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Families of K3 Surfaces in Smooth Fano 3-Folds with Picard Number 2

Abstract

We study certain families of K3 surfaces in Fano 3-folds that contain curves.

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References

  1. Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–535 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Batyrev, V.V.: Toroidal Fano 3-folds. Math. USSR, Izv. 19, 13–25 (1982)

    Article  MATH  Google Scholar 

  3. Eisenbud, D., Lange, H., Martens, G., Schreyer, F.-O.: The Clifford dimension of a projective curve. Compos. Math. 72, 173–204 (1989)

    MathSciNet  MATH  Google Scholar 

  4. Galkin, S.S.: Small toric degenerations of Fano 3-folds. Preprint (2007)

  5. Kobayashi, M., Mase, M.: Isomorphism among families of weighted K3 hypersurfaces. Tokyo J. Math. 35, 461–467 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  6. Martens, G.: Über den Clifford-index algebraischer Kurven. J. Reine Angew. Math. 336, 83–90 (1982)

    MathSciNet  MATH  Google Scholar 

  7. Martens, G.: On curves on K3 surfaces. In: Ballico, E., Ciliberto, C. (eds.) Algebraic Curves and Projective Geometry. Lecture Notes in Mathematics, vol. 1389, pp. 174–182. Springer, Berlin (1989)

    Chapter  Google Scholar 

  8. Mase, M.: Families of K3 surfaces in smooth Fano 3-folds. Comment. Math. Univ. St. Pauli 66, 103–114 (2012)

    MathSciNet  Google Scholar 

  9. Moĭšezon, B.G.: Algebraic homology classes on algebraic varieties. Math. USSR, Izv. 1, 209–251 (1967)

    Article  Google Scholar 

  10. Mori, S., Mukai, S.: Classification of Fano 3-folds with B 2≥2. Manuscr. Math. 36, 147–162 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  11. Mori, S., Mukai, S.: On Fano 3-folds with B 2≥2. In: Iitaka, S. (ed.) Algebraic Varieties and Analytic Varieties. Adv. Stud. Pure Math., vol. 1, pp. 101–129 (1983)

    Google Scholar 

  12. Mori, S., Mukai, S.: Classification of Fano 3-folds with B 2≥2, I. In: Nagata, M., et al. (eds.) Algebraic and Topological Theories: To the Memory of Dr. Takehiko Miyata, pp. 496–545. Kinokuniya, Tokyo (1986)

    Google Scholar 

  13. Mori, S., Mukai, S.: Classification of Fano 3-folds with B 2≥2. Manuscr. Math. 110, 407 (2003)

    MathSciNet  Article  Google Scholar 

  14. Šokurov, V.V.: Smoothness of the general anticanonical divisor on a Fano 3-fold. Math. USSR, Izv. 14, 395–405 (1980)

    Article  Google Scholar 

  15. Watanabe, K., Watanabe, M.: The classification of Fano 3-folds with torus embeddings. Tokyo J. Math. 5, 37–48 (1982)

    MathSciNet  Article  MATH  Google Scholar 

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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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Correspondence to Makiko Mase.

Appendix: Presenting Picard Lattices

Appendix: Presenting Picard Lattices

Theorem 4

The Picard lattices of families of K3 surfaces in smooth Fano 3-folds with Picard number 2 are determined as in Table 1.

Table 1 Picard lattices of families of K3 surfaces in smooth Fano 3-folds of ρ=2

Remark 7

  1. (1)

    U denotes the hyperbolic lattice.

  2. (2)

    For a lattice L=(Z r,〈,〉 L ) and a natural number m, denote L(m)=(Z r,m〈,〉 L ).

  3. (3)

    We denote by M 2,i the Picard lattice of a family of K3 surfaces in a smooth Fano 3-fold of Picard number 2 of No. i.

  4. (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. (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. (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 Lefschetz-type 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 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:XV 4 be the blow-up of V 4 along an elliptic curve C=AB with the exceptional divisor D, where \(A,B\in |\mathcal{O}_{V_{4}}(1)|\). Then we have

$${-}K_X = 2\pi^* H_{V_4}-D, $$

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

$$\begin{aligned} \pi^*H_{V_4}^2|_{{-}K_X} = & 2H_{V_4}^3 = 2\times 4 = 8, \\ \pi^*H_{V_4}.D|_{{-}K_X} = & H_{V_4}.C = 4, \\ D^2|_{{-}K_X} = & 2g(C)-2 = 0. \end{aligned}$$

By a base-change with , the Picard lattice and its discriminant are

$$M_{2,10}=U(4), \quad\quad \mathop{\mathrm{discr}}\nolimits M_{2,10}=-16. $$

No. 13 Let π:XQ be the blow-up of Q along a curve C of degC=6 and g(C)=2 with the exceptional divisor D. Then we have

$${-}K_X = 3\pi^* H - D, $$

where \(H=\mathcal{O}_{Q}(1)\) is a hyperplane section with H 3=2. Hence, one can compute

$$\begin{aligned} \pi^* H^2|_{{-}K_X} = & 3H^3 = 6,\\ \pi^* H.D|_{{-}K_X} = & H.C = 6, \\ D^2|_{{-}K_X} = & 2g(C)-2 = 2. \end{aligned}$$

By a base-change with , the Picard lattice and its discriminant are

$$M_{2,13}=\left( \begin{array}{c@{\quad}c} 6 & 0 \\ 0 & -4 \end{array} \right), \quad\quad \mathop{\mathrm{discr}}\nolimits M_{2,13}=-24. $$

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Remark 8

Families of K3 surfaces in smooth Fano 3-folds Nos. 28 and 33 are generically birationally corresponding [8]. Moreover, a family of K3 surfaces in smooth Fano 3-fold 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 3-Folds with Picard Number 2. Vietnam J. Math. 42, 295–304 (2014). https://doi.org/10.1007/s10013-014-0063-5

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Keywords

  • Family of K 3 surfaces
  • Picard lattice

Mathematics Subject Classification (2000)

  • 14J28
  • 14C22
  • 14E05
  • 14J10
  • 14J45
  • 14J70