X \times_Y \{y\}, where \{y\} has the natural structure of a scheme over Y as Spec of the residue field of y.}} {{defn|no=2|A stack F \times_G H given for f: F \to G, g:H \to G: an object over B is a triple (x, y, ψ), x in F(B), y in H(B), ψ an isomorphism f(x) \overset{\sim}\to g(y) in G(B); an arrow from (x, y, ψ) to (x', y, ψ') is a pair of morphisms \alpha: x \to x', \beta: y \to y' such that \psi' \circ f(\alpha) = g(\beta) \circ \psi. The resulting square with obvious projections does not commute; rather, it commutes up to natural isomorphism; i.e., it 2-commutes.}} {{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 \mathbb{Z} of integers; so that any scheme S is over \textrm{Spec} (\mathbb{Z}) , and in a unique way. }} {{defn|1= The morphism 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. 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. }} {{defn|1= The morphism 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 is 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 finitely many affine open sets \text{Spec }A where each A is finitely generated as a B-algebra. }} {{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) ⊆ 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. The morphism 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. }} {{defn|1= A morphism f is flat if it gives rise to a flat map on stalks. When viewing a morphism 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==

{{defn|1=Given a curve C, a divisor D on it and a vector subspace V \subset H^0(C, \mathcal{O}(D)), one says the linear system \mathbb{P}(V) is a grd if V has dimension r+1 and D has degree d. One says C has a grd if there is such a linear system.}} {{defn|1=The geometric genus of a smooth projective variety X of dimension n is \dim \Gamma(X, \Omega^n_X) = \dim \operatorname{H}^n(X, \mathcal{O}_X) (where the equality is Serre's duality theorem).}} {{defn|1=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.}} {{defn|1=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.}} {{defn|1=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.}} {{defn|no=2|1=A normal variety is said to be \mathbb{Q}-Gorenstein if the canonical divisor on it is \mathbb{Q}-Cartier (and need not be Cohen–Macaulay).}} ==H==

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

{{defn|1= Immersions 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. 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. 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. 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. }} {{defn|1=A locally free sheaf of a rank one. Equivalently, it is a torsor for the multiplicative group \mathbb{G}_m (i.e., line bundle).}} ==J==

{{defn|1=The Jacobian variety of a projective curve X is the degree zero part of the Picard variety \operatorname{Pic}(X).}} ==K==

{{defn|1= The morphism 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. }} ==M==

{{defn|1=See for example moduli space.{{quote box ==N==

{{defn|no=2|1=A smooth curve C \subset \mathbf{P}^r is said to be k-normal if the hypersurfaces of degree k cut out the complete linear series |\mathcal{O}_C(k)|. 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 X \subset \mathbf{P}^r 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.}} {{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 (I/I^2)^* = \mathcal{H}om_{\mathcal{O}_Y}(I/I^2, \mathcal{O}_Y). If the embedded of X into Y is regular, it is locally free and is called the normal bundle.}} {{defn|no=2|1=The normal cone to X is \operatorname{Spec}_X(\oplus_0^{\infty} I^n/I^{n+1}). if X is regularly embedded into Y, then the normal cone is isomorphic to \operatorname{Spec}_X(\mathcal{S}ym(I/I^2)), the total space of the normal bundle to X.}} {{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 \Gamma(X, L)^{\otimes n} \to \Gamma(X, L^{\otimes n}) is surjective.}} ==O==

{{defn|no=2|1=An open subscheme of a scheme X is an open subset U with structure sheaf \mathcal{O}_X|_U. }} ==P==

{{defn|1=The n-th plurigenus of a smooth projective variety is \dim \Gamma(X, \omega_X^{\otimes n}). See also Hodge number.}} {{defn|1= 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. }} {{defn|The projection formula says that, for a morphism f:X \to Y of schemes, an \mathcal{O}_X-module \mathcal{F} and a locally free \mathcal{O}_Y-module \mathcal{E} of finite rank, there is a natural isomorphism f_* (F \otimes f^* E) = (f_* F) \otimes E (in short, f_* is linear with respect to the action of locally free sheaves.) }} {{defn|no=2|1=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.}} {{defn|no=3|1=Projective morphisms are defined similarly to affine morphisms: is called projective if it factors as a closed immersion followed by the projection of a projective space \mathbb{P}^{n}_X := \mathbb{P}^n \times_{\mathrm{Spec}\mathbb Z} X to X . Note that this definition is more restrictive than that of EGA, II.5.5.2. The latter defines f to be projective if it is given by the global Proj of a quasi-coherent graded OX-algebra \mathcal S such that \mathcal S_1 is finitely generated and generates the algebra \mathcal S. Both definitions coincide when X is affine or more generally if it is quasi-compact, separated and admits an ample sheaf, e.g. if X is an open subscheme of a projective space \mathbb P^n_A over a ring A. }} {{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: \mathbf{P}(E) = \mathbf{Proj}(\operatorname{Sym}_{\mathcal{O}_X}(E^{\vee})). Note this definition is standard nowadays (e.g., Fulton's Intersection theory) but differs from EGA and Hartshorne (they don't take a dual).}} ==Q==

{{defn|no=1|1=Over an algebraically closed field, a variety is rational if it is birational to a projective space. For example, rational curves and rational surfaces are those birational to \mathbb{P}^1, \mathbb{P}^2.}} {{defn|no=2|Given a field k and a relative scheme X → S, a k-rational point of X is an S-morphism \operatorname{Spec}(k) \to X.}} {{defn|1=A rational normal curve is the image of \mathbf{P}^1 \to \mathbf{P}^d, \, (s:t) \mapsto (s^d : s^{d-1} t : \cdots : t^d). If d = 3, it is also called the twisted cubic.}} {{defn|1=A variety X over a field of characteristic zero 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.}} {{defn|no=1|1=A commutative ring R is reduced if it has no nonzero nilpotent elements, i.e., its nilradical is the zero ideal, \sqrt{(0)} = (0). Equivalently, R is reduced if \operatorname{Spec}(R) is a reduced scheme.}} {{defn|no=2|1=A scheme X is reduced if its stalks \mathcal{O}_{X,x} are reduced rings. Equivalently X is reduced if, for each open subset U \subset X, \mathcal{O}_X(U) is a reduced ring, i.e., X has no nonzero nilpotent sections.}} {{defn|1=A connected linear algebraic group G over a field k is a reductive group if and only if the unipotent radical R_u(G_{\overline{k}}) of the base change G_{\overline{k}} of G to an algebraic closure \overline{k} is trivial.}} {{defn|1=A closed immersion i: X \hookrightarrow Y 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, \mathcal{I}/\mathcal{I}^2 when \mathcal{I} is the ideal sheaf of X, is locally free.}} {{defn|1=Given a finite separable morphism \pi:X \to Y between smooth projective curves, if \pi is tamely ramified (no wild ramification), for example, over a field of characteristic zero, then the Riemann–Hurwitz formula relates the degree of π, the genera of X, Y and the ramification indices: 2g(X) - 2 = \operatorname{deg}(\pi) (2g(Y) - 2) + \sum_{y \in Y} (e_y - 1). Nowadays, the formula is viewed as a consequence of the more general formula (which is valid even if π is not tame): K_X \sim \pi^*K_Y + R where \sim means a linear equivalence and R = \sum_{P \in X} \operatorname{length}_{\mathcal{O}_P} (\Omega_{X/Y})P is the divisor of the relative cotangent sheaf \Omega_{X/Y} (called the different).}} {{defn|no=2|1=The general version is due to Grothendieck and called the Grothendieck–Riemann–Roch formula. It says: if \pi:X \to S 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 \operatorname{ch}(\pi_! E) \cdot \operatorname{td}(S) = \pi_*(\operatorname{ch}(E) \cdot \operatorname{td}(X)) where \pi_! = \sum_i (-1)^i R^i \pi_*, \operatorname{ch} means a Chern character and \operatorname{td} a Todd class of the tangent bundle of a space, and, over the complex numbers, \pi_* 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 \pi_*(e^{c_1(L)}(1-c_1(T^*X)/2)) = \operatorname{deg}(L) - g + 1.}} {{defn|1=Every infinitesimal deformation is trivial. For example, the projective space is rigid since \operatorname{H}^1(\mathbf{P}^n, T_{\mathbf{P}^n}) = 0 (and using the Kodaira–Spencer map).}} ==S==

{{defn|no=1|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.}} {{defn|1=A rational normal scroll is a ruled surface which is of degree n in a projective space \mathbb{P}^{n+1} for some n\in\mathbb{N}_{>1}.}} {{defn|1=The secant variety to a projective variety V \subset \mathbb{P}^r is the closure of the union of all secant lines to V in \mathbb{P}^r.}} {{defn|no=1|1= The higher-dimensional analog of étale morphisms are smooth morphisms. There are many different characterisations of smoothness. The following are equivalent definitions of smoothness of the morphism : • 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. • 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. }} {{defn|no=3|1=A smooth scheme over a field k is a scheme X that is geometrically smooth: X \times_k \overline{k} is smooth.}} {{defn|1=A divisor D on a smooth curve C is special if h^0(\mathcal{O}(K - D)), which is called the index of speciality, is positive.}} {{defn|no=1|1=In the context of an algebraic group G for certain properties P there is the derived property split-P. Usually P is a property that is automatic or more common over algebraically closed fields \overline{k}. If this property holds already for G defined over a not necessarily algebraically closed field k then G is said to satisfy split-P.}} {{defn|no=2|1=A linear algebraic group G defined over a field k is a torus if only if its base change G_{\overline{k}} to an algebraic closure \overline{k} is isomorphic to a product of multiplicative groups G_{m,\overline{k}}^{n}. G is a split torus if and only if it is isomorphic to G_{m,k}^{n} without any base change. G is said to split over an intermediate field k\subseteq L\subseteq\overline{k} if and only if its base change G_{L} to L is isomorphic to G_{m,L}^{n}.}} {{defn|no=4|1=A connected solvable linear algebraic group G defined over a field k is split if and only if it has composition series B=B_0\supset B_1\supset \ldots\supset B_t=\{1\} defined over k such that each successive quotient B_i/B_{i+1} is isomorphic to either the multiplicative group G_{m,k} or the additive group G_{m,a} over k.}} {{defn|no=6|1=In the classification of real Lie algebras split Lie algebras play an important role. There is a close connection between linear Lie groups, their associated Lie algebras and linear algebraic groups over k=\mathbb{R} resp. \mathbb{C}. The term split has similar meanings for Lie theory and linear algebraic groups.}} {{defn|1=Given a blow-up \pi: \widetilde{X} \to X along a closed subscheme Z and a morphism f: Y \to X, the strict transform of Y (also called proper transform) is the blow-up \widetilde{Y} \to Y of Y along the closed subscheme f^{-1} Z. If f is a closed immersion, then the induced map \widetilde{Y} \hookrightarrow \widetilde{X} is also a closed immersion.}} ==T==

{{defn|1=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).}} {{defn|1=A split torus is a product of finitely many multiplicative groups \mathbb{G}_m.}} ==U==

{{defn|no=2|Let \mathcal{M}_{g} be the moduli of smooth projective curves of genus g and \mathcal{C}_g = \mathcal{M}_{g, 1} that of smooth projective curves of genus g with single marked points. In literature, the forgetful map \pi: \mathcal{C}_{g} \to \mathcal{M}_{g} is often called a universal curve.}} {{defn|1= For a point y in 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. This is the geometric version (and generalization) of an unramified field extension in algebraic number theory. }} ==V==