Birational Geometry and Moduli Spaces (Springer INdAM Series) PDF
Preview Birational Geometry and Moduli Spaces (Springer INdAM Series)
Springer INdAM Series 39 Elisabetta Colombo · Barbara Fantechi Paola Frediani · Donatella Iacono Rita Pardini Editors Birational Geometry and Moduli Spaces Springer INdAM Series Volume 39 Editor-in-Chief GiorgioPatrizio,UniversitàdiFirenze,Florence,Italy SeriesEditors ClaudioCanuto,PolitecnicodiTorino,Turin,Italy GiulianellaColetti,UniversitàdiPerugia,Perugia,Italy GrazianoGentili,UniversitàdiFirenze,Florence,Italy AndreaMalchiodi,ScuolaNormaleSuperiore,Pisa,Italy PaoloMarcellini,UniversitàdiFirenze,Florence,Italy EmiliaMezzetti,UniversitàdiTrieste,Trieste,Italy GiocondaMoscariello,UniversitàdiNapoli“FedericoII”,Naples,Italy TommasoRuggeri,UniversitàdiBologna,Bologna,Italy SpringerINdAMSeries This series will publish textbooks, multi-authors books, thesis and monographs in English language resulting from workshops, conferences, courses, schools, seminars, doctoral thesis, and research activities carried out at INDAM - Istituto Nazionale di Alta Matematica, http://www.altamatematica.it/en. The books in the series will discuss recent results and analyze new trends in mathematics and its applications. THESERIESISINDEXEDINSCOPUS Moreinformationaboutthisseriesathttp://www.springer.com/series/10283 Elisabetta Colombo (cid:129) Barbara Fantechi (cid:129) Paola Frediani (cid:129) Donatella Iacono (cid:129) Rita Pardini Editors Birational Geometry and Moduli Spaces Editors ElisabettaColombo BarbaraFantechi DepartmentofMathematics SISSA-InternationalSchool UniversityofMilan forAdvancedStudies Milano,Italy Trieste,Italy PaolaFrediani DonatellaIacono DepartmentofMathematics DepartmentofMathematics UniversityofPavia Universita`degliStudidiBari Pavia,Italy Bari,Italy RitaPardini DepartmentofMathematics UniversityofPisa Pisa,Italy ISSN2281-518X ISSN2281-5198(electronic) SpringerINdAMSeries ISBN978-3-030-37113-5 ISBN978-3-030-37114-2(eBook) https://doi.org/10.1007/978-3-030-37114-2 ©SpringerNatureSwitzerlandAG2020 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,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This volumecollects contributionsfromspeakersat the INdAM Workshop“Bira- tionalGeometryandModuliSpaces”,whichwasheldinRomeon11–15June2018. The workshop was devoted to the interplay between birational geometry and modulispaces,twocentraltopicsinAlgebraicGeometrythathavealwaysattracted theinterestofnationalandinternationalresearchers. A longstandingproblem in geometry is the classification of geometric objects. Thestartingpointisageometricobject,suchasacurve,avarietyorasheaf,defined by some equationsor given by some conditions.Then, one can deformit to get a newgeometricobjectrelatedtotheoldone.Thisexpressestheconceptofafamily of an objectand the main purposeis the classification of these families, in such a way that the classifying space is a reasonable geometric space. This space is the so-calledmodulispaceanditsgeometricpointsparametrisetheobjectsthatweare considering.Oneisinterestedinclassifyinggeometricobjectsuptoisomorphism, so that the moduli space has a variety structure, or geometric objects with their automorphisms,thusmeaningthatthemodulispacehasastackstructure.Inorder to understand the local and global structure of moduli spaces, it is important to handlewelltechniquesfromdeformationtheory.Indeed,deformationtheorycanbe regardedasatooltounderstandthelocalgeometryofmodulispaces. Thereisalsointerestinclassifyinggeometricobjectsuptobirationalmorphism. Many results are known for the birational study of algebraic surfaces, while for higherdimensiontherearemanyopenproblemsthatcanbetackledviatheminimal model program. Great progress has been made in recent years, supporting the developmentofnewtechniquesandinnovativeideas. The workshop focused on these research aspects and their interactions and offered the possibility to disseminate the knowledge of advanced results and key techniquesusedtosolveopenproblemsinthearea. Thisvolumecoversmanytopicsinthewideresearchareaofbirationalgeometry and moduli spaces, and includes both surveys and original research papers. In particular,itcontainsworksonirreducibleholomorphicsymplecticmanifolds,Sev- eri varieties, degenerationof Calabi–Yau varieties, uniruledthreefolds, toric Fano v vi Preface threefolds, mirror symmetry, canonical bundle formula, the Lefschetz principle, birationaltransformation,anddeformationofdiagramsofalgebras. WeareindebtedtoFrancescoBastianelliandAntonioRapagnettafortheirhelp inorganisingtheworkshopandtoEmiliaMezzettiforhersupportasamemberof theScientificCommittee. OnbehalfoftheScientificandOrganisingCommittees,wewouldliketothank theFoundationCompositioMathematicafortheirsupport. WeareallverygratefultotheIstitutoNazionalediAltaMatematica“Francesco Severi”(INdAM)forthegeneroussupportandhelpthatmadeallthispossible. Finally,wewouldliketoofferspecialthankstoalltheparticipantsandspeakers oftheINdAMWorkshop,andespeciallytotheauthorswhoacceptedtheinvitation topublishhere. Milano,Italy ElisabettaColombo Trieste,Italy BarbaraFantechi Pavia,Italy PaolaFrediani Bari,Italy DonatellaIacono Pisa,Italy RitaPardini Contents NegativeRationalCurvesandTheirDeformationsonHyperkähler Manifolds.......................................................................... 1 EkaterinaAmerik ModuliSpacesofCubicThreefoldsandofIrreducibleHolomorphic SymplecticManifolds............................................................ 13 ChiaraCamere ANoteonSeveriVarietiesofNodalCurvesonEnriquesSurfaces......... 29 Ciro Ciliberto, Thomas Dedieu, Concettina Galati, andAndreasLeopoldKnutsen ATravelGuidetotheCanonicalBundleFormula ........................... 37 EnricaFlorisandVladimirLazic´ SomeExamplesofCalabi–YauPairswithMaximalIntersection andNoToricModel.............................................................. 57 Anne-SophieKaloghiros OnDeformationsofDiagramsofCommutativeAlgebras................... 77 EmmaLepriandMarcoManetti TheLefschetzPrincipleinBirationalGeometry:BirationalTwin Varieties........................................................................... 109 CésarLozanoHuertaandAlexMassarenti WhatistheMonodromyPropertyforDegenerationsofCalabi-Yau Varieties?.......................................................................... 133 LuigiLunardon ExamplesofIrreducibleSymplecticVarieties ................................ 151 ArvidPerego vii viii Contents AnExampleofMirrorSymmetryforFanoThreefolds...................... 173 AndreaPetracci ChernNumbersofUniruledThreefolds....................................... 189 StefanSchreiederandLucaTasin About the Editors ElisabettaColomboisAssociateProfessorofGeometryattheUniversityofMilan. Herresearchfieldiscomplexalgebraicgeometry,andshestudiesmainlycurvesand abelianvarietiesandtheirmoduli. Barbara Fantechi is Full Professor in Geometry at SISSA-ISAS in Trieste. Her research interests include deformation theory, derived algebraic geometry and stacks. Paola Frediani is Associate Professor of Geometry at the University of Pavia. Herresearchareaisalgebraicgeometry,inparticularmodulispacesofcurvesand abelianvarietiesandHodgetheory. Donatella Iacono is a Researcher in Geometry at the University of Bari. Her research focuses on deformation theory and differential graded Lie algebras in algebraicgeometry. RitaPardiniisFullProfessorofGeometryattheUniversityofPisa.Herresearch areaisalgebraicgeometry,especiallysurfacesandtheirmoduli,irregularvarieties andcoverings. ix
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