On the Geometry of Some Special Projective Varieties PDF
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Lecture Notes of the Unione Matematica Italiana Francesco Russo On the Geometry of Some Special Projective Varieties Winner of the UMI Book Prize 2015 Lecture Notes of 18 the Unione Matematica Italiana Moreinformationaboutthisseriesathttp://www.springer.com/series/7172 EditorialBoard CiroCiliberto(EditorinChief) CH-8057Zuerich,Switzerland DipartimentodiMatematica e-mail:[email protected] Universita’diRomaTorVergata FrancoFlandoli ViadellaRicercaScientifica DipartimentodiMatematicaApplicata 00133Roma(Italia) UniversitàdiPisa e-mail:[email protected] ViaBuonarroti1c SusannaTerracini(Co-editorinChief) 56127Pisa,Italy UniversitàdegliStudidiTorino e-mail:fl[email protected] DipartimentodiMatematica“GiuseppePeano” AngusMacIntyre ViaCarloAlberto10 QueenMaryUniversityofLondon 10123Torino,Italy SchoolofMathematicalSciences e-mail:[email protected] MileEndRoad AdolfoBallester-Bollinches LondonE14NS Departmentd’Àlgebra UnitedKingdom FacultatdeMatemàtiques e-mail:[email protected] UniversitatdeValència GiuseppeMingione Dr.Moliner,50 DipartimentodiMatematicaeInformatica 46100Burjassot(València) UniversitàdegliStudidiParma Spain ParcoAreadelleScienze,53/a(Campus) e-mail:[email protected] 43124Parma,Italy AnnalisaBuffa e-mail:[email protected] IMATI–C.N.R.Pavia MarioPulvirenti ViaFerrata1 DipartimentodiMatematica, 27100Pavia,Italy UniversitàdiRoma“LaSapienza” e-mail:[email protected] P.leA.Moro2 LuciaCaporaso 00185Roma,Italy DipartimentodiMatematica e-mail:[email protected] UniversitàRomaTre FulvioRicci LargoSanLeonardoMurialdo ScuolaNormaleSuperiorediPisa I-00146Roma,Italy PiazzadeiCavalieri7 e-mail:[email protected] 56126Pisa,Italy FabrizioCatanese e-mail:[email protected] MathematischesInstitut ValentinoTosatti Universitätstraße30 NorthwesternUniversity 95447Bayreuth,Germany DepartmentofMathematics e-mail:[email protected] 2033SheridanRoad CorradoDeConcini Evanston,IL60208 DipartimentodiMatematica USA UniversitàdiRoma“LaSapienza” e-mail:[email protected] PiazzaleAldoMoro5 CorinnaUlcigrai 00185Roma,Italy ForschungsinstitutfürMathematik e-mail:[email protected] HGG44.1 CamilloDeLellis Rämistrasse101 InstitutfuerMathematik 8092Zürich,Switzerland UniversitaetZuerich e-mail:[email protected] Winterthurerstrasse190 TheEditorialPolicycanbefoundatthebackofthevolume. Francesco Russo On the Geometry of Some Special Projective Varieties 123 FrancescoRusso DipartimentodiMatematicaeInformatica UniversitaJdegliStudidiCatania Catania,Italy ISSN1862-9113 ISSN1862-9121 (electronic) LectureNotesoftheUnioneMatematicaItaliana ISBN978-3-319-26764-7 ISBN978-3-319-26765-4 (eBook) DOI10.1007/978-3-319-26765-4 LibraryofCongressControlNumber:2015958350 MathematicsSubjectClassification:14N05,14M07,14M10,14M22,14E30,14J70,14E05 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland. TheUnioneMatematicaItaliana(UMI)hasestablisheda bi-annualprize,sponsoredbySpringer-Verlag,tohonor an excellent, original monograph presenting the latest developmentsinanactiveresearchareaofmathematics, towhichtheauthormadeimportantcontributionsinthe recentyears. Theprize-winningmonographsarepublishedinthisseries. Detailsabouttheprizecanbefoundat: http://umi.dm.unibo.it/en/unione-matematica-italiana-prizes/ book-prize-unione-matematicaitaliana/ This book has been awarded the 2015 Book Prize of the Unione MatematicaItaliana. Themembersofthescientificcommitteeofthe2015prizewere: LuciaCaporaso UniversitàRomaTre,Italy CiroCiliberto (PresidenteoftheUMI) UniversitàdegliStudidiRomaTorVergata,Italy GianniDalMaso ScuolaInternazionaleSuperiorediStudiAvanzati(SISSA), Trieste,Italy CamilloDeLellis UniversityofZurich(UZH),Switzerland AlessandroVerra UniversitàRomaTre,Italy aCledvane, GiuliaeLuca contantoamoreeaffetto Preface Providing an introduction to both classical and modern techniques in projective algebraic geometry, this monograph treats the geometrical properties of varieties embedded in projective spaces, their secant and tangent lines, the behavior of tangent linear spaces, the algebro-geometric and topological obstructions to their embeddingintosmallerprojectivespaces,andtheclassificationofextremalcases. ItalsoprovidesasolutiontoHartshorne’sConjectureonCompleteIntersectionsfor theclassofquadraticmanifoldsandnewshortproofsofpreviouslyknownresults, usingthe moderntoolsofMoriTheoryandofrationallyconnectedmanifoldsand followingtheideasandmethodscontainedin[103,104,154,156,160]. Thenewapproachtosomeoftheproblemsconsideredcanbesummarizedinthe principlethat, instead of studyinga special embeddedmanifolduniruledby lines, one analyzes the original geometrical properties of the manifold of lines passing throughageneralpointandcontainedinthemanifold.Oncethismanifoldoflines in its natural embedding, usually of lower codimension, is classified, one tries to reconstructtheoriginalmanifold,followingaprinciplewhichalsoappearsinother areasofgeometrysuchasprojectivedifferentialgeometryandcomplexgeometry. These classical themes in algebraic geometry enjoyed renewed interest at the beginningof the 1980s, following some conjectures posed by Hartshorne and the discoverybyFultonandHansenofanimportantconnectednesstheoremwithnew anddeepapplicationstothegeometryofalgebraicvarietiesfoundbyZak,see[44, 69,70,198]. Catania,Italy FrancescoRusso1 19October2015 1Partially supported by PRIN Grant Geometria delle varietà algebriche and by the Research Project FIR 2014 Aspetti geometrici e algebrici della Weak e Strong Lefschetz Property of the UNICT.TheauthorisamemberoftheGNSAGAgroupofINDAM. ix
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