A $1$-dimensional family of Enriques surfaces in characteristic $2$ covered by the supersingular $K3$ surface with Artin invariant $1$

Rational curves on the supersingular K3 surface with Artin invariant 1 in characteristic 3

Let S be an Enriques surface over an algebraically closed field k of arbitrary characteristic p. Recall that this means that S is a connected smooth projective surface whose canonical class is numerically trivial and second Betti number equal to 10 [4]. It is well-known that, generically over k=(E, an Enriques surface does not contain nonsingular rational curves. This can be seen, for example, by considering the period space for such surfaces I-3]. Also, it is known that if S contains such a curve, then, again generically, it contains infinitely many of them. This can be seen, for example, by viewing an Enriques surface as an elliptic surface whose jacobian surface is a rational elliptic surface. Assuming that the latter is general enough, its translation group is infinite and acts on S by automorphisms. Thus, the existence of one such curve implies the existence of infinitely many. In this paper we prove the following rather surprising result:

§0. Introduction A K3 surface X is a compact complex surface with KX ∼ 0 and H(X, OX) = 0. An Enriques surface is a compact complex surface with H(Y,OY ) = H(Y,OY ) = 0 and 2KY ∼ 0. The universal covering of an Enriques surface is a K3 surface. Conversely every quotient of a K3 surface by a free involution is an Enriques surface. Here a free involution is an automorphism of order 2 without any fixed points. The moduli space of Enriques surfaces is constructed using the periods of their covering K3 surfaces. Precisely speaking, an Enriques surface determines a lattice-polarized K3 surface and vice versa, so that the moduli space of Enriques surfaces can be described by the moduli space of lattice-polarized K3 surfaces. We note that even if we do not fix any polarization on Enriques surfaces, their covering K3 surfaces automatically have a lattice-polarization. Then, what happens if we drop the lattice-polarization of the covering K3 surface? We will call two Enriques quotients of a K3 surface distinct if they are not isomorphic to each other as varieties. In his paper [3], Kondo discovered a K3 surface with two distinct Enriques quotients. He computed the automorphism groups of the two quotients. Since then, as far as the author knows, no other examples have been found. In this paper we investigate this phenomenon. We show that K3 surfaces with more than one distinct Enriques quotients have 9-dimensional components (neither irreducible nor closed) in the period domain. Moreover we compute the exact number of distinct Enriques quotients at a very general point of

An Enriques surface over an algebraically closed field k of characteristic 4=2 is a nonsingular projective surface F with Hi(F, (gv)= H2(F, Or)=0, 2Kv=0. The unramified double cover of F defined by the torsion class K v is a K3-surface F, a nonsingular projective surface with HI(F,(gr)=0, Kr=0. The study of Enriques surfaces is equivalent to the study of K3-surfaces with a fixed-point-free involution z. In particular, the automor_phism group Aut(F) of F is isomorphic to the group Aut(ff, z)/(z), where Aut(F,z) is the centralizer of z in the automorphism group Aut(F) of ft. In the case k = ~ , the field of complex numbers, the study of Aut(F) is based on the Global Torelli Theorem for K3-surfaces proven by I. Piatetski-Shapiro and I. Shafarevich in [19]. It follows from this theorem that up to a finite group the group Aut(ff) is isomorphic to the quotient group O(Pic(F))/W, where O(Pic(F)) is the orthogonal group of the Picard lattice of ff and W its normal subgroup generated by the reflections into the classes of nonsingular rational curves. For a generic Enriques surface F this theorem allows to compute Aut(F) (see [3] and also [17], where this result is not stated explicitly). For an arbitrary F the relation between F and ff does not help, since it is very difficult to compute the action of z in Pie(if). However, by other means, we can prove the following analog of Piatetski-Shapiro and Shafarevich's result:

Let X and Y be supersingular K3 surfaces defined over an algebraically closed field. Suppose that the sum of their Artin invariants is 11. Then there exists a certain duality between their Neron–Severi lattices. We investigate geometric consequences of this duality. As an application, we classify genus one fibrations on supersingular K3 surfaces with Artin invariant 10 in characteristic 2 and 3, and give a set of generators of the automorphism group of the nef cone of these supersingular K3 surfaces. The difference between the automorphism group of a supersingular K3 surface X and the automorphism group of its nef cone is determined by the period of X. We define the notion of genericity for supersingular K3 surfaces in terms of the period, and prove the existence of generic supersingular K3 surfaces in odd characteristics for each Artin invariant larger than one.

Starting from an Enriques surface over \mathbb{Q}(t) constructed by Lafon, we give the first examples of smooth projective weakly special threefolds which fibre over the projective line in Enriques surfaces (resp. K3 surfaces) with nowhere reduced but non-divisible fibres and general type orbifold base. We verify that these families of Enriques surfaces (resp. K3 surfaces) are non-isotrivial and we compute their fundamental groups by studying the behaviour of local points along certain étale covers. The existence of these threefolds implies that the Weakly Special Conjecture formulated in 2000 contradicts the Orbifold Mordell Conjecture, and hence the abc conjecture. Using these examples, we can also easily disprove several complex-analytic analogues of the Weakly Special Conjecture. Finally, these examples show that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised in 2005.

In [Bor 96], Borcherds constructed a non-vanishing weight 4 modular form Φ on the moduli space of marked, polarized Enriques surface of degree 2 by considering the twisted denominator function of the fake monster Lie algebra associated to an automorphism of order 2 of the Leech lattice fixing an 8-dimensional subspace. In [JT 94] and [JT 96], we defined and studied a meromorphic (multi-valued) modular form of weight 2, which we call the K3 analytic discriminant, on the moduli space of marked, polarized, K3 surfaces of degree 2d; in certain cases, including when , where p k are distinct primes, our meromorphic form is actually a holomorphic form. Our construction involves a determinant of the Laplacian on a polarized K3 surface with respect to the Calabi-Yau metric together with the L 2 norm of the image of the period map with respect to a properly scaled holomorphic two form. Since the universal cover of any Enriques surface is a K3 surface, we can restrict the K3 analytic discriminant to the moduli space of degree 2 Enriques surfaces. The main result of this paper is the observation that the square of our degree 2 analytic discriminant, viewed as a function on the moduli space of degree 2 Enriques surfaces, is equal to the Borcherd's Φ function, up to a universal multiplicative constant. This result generalizes known results in the study of generalized Kac-Moody algebras and elliptic curves, and suggests further connections with higher dimensional Calabi-Yau varieties, specifically those which can be realized as complete intersections in some, possibly weighted, projective space.

This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with Jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction.

Let $L$ be a line bundle on a K3 or Enriques surface. We give a vanishing theorem for $H^1(L)$ that, unlike most vanishing theorems, gives necessary and sufficient geometrical conditions for the vanishing. This result is essential in our study of Brill-Noether theory of curves on Enriques surfaces (2006) and of Enriques-Fano threefolds (2006 preprint).

We construct, on a supersingular K3 surface with Artin invariant 1 in characteristic 2, a set of 21 disjoint smooth rational curves and another set of 21 disjoint smooth rational curves such that each curve in one set intersects exactly 5 curves from the other set with multiplicity 1 by using the structure of a generalized Kummer surface. As a corollary we have a concrete construction of a K3 surface with 21 rational double points of type A1 in characteristic 2.

Given din {mathbb {N}}, we prove that any polarized Enriques surface (over any field k of characteristic p ne 2 or with a smooth K3 cover) of degree greater than 12d^2 contains at most 12 rational curves of degree at most d. For d>2, we construct examples of Enriques surfaces of high degree that contain exactly 12 rational degree-d curves.

We show that supersingular K3 surfaces in characteristic p ≥ 5a re related by purely inseparable isogenies. This implies that they are unirational, which proves conjectures of Artin, Rudakov, Shafarevich, and Shioda. As a byproduct, we exhibit the moduli space of rigidified K3 crystals as an iterated P 1 -bundle over F p2. To complete the picture, we also establish Shioda-Inose type isogeny theorems for K3 surfaces with Picard rank ρ ≥ 19 in positive characteristic. Mathematics Subject Classification 14J28 · 14G17 · 14M20 · 14D22

We determine the Gromov–Witten invariants of the local Enriques surfaces for all genera and curve classes and prove the Klemm–Mariño formula. In particular, we show that the generating series of genus 1 invariants of the Enriques surface is the Fourier expansion of a certain power of Borcherds automorphic form on the moduli space of Enriques surfaces. We also determine all Vafa–Witten invariants of the Enriques surface. The proof uses the correspondence between Gromov–Witten theory and Pandharipande–Thomas theory. On the Gromov–Witten side, we prove the relative Gromov–Witten potentials of elliptic Enriques surfaces are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. On the sheaf side, we relate the Pandharipande–Thomas invariants of the Enriques–Calabi–Yau threefold in fiber classes to the 2-dimensional Donaldson–Thomas invariants by a version of Toda’s formula for local K3 surfaces. Altogether, we obtain sufficient modular constraints to determine all invariants from basic geometric computations.

We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general n-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations.

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth. In particular, he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a K3 surface to a Hirzebrunch surface such that the branch divisor is that effective divisor. Furthermore, he will decide the Galois group and give the way to construct that Abelian cover from an effective divisor on a Hirzebruch surface. Subsequently, he studies the same theme for Enriques surfaces.