Birational Geometry and Moduli Spaces (Springer INdAM Series)

204 Pages·2020·1.37 MB·English

by  Elisabetta Colombo (editor), Barbara Fantechi (editor), Paola Frediani (editor), Donatella Iacono (editor), Rita Pardini (editor)

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
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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
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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
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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.
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