Algebraic geometry. A concise dictionary

240 Pages·2014·1.443 MB·English

by  Rubei E.

ElenaRubei
AlgebraicGeometry
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ISSN:1615-715X,e-ISSN1615-7168
Elena Rubei
Algebraic
Geometry
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A Concise Dictionary

MathematicsSubjectClassification2010
14-00,14-01
Author
Prof.Dr.ElenaRubei
UniversitàdegliStudidiFirenze
DipartimentodiMatematicaeInformatica“U.Dini”
VialeMorgagni67/A
50134Firenze
Italy
[email protected]
ISBN978-3-11-031622-3
e-ISBN978-3-11-031623-0
Set-ISBN978-3-11-031600-1
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Notation
𝔸𝑛 Wedenoteby𝔸𝑛 theaffinespaceofdimension𝑛over𝐾.
𝐾 𝐾
𝐴𝑝(𝑋) Forany𝐶∞manifold𝑋,𝐴𝑝(𝑋)denotesthevectorspaceofthe𝐶∞𝑝-formson
𝑋.
𝐴𝑝,𝑞(𝑋) Foranyalmostcomplexmanifold𝑋(see“Almostcomplexmanifolds,holomor-
phicmaps,holomorphictangentbundles”),𝐴𝑝,𝑞(𝑋)denotesthevectorspace
ofthe𝐶∞(𝑝,𝑞)-formson𝑋.
𝑏(𝑋,𝑅) Foranytopologicalspace𝑋andring𝑅,𝑏(𝑋,𝑅)(Bettinumber)denotesthe
𝑖 𝑖
rankof𝐻(𝑋,𝑅);itisalsodenotedbyℎ(𝑋,𝑅);see“Singularhomologyand
𝑖 𝑖
cohomology”.
b.p.f. Theabbreviationb.p.f.standsfor“basepointfree”,see“Bundles,fibre-”.
𝐶𝐻𝑝(𝑋) Foranyalgebraicvariety𝑋,𝐶𝐻𝑝(𝑋)denotesthe𝑝-thChowgroupof𝑋;see
“Equivalence, algebraic, rational, linear -, Chow, Neron–Severi and Picard
groups”.
𝐶𝑙(𝑋) Foranyalgebraicvariety𝑋,𝐶𝑙(𝑋)denotesthedivisorclassgroupof𝑋;see
“Divisors”.
𝐶∞(𝑋,𝐸) Forany𝐶∞vectorbundle𝐸ona𝐶∞manifold𝑋,𝐶∞(𝑋,𝐸)denotesthesetof
the𝐶∞sectionsof𝐸;see“Bundles,fibre-”.
𝛿 𝛿 stands(asusual)fortheKroneckerdelta.
𝛼,𝛽 𝛼,𝛽
(𝐷) ForanyCartierdivisor𝐷,(𝐷)denotesthelinebundleassociatedto𝐷;see“Di-
visors”.
𝑓 ,𝑓∗,𝑓−1 See “Pull-back and push-forward of cycles”, “Direct and inverse image
∗
sheaves”,“Singularhomologyandcohomology”.
|𝐷|,L(𝐷) Foranydivisor𝐷,wedenoteby|𝐷|thecompletelinearsystemassociatedto𝐷,
see“Linearsystems”;seealso“Linearsystems”forthedefinitionofL(𝐷).
𝜑 Foranylinebundle𝐿onavariety𝑋,𝜑 denotesthemapassociatedto𝐿;see
𝐿 𝐿
“Bundles,fibre-”.
𝑔𝑟 𝑔𝑟 denotesalinearsystemonaRiemannsurfaceofdegree𝑑andprojective
𝑑 𝑑
dimension𝑟;see“Linearsystems”.
𝐺(𝑘,𝑉),𝐺(𝑘,ℙ) Foranyvectorspace𝑉,𝐺(𝑘,𝑉)denotestheGrassmannianof𝑘-planesin𝑉,see
“Grassmannians”;analogously,foranyprojectivespaceℙ,𝐺(𝑘,ℙ)denotesthe
Grassmannianofprojective𝑘-planesinℙ.
𝐻(𝑋,𝑅),ℎ(𝑋,𝑅) Foranytopologicalspace𝑋andanyring𝑅,𝐻(𝑋,𝑅)denotesthe𝑖-thhomology
𝑖 𝑖 𝑖
moduleof𝑋withcoefficientsin𝑅(see“Singularhomologyandcohomology”);
ℎ(𝑋,𝑅)denotesitsrank;ℎ(𝑋,𝑅)isalsodenotedby𝑏(𝑋,𝑅)(Bettinumber)
𝑖 𝑖 𝑖
𝐻𝑖(𝑋,𝑅),ℎ𝑖(𝑋,𝑅) Foranytopologicalspace𝑋andanyring𝑅,𝐻𝑖(𝑋,𝑅)denotesthe𝑖-thcohomol-
ogymoduleof𝑋withcoefficientsin𝑅(see“Singularhomologyandcohomol-
ogy”);ℎ𝑖(𝑋,𝑅)denotesitsrank.
viii | Notation
𝐻𝑖(𝑋,E) ForanysheafofAbeliangroupsEonatopologicalspace𝑋,𝐻𝑖(𝑋,E)denotes
the𝑖-thcohomologygroupofE,see“Sheaves”;ℎ𝑖(𝑋,E)denotesitsrank;if𝐸
isaholomorphicvectorbundleonacomplexmanifold𝑋oranalgebraicvec-
torbundleonanalgebraicvariety𝑋,wesometimeswrite𝐻𝑖(𝑋,𝐸)insteadof
𝐻𝑖(𝑋,O(𝐸)).
𝜒(𝑋) Foranytopologicalspace𝑋,𝜒(𝑋)denotestheEuler–Poincarécharacteristicof
𝑋,i.e.,thesum∑ (−1)𝑖𝑏(𝑋,ℤ),whenitisdefined.
𝑖=1,...,∞ 𝑖
𝜒(𝑋,E) ForanysheafofAbeliangroupsEonatopologicalspace𝑋,𝜒(𝑋,E)denotes
theEuler–PoincarécharacteristicofE,i.e.,∑ (−1)𝑖ℎ𝑖(𝑋,E),whenitisde-
𝑖=1,...,∞
fined.
𝐾 ,𝜔 For any complex manifold or smooth algebraic variety 𝑋, 𝐾 denotes the
𝑋 𝑋 𝑋
canonicalbundleand𝜔 thecanonicalsheaf,i.e.,O(𝐾 );see“Canonicalbun-
𝑋 𝑋
dle,canonicalsheaf”.
𝑀F,F𝑀 See“Serrecorrespondence”.
𝑀(𝑚×𝑛,𝐾) Wedenoteby𝑀(𝑚×𝑛,𝐾)thespaceofthe𝑚×𝑛matriceswithentriesin𝐾.
nef Theabbreviationnefstandsfor“numericallyeffective”;see“Bundles,fibre-”.
𝑁𝑆𝑝(𝑋) Foranyalgebraicvariety𝑋,𝑁𝑆𝑝(𝑋)denotesthe𝑝-thNeron–Severigroupof𝑋;
see“Equivalence,algebraic,rational,linear-,Chow,Neron–SeveriandPicard
groups”.
O ,O (𝑠) If𝑋isacomplexmanifold,O (orsimplyO)denotesthesheafofholomor-
𝑋 𝑋 𝑋
phicfunctions;if𝑋isanalgebraicvariety,itdenotesthesheafoftheregular
functions;moregenerally,itdenotesthestructuresheafofaringedspace;see
“Space,ringed-”;forthedefinitionofO (𝑠),see“Hyperplanebundles,twist-
𝑋
ingsheaves”.
O(𝐸),𝛺𝑝(𝐸) Let𝐸beanalgebraicvectorbundleonanalgebraicvarietyoraholomorphic
vector bundle on a complex manifold; then O(𝐸) denotes the sheaf of the
(regular,resp.holomorphic)sectionsof𝐸;see“Bundles,fibre-”.Wedenote
𝛺𝑝⊗O(𝐸)by𝛺𝑝(𝐸).
𝛺𝑝 Foranycomplexmanifold𝑋,𝛺𝑝denotesthesheafoftheholomorphic𝑝-forms;
for any algebraic variety, 𝛺𝑝 denotes the sheaf of the regular 𝑝-forms; see
“Zariskitangentspace,differentialforms,tangentbundle,normalbundle”.
ℙ𝑛 Wedenotebyℙ𝑛 theprojectivespaceofdimension𝑛over𝐾.
𝐾 𝐾
𝑝 (𝑋),𝑝 (𝑋),𝑃(𝑋) Thesymbols𝑝 (𝑋),𝑝 (𝑋)and𝑃(𝑋)denoterespectivelythearithmeticgenus,
𝑎 𝑔 𝑖 𝑎 𝑔 𝑖
thegeometricgenusandthe𝑖-thplurigenusof𝑋(for𝑋varietyormanifold);
see“Genus,arithmetic,geometric,real,virtual-”,“Plurigenera”.
𝑃𝑖𝑐(𝑋) Foranyalgebraicvariety𝑋,𝑃𝑖𝑐(𝑋)denotesthePicardgroupof𝑋;see “Equiv-
alence,algebraic,rational,linear-,Chow,Neron–SeveriandPicardgroups”.
PID PIDstandsfor“principalidealdomain”,i.e.,foranintegraldomainsuchthat
everyidealisprincipal.
𝜋(𝑋,𝑥) Foranytopologicalspace𝑋andany𝑥 ∈ 𝑋,𝜋(𝑋,𝑥)denotesthe𝑖-thfunda-
𝑖 𝑖
mentalgroupof𝑋atthebasepoint𝑥;see“Fundamentalgroup”.
Notation | ix
𝑞(𝑋) Foranycomplexmanifoldoralgebraicvariety𝑋,𝑞(𝑋)denotestheirregularity
of𝑋;see“Irregularity”.
𝑅𝑞𝑓 ,𝑅𝑞 See“Directandinverseimagesheaves”
∗ 𝑓
𝑠𝑎𝑡(𝐼) Foranyideal𝐼,𝑠𝑎𝑡(𝐼)denotesthesaturationof𝐼;see“Saturation”.
𝛴 Forany𝑑∈ℕ,wedenoteby𝛴 thegroupofthepermutationson𝑑elements.
𝑑 𝑑
𝑆𝑝𝑒𝑐(𝑅),𝑃𝑟𝑜𝑗(𝑆) Foranyring𝑅,𝑆𝑝𝑒𝑐(𝑅)denotesitsspectrum,see“Schemes”.See“Schemes”
alsoforthedefinitionof𝑃𝑟𝑜𝑗(𝑆)for𝑆gradedring.
𝑆× 𝑆󸀠 Thesymbol𝑆× 𝑆󸀠denotesthefibredproductof𝑆and𝑆󸀠over𝐵;see“Fibred
𝐵 𝐵
product”.
⋅ Thesymbol⋅denotestheintersectionofcycles;see“Intersectionofcycles”.
Sometimesitisomitted.
√𝐽 Foranyideal𝐽inaring𝑅,wedenoteby√𝐽theradicalof𝐽,i.e.,
√𝐽={𝑥∈𝑅|∃𝑛∈ℕ −{0} s.t.𝑥𝑛∈𝐽}.
𝑈𝑉,𝑈+𝑉,(𝑈:𝑉) Let𝑅beacommutativeringand𝑈and𝑉betwoidealsin𝑅.Wedefine𝑈𝑉to
betheideal{∑ 𝑢𝑣|𝑘∈ℕ, 𝑢 ∈𝑈, 𝑣 ∈𝑉}.Moreover,wedefine𝑈+𝑉=
𝑖=1,...,𝑘 𝑖 𝑖 𝑖 𝑖
{𝑢+𝑣|𝑢∈𝑈𝑣∈𝑉},and(𝑈:𝑉):={𝑥∈𝑅|𝑥𝑉⊂𝑈}.
𝑉∨ Foranyvectorspace𝑉,wedenoteby𝑉∨itsdualspace.
⊔ ⊔denotesthedisjointunion.
𝑓:(𝑋,𝑆)→(𝑌,𝑇) If𝑆⊂𝑋and𝑇⊂𝑌,thenotation𝑓:(𝑋,𝑆)→(𝑌,𝑇)standsforamap𝑓:𝑋→𝑌
suchthat𝑓(𝑆)⊂𝑇.
Note.Theendofthedefinitions,theorems,andpropositionsisindicatedbythesymbol .