Table Of Content1842
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
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Berlin
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Trygve Johnsen
Andreas Leopold Knutsen
K3 Projective Models
in Scrolls
1 3
Authors
TrygveJohnsen
DepartmentofMathematics
UniversityofBergen
Johs.Brunsgt.12
5008Bergen,Norway
e-mail:[email protected]
AndreasLeopoldKnutsen
DepartmentofMathematics
UniversityofOslo
Box1053,Blindern
0316Oslo,Norway
e-mail:[email protected]
LibraryofCongressControlNumber:2004103750
MathematicsSubjectClassification(2000):14J28,14H51
ISSN0075-8434
ISBN3-540-21505-0Springer-VerlagBerlinHeidelbergNewYork
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Preface
The cover picture shows a smooth quartic surface in space, the simplest ex-
ample of a projective model of a K3 surface. In the following pages we will
encounter many more examples of models of such surfaces.
The purpose of this volume is to study and classify projective models
of complex K3 surfaces polarized by a line bundle L such that all smooth
curves in |L| have non-general Clifford index. Such models are in a natural
way contained in rational normal scrolls.
These models are special in moduli in the sense that they do not repre-
sent the general member in the countable union of 19-dimensional families
of polarized K3 surfaces. However, they are of interest because they fill up
the set of models in Pg for g ≤10 not described as complete intersections in
projectivespaceorinahomogeneousspaceasdescribedbyMukai,withafew
classificable exceptions.
Thusourstudyenablesus toclassifyanddescribeallprojectivemodelsof
K3 surfaces of genus g ≤10, which is the main aim of the volume.
Acknowledgements. We thank Kristian Ranestad, who suggested to study
certain projective models of K3 surfaces in scrolls that had shown up in
connection with his work on varieties of sums of powers (see [I-R1], [I-R2]
and [R-S]). This idea was the starting point of our work.
We are also grateful to M. Coppens, G. Fløystad, S. Ishii, S. Lekaus,
R. Piene, J. Stevens, S. A. Strømme, B. Toen and J. E. Vatne for useful
conversations, and to G. M. Hana for pointing out several mistakes in an
earlier version of the manuscript.
We also thank Alessandra Sarti who made us the nice picture we have
used on the cover.
Most of this book was written while the authors were visitors at the De-
partmentofMathematics,UniversityofUtah,andattheMax-Planck-Institut
VI Preface
für Mathematik,Bonn,inthe year2000.We thank both institutions for their
hospitality.
The second author was supported by a grant from the Research Council
of Norway.
Bergen/Oslo,March 2004 Trygve Johnsen
Andreas Leopold Knutsen
Contents
1 Introduction............................................... 1
1.1 Background............................................. 1
1.2 Related literature........................................ 7
1.3 How the book is organised................................ 7
1.4 Notation and conventions................................. 9
2 Surfaces in Scrolls ......................................... 15
2.1 Rational normal scrolls................................... 15
2.2 Specializing to K3 surfaces ............................... 17
3 The Clifford index of smooth curves in |L| and the
definition of the scrolls T(c,D,{Dλ})...................... 19
3.1 Gonality and Clifford index of curves ...................... 19
3.2 The result of Green and Lazarsfeld ........................ 20
3.3 Clifford divisors ......................................... 21
3.4 Getting a scroll ......................................... 28
4 Two existence theorems ................................... 31
5 The singular locus of the surface S(cid:1) and the scroll T ....... 35
5.1 The singular locus of ϕ (S)............................... 35
L
5.2 The singular locus of T and perfect Clifford divisors ......... 36
6 Postponed proofs .......................................... 47
7 Projective models in smooth scrolls........................ 59
8 Projective models in singular scrolls ....................... 63
8.1 Blowing up S ........................................... 64
8.2 The smooth scroll T0..................................... 67
8.3 Techniques for finding Betti-numbers of the ϕ (D ) ......... 70
L λ
8.4 Resolutions for projective models .......................... 82
8.4.1 Pushing down resolutions........................... 86
8.5 Rolling factors coordinates................................ 89
8.6 Some examples.......................................... 92
VIII Contents
9 More on projective models in smooth scrolls of K3
surfaces of low Clifford-indices............................. 99
9.1 Projective models with c=1..............................100
9.2 Projective models with c=2..............................105
9.2.1 An interpretation of b1 and b2.......................108
9.2.2 Possible scroll types for c=2 .......................109
9.3 Projective models with c=3..............................114
9.4 Higher values of c .......................................117
10 BN general and Clifford general K3 surfaces ..............121
10.1 The results of Mukai.....................................121
10.2 Notions of generality.....................................123
10.3 The case g =8 ..........................................124
10.4 The case g =10 .........................................127
11 Projective models of K3 surfaces of low genus .............129
11.1 A new decomposition of R................................130
11.2 Perfect Clifford divisors for low c ..........................131
11.3 The possible scroll types..................................135
11.4 Some concrete examples..................................138
11.5 The list of projective models of low genus...................143
12 Some applications and open questions .....................155
12.1 BN generality ..........................................155
12.2 Applications to Calabi-Yau threefolds ......................156
12.3 Analogies with Enriques surfaces ..........................157
References.....................................................159
Index..........................................................163
1
Introduction
1.1 Background
A K3 surface is a smooth compact complex connected surface with trivial
canonical bundle and vanishing first Betti number. The mysterious name
K3 is explained by A. Weil in the comment on his Final report on contract
AF18(603)-57 (see [We] p 546):
Dans la seconde partie de mon rapport, il s’agit des variétés kähler-
iénnes dites K3, ainsi nommées en l’honneur de Kummer, Kodaira,
Kähler et de la belle montagne K2 au Cachemire.
ItiswellknownthatallK3surfacesarediffeomorphic,andthatthereisa
20-dimensionalfamily ofanalyticalisomorphismclassesofK3 surfaces.How-
ever, the general element in this family is not algebraic, in fact the algebraic
onesformacountableunionof19-dimensionalfamilies.Moreprecisely,forany
n > 0 there is a 19-dimensional irreducible family of K3 surfaces equipped
with a base point free line bundle of self-intersection n. Moreover,the family
ofK3surfaceshaving≥k linearlyindependentdivisors(i.e.thesurfaceswith
Picard number ≥ k, where the Picard number is by definition the rank of
the Picardgroup)forms a dense countable union of subvarietiesof dimension
20−k in the family ofall K3 surfaces.In particular,onthe generalalgebraic
K3surfacealldivisorsarelinearlyequivalentto somerationalmultiple ofthe
hyperplane class (see [G-H, pp. 590–594]).
A pair (S,L) of a K3 surface S and a base point free line bundle L with
L2 = 2g − 2 will be called a polarized K3 surface of genus g. Note that
g = h0(L)−1 and that g is the arithmetic genus of any member of |L|. The
sections of L give a map ϕ of S to Pg, and the image is called a projective
L
model of S. When ϕ is birational,the image is a surface of degree 2g−2 in
L
Pg. It is also easy to see that a projective model of genus 2 is a 2 : 1 map
S →P2 branched along a sextic curve.
A very central point in the theory of projective models of K3 surfaces is
that by the adjunction formula every smooth hyperplane section of a projec-
T.JohnsenandA.L.Knutsen:LNM1842,pp.1–14,2004.
(cid:1)c Springer-VerlagBerlinHeidelberg2004
2 1 Introduction
tive model of S (these are the images by ϕ of the smooth members of |L|)
L
are canonical curves, i.e. curves for which ω (cid:4)O (1).
C C
The firstexamples of projectivemodels ofK3 surfacesare the ones which
are complete intersections in projective space. Using the fact that for a com-
pleteintersectionsu(cid:1)rfaceS ofn−2hypersurfacesinPnofdegreesd1,...,dn−2
we have ω (cid:4) O ( d −n−1) and h1(O ) = 0 (see e.g. [Hrts, Exercises
S S i S
II,8.4 and III,5,5]), we find that there are exactly three types of K3 com-
plete intersections, namely a hyperquartic in P3, a complete intersection of
a hyperquadric and a hypercubic in P4 and a complete intersection of three
hyperquadrics in P5.
In fact one can show that any birational projective model of genus 3 is
a quartic surface and of genus 4 a complete intersection of a quadric and a
cubichypersurface.Butalreadyforgenus5thesituationisnotassimple:The
generalmodelis acompleteintersectionofthree hyperquadrics,butthereare
models which are not. In fact, take a 3-dimensional smooth rational normal
scrollX ofdegree3inP5,whichcanbeseenastheunionofP2sparametrized
byP1,i.e.aP2-bundleoverP1.Intersectthisscrollbya(sufficientlygeneral)
cubic hypersurfaceC containingoneofthe P2-fibers,callitF,thenthe inter-
section is C∩X =F ∪S, where S is a smooth surface of degree 8 in P5, i.e.
a K3 surface.The idealof this surface cannotbe generatedonly by quadrics,
whenceS isnotacompleteintersectionofthreehyperquadrics.Notethatthe
intersection of C with a general P2-fiber is a smooth curve of degree 3 in P2,
whichis elliptic by the genusformula,so S containsa pencilof elliptic curves
of degree 3. In particular, since such a curve cannot be linearly equivalent
to a multiple of the hyperplane section, S contains two linearly independent
divisors, whence these surfaces can at most fill up an 18-dimensional family
(in fact we will show that they do fill up an 18-dimensionalfamily). Another
interesting point is that the elliptic pencil on the surface cuts out a g1 (i.e. a
3
linear system of dimension 1 and degree 3) on each hyperplane section of S.
Conversely, by a classical theorem of Enriques-Petri, the homogeneous ideal
of a canonical curve with a g1 is generated by both quadrics and cubics, so
3
any projectivemodelin P5 ofa K3surface whosehyperplane sectionshavea
g1 cannot be the complete intersection of three hyperquadrics.
3
For 6≤g ≤10 and g =12 it is shown by Mukai in [Mu1] and [Mu2] that
the general projective models are complete intersections in certain homoge-
neousvarietiescontainedinprojectivespacesoflargerdimensionthan g.The
ambientvarietiesareconstructedusingspecialdivisorsonthehyperplanesec-
tions,andthegeneralmodelshavethepropertythattheirhyperplanesections
do not carry certain particular grs induced from divisors on the surface.
d
ThattheprojectivemodelofaK3surfacesomehowhastodowithspecial
divisors carried by the curves in |L| dates back to the classical paper [SD]
of Saint-Donat, which has become the main reference for all later work on
projective models of or curves on K3 surfaces.
Asremarkedin[SD]itisclearfromZariski’sMainTheorem(seee.g.[Hrts,
V, Thm. 5.2]) that
