Abstract
We study certain families of K3 surfaces in Fano 3folds that contain curves.
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References
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Acknowledgements
The author thanks Professors Komeda and Obuchi for useful suggestions about algebraic curves, Professor Kobayashi for his helpful comments and encouragement, and organizers of “Topology of Singularities and Related Topics III” for giving an opportunity of talking.
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Appendix: Presenting Picard Lattices
Appendix: Presenting Picard Lattices
Theorem 4
The Picard lattices of families of K3 surfaces in smooth Fano 3folds with Picard number 2 are determined as in Table 1.
Remark 7

(1)
U denotes the hyperbolic lattice.

(2)
For a lattice L=(Z ^{r},〈,〉_{ L }) and a natural number m, denote L(m)=(Z ^{r},m〈,〉_{ L }).

(3)
We denote by M _{2,i } the Picard lattice of a family of K3 surfaces in a smooth Fano 3fold 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 D _{1},D _{2} such that D _{1}.D _{2}=0 in the following cases: of numbers i=1,36,5,34,4,28,33,11,31,9, 27,19,24,26,14, 17,20 since Picard lattices of K3 surfaces are even lattices but the discriminants of lattices M _{2,i } are odd numbers; of number 12, where the discriminant of the lattice M _{2,12} is −20=−2×10 by an easy calculation; of number 7, where the discriminant of the lattice M _{2,7} is −64=−2×32=−4×16=−8×8 by an easy calculation.

(6)
It is clear that lattices with different Gram matrices are mutually distinct even if they have the same discriminants, for which we omit the proof.
Proof
The Picard lattices are obtained by an explicit computation. By Moĭs̆ezon’s version of Lefschetztype theorem [9], we need to compute the intersection matrix of the lattice generated by \(D_{1}_{{}K_{X}}, D_{2}_{{}K_{X}}\), where D _{1},D _{2} is a generator of the Picard group of a Fano 3fold. When a Fano 3fold 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→V _{4} be the blowup of V _{4} along an elliptic curve C=A∩B with the exceptional divisor D, where \(A,B\in \mathcal{O}_{V_{4}}(1)\). Then we have
where \(H_{V_{4}}=\mathcal{O}_{V_{4}}(1)\) is a hyperplane section of V _{4} with \(H_{V_{4}}^{3}=4\). Hence, one can compute
By a basechange with , the Picard lattice and its discriminant are
No. 13 Let π:X→Q be the blowup of Q along a curve C of degC=6 and g(C)=2 with the exceptional divisor D. Then we have
where \(H=\mathcal{O}_{Q}(1)\) is a hyperplane section with H ^{3}=2. Hence, one can compute
By a basechange with , the Picard lattice and its discriminant are
□
Remark 8
Families of K3 surfaces in smooth Fano 3folds Nos. 28 and 33 are generically birationally corresponding [8]. Moreover, a family of K3 surfaces in smooth Fano 3fold No. 4 is generically corresponding to these two families, as is proved in Theorem 3.
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Mase, M. Families of K3 Surfaces in Smooth Fano 3Folds with Picard Number 2. Vietnam J. Math. 42, 295–304 (2014). https://doi.org/10.1007/s1001301400635
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DOI: https://doi.org/10.1007/s1001301400635
Keywords
 Family of K _{3} surfaces
 Picard lattice
Mathematics Subject Classification (2000)
 14J28
 14C22
 14E05
 14J10
 14J45
 14J70