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  • Elena RubeiAlgebraic Geometry

  • Also of InterestKnotsBurde/Zieschang/Heusener, 2013ISBN 978-3-11-027074-7, e-ISBN 978-3-11-027078-5

    Topology of Algebraic CurvesDegtyarev, 2012ISBN 978-3-11-025591-1, e-ISBN 978-3-11-025842-4

    Lectures on the Topology of 3-ManifoldsSaveliev, 2011ISBN 978-3-11-025035-0, e-ISBN 978-3-11-025036-7

    Abstract AlgebraCarstensen/Fine/Rosenberger, 2011ISBN 978-3-11-025008-4, e-ISBN 978-3-11-025009-1

    |Advances in GeometryGrundhfer/Strambach (Managing Editors)ISSN: 1615-715X, e-ISSN 1615-7168

  • Elena Rubei

    AlgebraicGeometry

    |A Concise Dictionary

  • Mathematics Subject Classication 201014-00, 14-01

    AuthorProf. Dr. Elena RubeiUniversit degli Studi di FirenzeDipartimento di Matematica e Informatica U. DiniViale Morgagni 67/A50134 FirenzeItalyrubei@math.uni.it

    ISBN 978-3-11-031622-3e-ISBN 978-3-11-031623-0Set-ISBN 978-3-11-031600-1

    Library of Congress Cataloging-in-Publication DataA CIP catalog record for this book has been applied for at the Library of Congress.

    Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograe;detailed bibliographic data are available in the Internet at http://dnb.dnb.de.

    2014 Walter de Gruyter GmbH, Berlin/BostonTypesetting: PTP Protago TEX-Production GmbH, Berlin (www.ptp-berlin.de)Printing and binding: CPI buch bcher.de GmbH, BirkachPrinted on acid-free paperPrinted in Germany

    www.degruyter.com

  • Notation

    We denote by the ane space of dimension over.() For any manifold , () denotes the vector space of the -forms on.,() For any almost complexmanifold (see Almost complexmanifolds, holomor-phic maps, holomorphic tangent bundles), ,() denotes the vector spaceof the (, )-forms on .(, ) For any topological space and ring , (, ) (Betti number) denotes therank of (, ); it is also denoted by (, ); see Singular homology andcohomology.

    b.p.f. The abbreviation b.p.f. stands for base point free, see Bundles, bre -.() For any algebraic variety , () denotes the -th Chow group of ; seeEquivalence, algebraic, rational, linear -, Chow, NeronSeveri and Picardgroups.() For any algebraic variety , () denotes the divisor class group of ; seeDivisors.(, ) For any vector bundle on a manifold , (, ) denotes the set ofthe sections of ; see Bundles, bre -., , stands (as usual) for the Kronecker delta.() For any Cartier divisor , () denotes the line bundle associated to ; see Di-visors., , 1 See Pull-back and push-forward of cycles, Direct and inverse imagesheaves, Singular homology and cohomology.||, L() For any divisor, we denote by || the complete linear system associated to,see Linear systems; see also Linear systems for the denition of L(). For any line bundle on a variety , denotes the map associated to ; seeBundles, bre -. denotes a linear system on a Riemann surface of degree and projectivedimension ; see Linear systems.(, ), (,) For any vector space,(, ) denotes the Grassmannian of -planes in, seeGrassmannians; analogously, for any projective space ,(,) denotes theGrassmannian of projective -planes in .(, ), (, ) For any topological space and any ring,(, ) denotes the -th homologymodule ofwith coecients in (see Singular homology and cohomology);(, ) denotes its rank; (, ) is also denoted by (, ) (Betti number)(, ), (, ) For any topological space and any ring ,(, ) denotes the -th cohomol-ogy module of with coecients in (see Singular homology and cohomol-ogy); (, ) denotes its rank.

  • viii | Notation

    (,E) For any sheaf of Abelian groups E on a topological space ,(,E) denotesthe -th cohomology group of E, see Sheaves; (,E) denotes its rank; if is a holomorphic vector bundle on a complex manifold or an algebraic vec-tor bundle on an algebraic variety , we sometimes write (, ) instead of(,O()).() For any topological space, () denotes the EulerPoincar characteristic of, i.e., the sum =1,...,(1)(,), when it is dened.(,E) For any sheaf of Abelian groups E on a topological space , (,E) denotesthe EulerPoincar characteristic of E, i.e.,=1,...,(1)(,E), when it is de-ned., For any complex manifold or smooth algebraic variety , denotes thecanonical bundle and the canonical sheaf, i.e.,O(); see Canonical bun-dle, canonical sheaf.F, F See Serre correspondence.( , ) We denote by( , ) the space of the matrices with entries in .

    nef The abbreviation nef stands for numerically eective; see Bundles, bre -.() For any algebraic variety,() denotes the -th NeronSeveri group of;see Equivalence, algebraic, rational, linear -, Chow, NeronSeveri and Picardgroups.

    O, O() If is a complex manifold, O (or simply O) denotes the sheaf of holomor-phic functions; if is an algebraic variety, it denotes the sheaf of the regularfunctions; more generally, it denotes the structure sheaf of a ringed space; seeSpace, ringed -; for the denition of O(), see Hyperplane bundles, twist-ing sheaves.

    O(), () Let be an algebraic vector bundle on an algebraic variety or a holomorphicvector bundle on a complex manifold; then O() denotes the sheaf of the(regular, resp. holomorphic) sections of ; see Bundles, bre -. We denote O() by (). For any complexmanifold, denotes the sheaf of the holomorphic -forms;for any algebraic variety, denotes the sheaf of the regular -forms; seeZariski tangent space, dierential forms, tangent bundle, normal bundle. We denote by the projective space of dimension over.(), (), () The symbols (), () and () denote respectively the arithmetic genus,the geometric genus and the -th plurigenus of (for variety or manifold);see Genus, arithmetic, geometric, real, virtual -, Plurigenera.() For any algebraic variety , () denotes the Picard group of ; see Equiv-alence, algebraic, rational, linear -, Chow, NeronSeveri and Picard groups.

    PID PID stands for principal ideal domain, i.e., for an integral domain such thatevery ideal is principal.(, ) For any topological space and any , (, ) denotes the -th funda-mental group of at the basepoint ; see Fundamental group.

  • Notation | ix

    () For any complex manifold or algebraic variety , () denotes the irregularityof; see Irregularity., See Direct and inverse image sheaves() For any ideal , () denotes the saturation of ; see Saturation. For any , we denote by the group of the permutations on elements.(), () For any ring , () denotes its spectrum, see Schemes. See Schemesalso for the denition of () for graded ring. The symbol denotes the bred product of and over ; see Fibredproduct. The symbol denotes the intersection of cycles; see Intersection of cycles.Sometimes it is omitted. For any ideal in a ring , we denote by the radical of , i.e., = { | {0} s.t. }., + , ( : ) Let be a commutative ring and and be two ideals in . We dene tobe the ideal {=1,..., | , , } .Moreover, we dene+ ={ + | }, and ( : ) := { | }. For any vector space , we denote by its dual space. denotes the disjoint union. : (, ) (, ) If and , the notation : (, ) (, ) stands for a map : such that () .

    Note. The end of the denitions, theorems, and propositions is indicated by the symbol .

  • AAbelian varieties. See Tori, complex - and Abelian varieties.

    Adjunction formula. ([72], [93], [107], [129], [140]). Let be a complex manifoldor a smooth algebraic variety and let be a submanifold, respectively a smooth closedsubvariety. We have = | ,,where , is the normal bundle and and are the canonical bundles respec-tively of and (see Canonical bundle, canonical sheaf, Zariski tangent space,dierential forms, tangent bundle, normal bundle).If, in addition, is a hypersurface, the formula becomes = | ()|,where () is the bundle associated to the divisor , since, in this case the bundle,is the bundle given by (see Bundles, bre - for the denition of bundles associatedto divisors).

    Albanese varieties. ([93], [163], [166]). The Albanese variety is a generalizationof the Jacobian of a compact Riemann surface (see Jacobians of compact Riemannsurfaces) for manifolds of higher dimension.Let be a compact Khler manifold of dimension (see Hermitian and Khlerianmetrics and Hodge theory). The Albanese variety of is the complex torus (seeTori, complex - and Abelian varieties)() := 0(, 1)(1(,)) ,where : 1(,) 0(, 1) is dened by () = for any 1(,).The Albanese map : ()is dened in the following way: we x a point 0 of (base point) and we dene() = 0for any , where0 is the integral along a path joining 0 and (thus, obviously, itdenes a linear function on0(, 1) only up to elements of (1(,))). If we choosea basis 1, . . . , of0(, 1) we can describe in the following way:() = ( 0 1, . . . , 0 ).

  • 2 | Algebras

    For any compact Khler manifold of dimension , the Albanese variety of is iso-morphic to the -th intermediate Griths Jacobian of . The Albanese variety of asmooth complex projective algebraic variety is an Abelian variety, that is, it can beembedded in a projective space. See Jacobians, Weil and Griths intermediate -,Tori, complex - and Abelian varieties.

    Algebras. We say that is an algebra over a ring if it is an -module and a ringwith unity (with the same sum) and, for all , and , we have() = () = ().Algebraic groups. ([27], [126], [228], [235], and references in Tori, complex - andAbelian varieties). An algebraic group is a set that is both an algebraic variety (seeVarieties, algebraic -, Zariski topology, regular and rational functions, morphismsand rational maps) and a group and the two structures are compatible, that is, themap ,(, ) 1is a morphism between algebraic varieties.

    Structure theorem for algebraic groups (Chevalleys theorem). Let be an algebraicgroup over an algebraically closed eld. Then there exists a (unique) normal anesubgroup such tha