Real K3 surfaces with non-symplectic involution and applications

Abstract

called right DPN-pairs. Classication of K3 surfaces with non-symplectic involution ðX; Þ is equivalent to classication of right DPN-pairs A & Y .In this paper we consider a description of connected components of moduli of real K3 surfaces with non-symplectic involution and real right DPN-pairs. Equivalently, we consider the classication of triplets ðX; ; 'Þ where X is a K3 surface, is its holomorphic non-symplectic involution and ' is its antiholomorphic involution. The involutions and ' must commute: ' ¼ '. Equivalently, we consider the classication of real right DPN-pairs A & Y with an anti-holomorphic involution of Y where ðAÞ ¼ A.This classication depends on the denition of degeneration of ðX; ; 'Þ. We choose one of the weakest types of degeneration (that is, giving one of the richest discriminants). Geometrically (over C) this is equivalent to the appearance of an exceptional curve F with square À2 on the quotient Y . Over R this curve should be either real, ðF Þ ¼ F , or such that ðF Þ Á F 6 1. See x 2.1.We mention that our denition of degeneration is weaker (it gives a much richer discriminant) than the one used in [4] for the description of connected components of moduli of real Enriques surfaces. Thus, even for real Enriques surfaces, our results are stronger (more exact). For example, one of the dierences of our results from [4] is in the appearance of some exactly-dened group G of equivalences.On the one hand, our type of degeneration gives a rich discriminant. For example, it is enough to describe connected components of moduli of real bidegree ð4; 4Þ curves on the hyperboloid which was one of the main purposes of this paper. A degeneration of this case gives non-singular curves A 2 jÀ2K F 2 j which we must exclude. Here we denote as F n , with n > 0, a relatively minimal rational surface with the section s having s 2 ¼ Àn. See Proposition 4. On the other hand, using the Global Torelli Theorem [25] and epimorphicity of the Torelli map [14] for K3 surfaces we prove that connected components of moduli are described by suciently simple invariants which are isomorphism classes of integral involutions with conditions. They are triplets ðL; '; ðS; ÞÞ. Here L ¼ H 2 ðX; ZÞ is the even unimodular homology lattice with the action of ', S ¼ L and ¼ 'jS. The pair ðS; Þ determines the type of real K3 surfaces with non-symplectic involution ðX; ; 'Þ whose connected components of moduli we want to describe. See Theorems 1, 6 and 7. All genus invariants of integral involutions with condition were classied in [22]. See x 2.3 for more about these invariants. They give very universal invariants which work in all the cases that we consider. This paper can be considered as one of the applications of this classication and the invariants that it uses.We mention that a similar arithmetic description of connected components of moduli had already been obtained in [20] for real polarized K3 surfaces. A similar result was obtained in [4] for real Enriques surfaces with the much stronger type of degeneration used in [4] (or with ...

On real anti-bicanonical curves with one double point on the 4-th real Hirzebruch surface

“…This paper is a continuation of [10] and [11]. In [10] the moduli spaces of "(DR)-nondegenerate" real K3 surfaces with non-symplectic holomorphic involutions (namely, real 2-elementary K3 surfaces) are formulated and it is shown that the connected components of such a moduli space are in one to one correspondence with the isometry classes of integral involutions of the K3 lattice of certain type (see Theorem 2.11 below and [10] for more precise statements).…”

“…Using the same method as above, we obtain the real isotopic classifications of real nonsingular anti-bicanonical curves on the real Hirzebruch surfaces RF 2 and RF 3 (see also Theorem 2.13 and Remark 2.26 of this paper) in [11]. Especially, all the connected components of the moduli space of real 2-elementary K3 surfaces of type (S, θ) ∼ = ((3, 1, 1), −id), which are defined below, are enumerated in [11].…”

“…In [10] the moduli spaces of "(DR)-nondegenerate" real K3 surfaces with non-symplectic holomorphic involutions (namely, real 2-elementary K3 surfaces) are formulated and it is shown that the connected components of such a moduli space are in one to one correspondence with the isometry classes of integral involutions of the K3 lattice of certain type (see Theorem 2.11 below and [10] for more precise statements). As its applications, we obtain the real isotopic classifications (more precisely, deformation classifications) of real nonsingular sextic curves on the real projective plane RP 2 , real nonsingular curves of bidegree (4, 4) on the real P 1 × P 1 (hyperboloid, ellipsoid), and real nonsingular anti-bicanonical curves on the real Hirzebruch surfaces RF m (m = 1, 4).…”

“…Any 2-elementary K3 surface (X, τ ) of type S ∼ = (3, 1, 1) has an elliptic fibration and a unique reducible fiber E + F , where the curves E and F are defined in [11] (see also Subsection 2.2 of this paper). There are two types of F , which are reducible and irreducible.…”

“…Especially, all the connected components of the moduli space of real 2-elementary K3 surfaces of type (S, θ) ∼ = ((3, 1, 1), −id), which are defined below, are enumerated in [11]. Here we should remark that any real 2-elementary K3 surfaces of type (S, θ) ∼ = ((3, 1, 1), −id) are (DR)-nondegenerate in the sense of [10].…”