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Vietnam J MathDOI 10.1007/s10013-014-0063-5
Families of K3 Surfaces in Smooth Fano 3-Foldswith Picard Number 2
Makiko Mase
Received: 28 July 2013 / Accepted: 29 October 2013© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore2014
Abstract We study certain families of K3 surfaces in Fano 3-folds that contain curves.
Keywords Family of K3 surfaces · Picard lattice
Mathematics Subject Classification (2000) 14J28 · 14C22 · 14E05 · 14J10 · 14J45 ·14J70
1 Introduction
Let S be a compact complex connected two-dimensional variety, which we call a surface.A surface S is called a Gorenstein K3 surface if the canonical divisor KS is trivial, theirregularity h1(OS) is zero, and S has at worst ADE singularities. When a Gorenstein K3surface is nonsingular, we call it simply a K3 surface. The Picard lattice Pic(S) of a K3surface S is the Picard group of S with a lattice structure induced from that of H 2(S,Z).The rank of a Picard group is called the Picard number and is denoted by ρ(S).
Let X be a Fano 3-fold, that is, a three-dimensional algebraic variety admitting a canon-ical divisor whose anticanonical divisor −KX is ample. Smooth Fano 3-folds with Picardnumber ≥ 2 are classified into 89 classes by Mori and Mukai [10–13].
It is proved by Šokurov [14] that general anticanonical members of smooth Fano 3-foldsare smooth. Furthermore, generic anticanonical members of a smooth Fano 3-fold are K3surfaces since any member in the complete anticanonical linear system of a Fano 3-fold hasa trivial canonical divisor and irregularity zero by adjunction formula, Lefschetz’s hyper-plane section theorem, and Kodaira vanishing.
Let D1, . . . ,Dr be a generator of the Picard group of a smooth Fano 3-fold X. A theoremby Moı̆s̆ezon [9] induces that for any generic anticanonical member S ∈ |−KX| of X, thePicard lattice Pic(S) is generated by divisors D1|−KX
, . . . ,Dr |−KXon S, the restriction of a
M. Mase (B)Department of Mathematics and Information Sciences, Tokyo Metropolitan University,1-1 Minami-Osawa, Hachioji-shi, Tokyo 192-0397, Japane-mail: [email protected]
M. Mase
generator of the Picard group of X to the anticanonical divisor −KX . Hence, in particular,the Picard number of a generic anticanonical member in a smooth Fano 3-fold X is the sameas that of X.
In general, the complete anticanonical linear system |−KX| of a Fano 3-fold X param-eterizes (possibly degenerate) K3 surfaces, and there is a family FX of K3 surfaces in X
parameterized by |−KX|. We define the Picard lattice of a family FX as the Picard lattice ofgeneric member in FX .
We are interested in studying a symmetry that may appear among families of K3 surfacestogether with relation to mirror symmetry theory. It is proved by Mase [8] that the Picardlattices of families of K3 surfaces in smooth toric Fano 3-folds are mutually distinct. How-ever, there are isometric Picard lattices among those of families of K3 surfaces in smooth,not necessarily toric Fano 3-folds. An example of such a pair of families is also studied in[8], and it is concluded that these families are essentially the “same,” which means that thegeneric members in these families are birationally corresponding. The result is a general-ization of birational correspondence among families of weighted K3 surfaces by Kobayashiand Mase [5] to nontoric K3 hypersurface case since weighted projective spaces are all toricvarieties, that is, varieties admitting a torus-action with Zariski open torus orbit.
The main results of this article are the following.Let X′,X be smooth Fano 3-folds obtained by blowing-up P 3 along a line and a smooth
plane cubic curve, respectively. Let Y be a small toric degeneration of X. Denote byF ′,F, ˜F families of K3 surfaces parameterized by |−KX′ |, |−KX|, |−KY |, respectively.A correspondence between families F ′ and F is studied in [8], and we find a relation withanother family ˜F .
Main Theorem 1 There exists a birational map Φ : F ′ → ˜F that corresponds generalmembers in F ′ with those in ˜F .
Let K = (3) ∩ (3) be a smooth irreducible curve that is an intersection of two smoothcubic surfaces in P 3, and X′′ be a smooth Fano 3-fold obtained by blowing-up P 3 along K.Denote by F ′′ a family of K3 surfaces parameterized by |−KX′′ |. We also have a correspon-dence of F ′ with the family F ′′.
Main Theorem 2 There exists a common subfamily G of F ′ and F ′′ such that GorensteinK3 surfaces in F ′ and F ′′ are birationally correspondent via G.
For a parameter t in a parameter space of a family of K3 surfaces, denote by St thecorresponding member in the family. Main Theorem 1 means that a general member St inF ′ is birationally equivalent to the member Sφ(t) in ˜F , where φ is an isomorphism betweenthe parameter spaces of F ′ and ˜F . Turning to Main Theorem 2, we mean that there existsubfamilies G′ of F ′ and G′′ of F ′′, and an isomorphism φ between the parameter spaces ofG′ and G′′ such that a member St in G′ is birationally equivalent to the member Sφ(t) in G′′.
Main Theorem 1 is proved in Sect. 3.1, and Main Theorem 2 is proved in Sect. 3.2. InAppendix, we give all the Picard lattices of families of K3 surfaces in smooth Fano 3-foldswith the Picard number 2. In addition, we study which Picard lattices are isometric or notand give a proof for each case.
Families of K3 Surfaces in Smooth Fano 3-Folds with Picard Number 2
2 Preliminary
We refer [12] for a standard computation of intersection numbers of divisors on K3 surfacesin smooth Fano 3-folds.
We fix the notation according to [10] for the reader’s convenience.
Vd is a Fano 3-fold of index 2 and Picard number 1 with (−KVd/2)3 = d (d = 1,2,3,4,5).
That is, −KVd= 2HVd
, where HVd= OVd
(1) with H 3Vd
= d .V7 is the blow-up of P 3 at a point.W is a divisor in P 2 × P 2 of bidegree (1,1).Q is a smooth quadratic hypersurface in P 4.
3 K3 Surfaces Containing a Line, a Smooth Plane Cubic, and a Martens Curve
3.1 Families F ′ and ˜F
Let l and C be respectively a line and a smooth plane cubic curve in P 3; we assume thatl and C are in the same plane H on P 3. Let X′ and X be the blow-ups of P 3 along l andC, respectively. Then, by classification [11], X′ is a smooth toric Fano 3-fold, and X is asmooth nontoric Fano 3-fold. Denote by F ′,F the families of K3 surfaces parameterizedby the complete anticanonical linear systems |−KX′ | and |−KX|, respectively. The Picardlattices of F ′ and F are isometric to
(
Z2,( 4 3
3 0
))
. Recall the following:
Theorem 1 [8] For every smooth member S ′ ∈ F ′ (resp. S ∈ F ), there exists a GorensteinK3 surface S1 ∈ F (resp. S ′
1 ∈ F ′) such that S ′ and S1 (resp. S and S ′1) are birational.
Definition 1 [4] Let Y be a normal Gorenstein toric Fano 3-fold. Y is called a small toricdegeneration of smooth Fano 3-folds if there exists a projective flat morphism π : X →�1 := {t ∈ C | |t | < 1}, where X is an irreducible complex manifold, such that
(1) For all t ∈ �1\{0}, the fiber Xt = π−1(t) is a smooth Fano 3-fold;(2) The central fiber X0 = π−1(0) has at worst Gorenstein terminal singularities, and
X0 � Y ;(3) For all t ∈ �1, the restriction map Pic(X) → Pic(Xt) of the Picard groups is an isomor-
phism.
Remark 1 Gorenstein terminal singularities in three-dimensional toric varieties are knownto be nodes defined by XZ − YW = 0.
Let ΣX′ ([2, 15]) in R3 be the fan of X′ with 1-simplices generated by
(1,0,0), (0,1,0), (0,0,1), (−1,−1,0), (−1,−1,−1).
The polar dual �∗X′ of the polytope corresponding to ΣX′ has vertices (Fig. 1)
t (−1,−1,3), t (−1,−1,−1), t (2,−1,0), t (−1,2,0), t (2,−1,−1), t (−1,2,−1).
Let ΣY in R3 be a fan with 1-simplices generated by
(1,0,0), (0,1,0), (0,0,1), (−1,−1,0), (−1,0,−1), (0,−1,−1), (−1,−1,−1).
M. Mase
Fig. 1 The convex polytopes �X′ and �∗X′
Fig. 2 The convex polytopes �Y and �∗Y
Then Y := P (ΣY ) is the small toric degeneration of the nontoric smooth Fano 3-fold X [4].The toric Fano 3-fold Y has three nodes, and the Picard number and degree of Y are thesame as those of Xs. The polar dual �∗
Y of the polytope corresponding to ΣY has vertices(Fig. 2)
t (−1,−1,2), t (−1,−1,−1), t (2,−1,−1), t (−1,2,−1),
t (1,0,0), t (0,1,0), t (0,0,1).
Remark 2 Both polytopes �∗X′ and �∗
Y are reflexive so that general anticanonical membersof X′ and Y are Gorenstein K3 surfaces [1].
By Theorem 1.2 of [14], a general anticanonical member in the complete anticanonicallinear system |−KY | of a Fano 3-fold Y is a K3 surface. Hence, the linear system |−KY |parameterizes a family ˜F of K3 surfaces in Y .
Remark 3 In Fig. 3, we label the vertices of common polytope with monomials. The (polardual) polytope �∗
P 3 of P 3 has a natural labeling of its vertices with monomials. By using
the fact that the Fano 3-folds X′ and Y are obtained by blowing-ups of P 3 along a line andapparently a union of three lines in P 3, respectively, one can label lattice points of polytopes�∗
X′ and �∗Y with monomials. Indeed, �∗
X′ is obtained by taking off an edge passing throughvertices of �∗
P 3 labeled as x42 and x4
3 , and �∗Y is obtained by taking off three edges through
x42 and x4
3 , x42 and x4
1 , and x41 and x4
3 .
Families of K3 Surfaces in Smooth Fano 3-Folds with Picard Number 2
Fig. 3 �∗Y
is a subpolytope of�∗
X′
Proposition 1 There exists a monomial mapping (in torus) between the sets of monomi-als in H 0(X′,OX′(−KX′)) and in H 0(Y,OY (−KY )). This mapping exists uniquely up toisomorphism.
Proof By an elementary calculation, the polytope �∗Y is a subpolytope of the polytope �∗
X′(Fig. 3). Hence, we have a monomial map by
H 0(X′,OX′(−KX′)) x40 x0x
31 x0x
33 x0x
32 x1x
22x3
H 0(Y,OY (−KY )) x40 x0x
31 x0x
33 x0x
32 x1x
22x3.
Here, (x0 : x1 : x2 : x3) is a coordinate system of P 3. This transformation exists uniquely upto linear isomorphism. �
Theorem 2 There exists a birational map Φ : F ′ → ˜F that corresponds general membersin F ′ with those in ˜F .
Proof Note that both polytopes �∗Y and �∗
X′ contain only lattice point t (0,0,0) in the rela-tive interior. Hence, the monomial mapping Φ∗ as in Proposition 1 induces a natural map-ping between anticanonical members that are in common in Y and X′. Moreover, since theminimal model exists uniquely, if for F ∈ |−KX′ |, Φ∗F = F̃ ∈ |−KY |, then the minimal
models ˜(F = 0) and ˜
(F̃ = 0) are isomorphic. Therefore, Φ is a birational mapping. �
3.2 Families F ′ and F ′′
We generalize Lemma 4.1 in [8]:
Lemma 1 Let ϕ : Y → P 3 be the blow-up of P 3 along a smooth irreducible curve Σ in P 3
of degree d with the exceptional divisor Σ ′. Assume that Y is a smooth Fano 3-fold. If M isa Gorenstein K3 surface in |−KY |, then
(i) ϕ(M) contains the curve Σ ,(ii) if M and ϕ(M) are both smooth, then ϕ|M is isomorphic.
Proof (i) Let IΣ be the ideal sheaf of Σ . Since −KY = ϕ∗(−KP 3)−Σ ′, it is easy to see thatH 0(Y,OY (−KY )) ∼= H 0(P 3,OP 3(4H) ⊗ IΣ). Therefore, for all Gorenstein K3 surfacesM ∈ |−KY |, ϕ(M) contains the curve Σ .
M. Mase
(ii) Let M ∈ |−KY | and ϕ(M) ∈ |−KP 3 − Σ | be both smooth. Let l be any P 1-fiber ofthe P 1-fibration Σ ′ → Σ . It is easily verified that −KY .l = 1, and since M is minimal, M
intersects l transversally. Hence, by definition, it is clear that ϕ is an immersion and ϕ|M isinjective. Therefore, ϕ is an embedding of M into P 3. �
Let K = (3) ∩ (3) be a smooth irreducible curve that is an intersection of two smoothcubic surfaces in P 3. Martens [6] studies such curves:
Remark 4 (Bemerkung 1 and its following additional remark [6]) A (smooth) completeintersection of two cubic surfaces in P 3 is a curve of degree 9 and genus 10, and the Cliffordindex of this curve is 3. If Σ is a curve of degree 9 and genus 10, then Σ is a completeintersection of two cubic surfaces and is 6-gonal.
In this paper, we call such K a Martens curve. Moreover, Martens studies K3 surfacesthat contain a curve of degree 9 and genus 10 and obtains the following.
Remark 5 [3, 7] A smooth quartic surface in P 3, which is a K3 surface, having K in it alsocontains a line, and the Picard lattice of such a general K3 surface is
(
Z2,
(
4 11 −2
))
�(
Z2,
(
4 33 0
))
.
Let μ : X′′ → P 3 be the blow-up of P 3 along a Martens curve K. Then, by the classifi-cation [11], X′′ is a smooth (nontoric) Fano 3-fold. Denote by F ′′ a family of K3 surfacesthat is parameterized by the complete anticanonical linear system |−KX′′ |. By Lemma 1,K3 surfaces in F ′′ contain K. Then, we have the following:
Theorem 3 There exists a common subfamily G of F ′ and F ′′ such that Gorenstein K3surfaces in F ′ and F ′′ are birationally correspondent via G.
Proof Let G be a family of K3 surfaces parameterized by |−KP 3 − K − l|. Since the linesin P 3 have no moduli, we have |−KP 3 − K − l| � |−KP 3 − K − l′| for all line l′. Hence,by Remark 5, any general Gorenstein K3 surface in G is isomorphic to a Gorenstein K3surface in F ′′.
Since |−KP 3 − K − l| is a sublinear system of |−KP 3 − l|, any Gorenstein K3 surfacein G is isomorphic to some Gorenstein K3 surface in F ′. Thus, the Gorenstein K3 surfacesin F ′ and F ′′ are corresponding. �
Remark 6 In [8] and Theorem 3, we only deal with families of K3 surfaces in smooth Fano3-folds. In Sect. 3.1, we extend our interest into those in toric nonsmooth Fano 3-fold bystudying a family of K3 surfaces in a Gorenstein terminal Fano 3-fold.
Acknowledgements The author thanks Professors Komeda and Obuchi for useful suggestions about alge-braic curves, Professor Kobayashi for his helpful comments and encouragement, and organizers of “Topologyof Singularities and Related Topics III” for giving an opportunity of talking.
Appendix: Presenting Picard Lattices
Theorem 4 The Picard lattices of families of K3 surfaces in smooth Fano 3-folds withPicard number 2 are determined as in Table 1.
Families of K3 Surfaces in Smooth Fano 3-Folds with Picard Number 2
Table 1 Picard lattices of families of K3 surfaces in smooth Fano 3-folds of ρ = 2
No. Fano 3-fold Picard lattice
1 blow-up of V1 along an elliptic curve that is A ∩ B, where A,B ∈ |−KV1/2| U
2 double cover of P 1 × P 2 branched along (2,4)-curve in P 1 × P 2
(
−2 0
0 2
)
3 blow-up of V2 along an elliptic curve that is A ∩ B,A,B ∈ |−KV2/2| U(2)
36 P (OP 2 ⊕O
P 2 (2))
(
2 1
1 −2
)
35 V7 = P (OP 2 ⊕O
P 2 (1))
(
4 0
0 −2
)
5 blow-up of V3 ⊂ P 4 along a plane cubic U(3)
34 P 1 × P 2
(
2 3
3 0
)
4 blow-up of P 3 along (3) ∩ (3)
(
4 3
3 0
)
28 blow-up of P 3 along a plane cubic
(
4 3
3 0
)
33 blow-up of P 3 along a line
(
4 3
3 0
)
6 (2,2)-divisor in P 2 × P 2
(
−6 0
0 2
)
32 (1,1)-divisor in P 2 × P 2
(
−6 0
0 2
)
30 blow-up of P 3 along a plane conic
(
6 0
0 −2
)
15 blow-up of P 3 along (2) ∩ (3)
(
6 0
0 −2
)
11 blow-up of V3 ⊂ P 4 along a line
(
6 1
1 −2
)
31 blow-up of Q ⊂ P 4 along a line
(
6 1
1 −2
)
10 blow-up of V4 ⊂ P 5 along an elliptic curve which is the intersection of twohyperplane sections
U(4)
18 double cover of P 1 × P 2 branched along (2,2)-curve in P 1 × P 2
(
−8 0
0 2
)
29 blow-up of Q ⊂ P 4 along a conic
(
8 0
0 −2
)
8 double cover of V7 branched along B ∈ |−KV7 | where B ∩ D is smooth, D is the
exc. div. of V7 → P 3
(
4 0
0 −4
)
25 blow-up of P 3 along an elliptic quartic
(
4 0
0 −4
)
9 blow-up of P 3 along C with deg = 7,genus = 5
(
4 3
3 −2
)
M. Mase
Table 1 (Continued)
No. Fano 3-fold Picard lattice
27 blow-up of P 3 along a twisted cubic
(
4 3
3 −2
)
19 blow-up of V4 ⊂ P 5 along a line
(
4 3
3 −2
)
12 blow-up of P 3 along C with deg = 6,genus = 3
(
4 2
2 −4
)
16 blow-up of V4 ⊂ P 5 along a conic
(
10 0
0 −2
)
24 (1,2)-divisor in P 2 × P 2
(
2 1
1 −10
)
26 blow-up of V5 ⊂ P 6 along a line
(
−2 1
1 10
)
13 blow-up of Q ⊂ P 4 along C with deg 6,genus2
(
6 0
0 −4
)
22 blow-up of V5 ⊂ P 6 along a conic
(
−6 0
0 4
)
14 blow-up of V5 ⊂ P 6 along an elliptic curve that is the intersection of twohyperplane sections
U(5)
17 blow-up of Q ⊂ P 4 along an elliptic curve of deg 5
(
−4 5
5 0
)
21 blow-up of Q ⊂ P 4 along a twisted quartic
(
14 0
0 −2
)
23 blow-up of Q ⊂ P 4 along A ∩ B, where A ∈ |OQ(1)|,B ∈ |OQ(2)|(
14 0
0 −2
)
20 blow-up of V5 ⊂ P 6 along a twisted cubic
(
10 3
3 −2
)
7 blow-up of Q ⊂ P 4 along A ∩ B, where A,B ∈ |OQ(2)|(
6 8
8 0
)
Remark 7
(1) U denotes the hyperbolic lattice.(2) For a lattice L = (Zr , 〈, 〉L) and a natural number m, denote L(m) = (Zr ,m〈, 〉L).(3) We denote by M2,i the Picard lattice of a family of K3 surfaces in a smooth Fano 3-fold
of Picard number 2 of No. i.(4) In the above table, we follow the numbering and notation of [11]. However, we reorder
them as discriminants (which we omit here as they are to be computed easily) growing.(5) There do not exist divisors D1,D2 such that D1.D2 = 0 in the following cases: of num-
bers i = 1,36,5,34,4,28,33,11,31,9, 27,19,24,26,14, 17,20 since Picard latticesof K3 surfaces are even lattices but the discriminants of lattices M2,i are odd num-bers; of number 12, where the discriminant of the lattice M2,12 is −20 = −2 × 10by an easy calculation; of number 7, where the discriminant of the lattice M2,7 is−64 = −2 × 32 = −4 × 16 = −8 × 8 by an easy calculation.
Families of K3 Surfaces in Smooth Fano 3-Folds with Picard Number 2
(6) It is clear that lattices with different Gram matrices are mutually distinct even if theyhave the same discriminants, for which we omit the proof.
Proof The Picard lattices are obtained by an explicit computation. By Moı̆s̆ezon’s versionof Lefschetz-type theorem [9], we need to compute the intersection matrix of the latticegenerated by D1|−KX
,D2|−KX, where D1,D2 is a generator of the Picard group of a Fano
3-fold. When a Fano 3-fold is a toric variety, another proof is given in [8].We give two typical examples how to compute the Picard lattices.
No. 10 Let π : X → V4 be the blow-up of V4 along an elliptic curve C = A ∩ B with theexceptional divisor D, where A,B ∈ |OV4(1)|. Then we have
−KX = 2π∗HV4 − D,
where HV4 = OV4(1) is a hyperplane section of V4 with H 3V4
= 4. Hence, one can compute
π∗H 2V4
|−KX= 2H 3
V4= 2 × 4 = 8,
π∗HV4 .D|−KX= HV4 .C = 4,
D2|−KX= 2g(C) − 2 = 0.
By a base-change with( 1 0
−1 1
)
, the Picard lattice and its discriminant are
M2,10 = U(4), discrM2,10 = −16.
No. 13 Let π : X → Q be the blow-up of Q along a curve C of degC = 6 and g(C) = 2with the exceptional divisor D. Then we have
−KX = 3π∗H − D,
where H = OQ(1) is a hyperplane section with H 3 = 2. Hence, one can compute
π∗H 2|−KX= 3H 3 = 6,
π∗H.D|−KX= H.C = 6,
D2|−KX= 2g(C) − 2 = 2.
By a base-change with( 1 −1
0 1
)
, the Picard lattice and its discriminant are
M2,13 =(
6 00 −4
)
, discrM2,13 = −24. �
Remark 8 Families of K3 surfaces in smooth Fano 3-folds Nos. 28 and 33 are genericallybirationally corresponding [8]. Moreover, a family of K3 surfaces in smooth Fano 3-foldNo. 4 is generically corresponding to these two families, as is proved in Theorem 3.
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