Singular Algebraic Curves With an Appendix by Oleg Viro

569 Pages·2018·8.09 MB·English

by  Gert-Martin Greuel

Springer Monographs in Mathematics
Gert-Martin Greuel
Christoph Lossen
Eugenii Shustin
Singular
Algebraic
Curves
With an Appendix by Oleg Viro
Springer Monographs in Mathematics
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Isabelle Gallagher, Paris, France
Minhyong Kim, Oxford, UK
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Gert-Martin Greuel Christoph Lossen
(cid:129)
Eugenii Shustin
Singular Algebraic Curves
With an Appendix by Oleg Viro
123
Gert-Martin Greuel EugeniiShustin
Fachbereich Mathematik Schoolof Mathematical Sciences
TU Kaiserslautern TelAviv University
Kaiserslautern, Germany TelAviv, Israel
Christoph Lossen
Fachbereich Mathematik
TU Kaiserslautern
Kaiserslautern, Germany
ISSN 1439-7382 ISSN 2196-9922 (electronic)
SpringerMonographs inMathematics
ISBN978-3-030-03349-1 ISBN978-3-030-03350-7 (eBook)
https://doi.org/10.1007/978-3-030-03350-7
LibraryofCongressControlNumber:2018960191
Mathematics Subject Classification (2010): 14-02, 14Hxx, 14H10, 14H20, 14F17, 14B07, 14M25,
14J60,14P25,14Q05
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Für Ursula
Gert-Martin Greuel
Für Carmen, Katrin, Carolin und Benjamin
Christoph Lossen
To Mila and Boris
Eugenii Shustin
Preface
ThetitleSingularAlgebraicCurvesdesignatesawideresearchfieldwhichhasbeen
of constant interest and importance from Descartes, Pascal, Newton to nowadays.
Wedonotpursuethe“missionimpossible”togiveacompleteaccountofthistopic,
but concentrate mainly on the geometry of deformations and families of singular
algebraiccurvesonalgebraicsurfacesdefinedoverthecomplexorrealfield.More
precisely, this book presents in detail the theory of zero-dimensional schemes
relatedtoplanarcurvesingularities,andthecohomologyvanishingtheoryforideal
sheavesofzero-dimensionalschemes,whichbuildsabridgebetweenthelocaland
global geometry of singular algebraic curves. Moreover, the cohomological
approach combines classical and modern results on the geometry of deformations
and families of singular algebraic curves, on applications and computational
aspects.
The problems we address include such basic and natural questions as
(cid:129) Whatkindofsingularitiesandhowmany ofthemcanbeonanalgebraiccurve
of a given degree in the plane or in a given linear system on an algebraic
surface?
(cid:129) Which simultaneous deformations of singular points of a curve are possible if
the curve varies in a given linear system?
(cid:129) What are the geometric properties of equisingular families of curves, e.g.,
dimension, smoothness of “expected dimension,” irreducibility?
We refer to these questions as existence problem, deformation problem,
T-smoothnessproblem,andirreducibilityproblem,respectively.TheT-smoothness
property(i.e.,smoothofexpecteddimension),introducedinthismonograph,means
thatthesingularitiesimpose independentconditionsinaratherstrongsenseonthe
equisingular family. It plays a central role in our approach to answer the above
questions.
Thesequestionsappearedtoberatherhard.AfterSeveri’sfundamentaltheorem,
whichclaimsthatthefamilyofplanecurvesofagivendegreewithagivennumber
of nodes is smooth of “expected dimension,” and the nodes can be independently
vii
viii Preface
smoothed out, not much progress, except for a very few special cases, has been
done till the middle of the 1980s. However, the general development of coho-
mology of sheaves, deformation theory, local singularity theory, and, especially
equisingularity theory, in this period (just to mention Zariski, Schlessinger, Wahl,
Arnol’d and his school) played a crucial role in the further study. Harris’ theorem
(1985) on the irreducibility offamilies of plane irreducible curves of given degree
and genus was a major breakthrough, though the method, based on moduli of
curves and very restrictive deformation properties, applies, in fact, only in this
situation.
Themainideaoftheapproachwefocuson,whichcanbetracedbacktoSeveri’s
originalwork,istoreformulatetheabovequestionsasH1-vanishingconditionsfor
the ideal sheaves of certain zero-dimensional schemes and apply H1-vanishing
criteria. The key point here is a choice and an appropriate use of the cohomology
vanishingtheory,whichisanontrivialanddelicatetask.Themainadvantageisthat
thisapproachprovidesauniformpointofviewonalltheproblemsstated,itapplies
tocurveswitharbitrarysingularitiesandonvarioussurfaces,and,mostimportant,it
answers the above questions in the form of relations for numerical invariants of
curves and singularities, which cover an asymptotically proper range of possible
values of invariants. We especially stress on the asymptotic properness of the
results, because pathological properties of singular curves and their families have
beenobservedalreadyinthecasenexttonodalcurves,planecurveswithnodesand
cusps, and this, actually, does not allow one to get complete answers for the
non-nodal case. The numerical relations obtained are asymptotically close to the
borderbetweentheregularandpathologicalpropertiesofsingularalgebraiccurves.
We illustrate this just in two examples:
(cid:129) The first example concerns the existence of plane curves with prescribed sin-
gularities. Given analytic or topological singularity types S ;...;S satisfying
1 r
the inequality lðS Þþ (cid:2)(cid:2)(cid:2) þlðS Þ (cid:3) 1ðd2(cid:4)2dþ3Þ, we prove that then there
1 r 9
existsanirreducibleplanecurveofdegreedwithexactlyrsingularpointsofthe
given types, respectively. This sufficient condition is quadratic in d and differs
from the classically known necessary one, lðS Þþ (cid:2)(cid:2)(cid:2) þlðS Þ (cid:3) ðd(cid:4)1Þ2,
1 r
only by a constant factor of d2 in the right-hand side.
(cid:129) Theotherexampleconcernsthegeometryoffamiliesofplanecurveswithnodes
andcusps.Classicallyknownisthatthefamilyofplanecurvesofdegreed with
nnodesandkcuspsissmoothofexpecteddimension,ifk\3d.Thisisarather
restrictive condition, since asymptotically, k can be of order d2. We prove that
theinequality4nþ9k (cid:3) ðdþ3Þ2 issufficientforthesmoothnessandexpected
dimension of the family. Moreover, the coefficients of n and k in the latter
inequalityaresharpfork (cid:5) 3;i.e.,thereexistseriesofobstructedfamilieswith
9k ¼d2þOðdÞ, or k ¼3d and 4n¼d2 þOðdÞ.
Preface ix
We pay also attention to other approaches in the treatment of the problems
stated, but often do not give details. Some of them are already presented in recent
monographs. But the main reason for ourselection isthat it allows usto provide a
self-contained, logically connected exposition, developing methods and explaining
ideas of the main approach, and coming to concrete results and applications.
Readingofthemonographrequiressomefamiliaritywithsingularitytheoryand
(local) deformation theory of complex analytic spaces as well as with basic alge-
braic geometry. The necessary material can be found in various sources, for
example, in Hartshorne’s book on algebraic geometry [HaR2] and in our book
[GLS6],whichoriginallywasconceivedasthefirstpartofthepresentmonograph.
In particular, this book includes numerous references to statements and definitions
introduced in [GLS6]; moreover, references to many known facts and statements
usedhere aresupplied withadouble reference—onetotheoriginalsource andthe
other to [GLS6], where one can find both a precise formulation and a proof.
Mostofthestatementsinthetextarecompletelyproven;theothersaresupplied
byreferencestomonographsandarticles.ThemainnewresultsappearinChap.4,
where we give explicit references to the original sources, while the statements
without reference are new in this book. The main notions and theorems are illus-
tratedbyexamples,oftenconcretenumerical,elaboratedwiththecomputeralgebra
system SINGULAR.
The contents are divided into four chapters.
Chapter 1 is devoted to zero-dimensional schemes associated with singular
points of planar curves. We especially focus on the related singularity invariants,
both the classical ones and some new invariants, which appear in the last chapters
of the monograph. The reader interested in the classical problems on curves on
surfaces can start with Chap. 2 and look for details on zero-dimensional schemes
only when needed.
Chapter 2 on global deformation theory is concerned with three specific topics
whichplayafundamentalroleinthetheoryofequisingularfamiliesofcurves.The
significance of the classical theorems of Bézout, Noether, Bertini, Plücker,
Riemann–Hurwitz for the algebraic geometry of curves on the one hand and the
globaldeformationtheoryofanalyticspacesontheotherhandgomuchbeyondour
particular goals. These topics belong to the basic knowledge of any expert in
singularity theory. In addition, we introduce and discuss the notion of T-smooth-
ness for rather general singularity types of isolated hypersurface singularities. The
seminal patchworking construction, originated in works of Viro in the early 80s,
arouses a growing interest thanks to various applications to real and complex
algebraicgeometry, singularitytheory,andalgebra.It plays anultimate roleinour
construction of curves with prescribed singularities.
InChap.3,weconsiderseveralapproachestothecohomologyvanishingtheory
for ideal sheaves of zero-dimensional schemes on algebraic surfaces, with special
emphasis on numerical criteria. We expose the classical and modern contributions
tothetopicaswellasourownresults,notablyintendedtomakegeneralstatements
x Preface
applicabletoarbitrary singularitiesandzero-dimensionalschemes.Asanexample,
we mention the Castelnuovo function theory, which was brought to the study of
curves with nodes and cusps by Barkats and which later allowed us to obtain the
best known H1-vanishing criteria for ideal sheaves of the most interesting
zero-dimensional schemes.
The monograph culminates in Chap. 4, where we answer the questions on the
non-emptiness, dimension, smoothness, and irreducibility of equisingular families
of curves in the plane and on other algebraic surfaces. Some results about families
ofsingularcurves,especiallythoserelatedtothenon-emptinessandthesmoothness
problem,aresuppliedwithnaturalgeneralizationstofamiliesofhigher-dimensional
projective hypersurfaces with isolated singularities. The chapter starts with an
overview of thenew resultsand methods presentedin this monograph. We end up
with a discussion of open problems and conjectures, related to the geometry of
equisingular families of curves and hypersurfaces as well as to the tropical enu-
merativegeometryinwhichthepatchworkingconstructionappearsasanimportant
ingredient, providing a link between classical and tropical algebraic-geometric
objects.
Each chapter is supplied with a section devoted to historical notes and refer-
ences,wherewetracebackthekeyproblemsandideasrelatedtothecontentsofthe
chapter and give a general overview of the discussed topics.
Appendix A Patchworking Real Algebraic Varieties, written by Oleg Viro in
1983, shows the origin of the patchworking construction and its striking applica-
tions to Hilbert’s 16th problem. The ideas of this work influenced deformation
theory and singularity theory and served as one of the roots of the tropical
geometry.WeareverygratefultoOlegVirowhokindlypermittedustoincludehis
original article into the monograph.
Acknowledgements
Our work at the monograph has been supported by the Hermann Minkowski–
MinervaCenterforGeometryatTelAvivUniversityandbygrantno.G-616-15.6/
99 from the German-Israeli Foundation for Research and Development. We have
significantly advanced in our project during our two “Research-in-Pairs” stays at
theMathematischesForschungsinstitutOberwolfach.C.Lossenwasalsosupported
by the DFG grant no. Lo 864/1-1, E. Shustin was also supported by the Bessel
Research Award from the Alexander von Humboldt Foundation and by the grants
no.448/09and176/15fromtheIsraeliScienceFoundation.Weareindebtedtoour
colleagues for important corrections and suggestions, notably, to Adrian Langer,