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.

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) 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 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 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 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:X→V ₄ be the blow-up 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

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

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:X→Q 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.