Nodal rational curves on Enriques surfaces of base change type

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.

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:

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.

In this note we study the local negativity for certain configurations of smooth rational curves in smooth surfaces with numerically trivial canonical class. We show that for such rational curves there is a bound for the so-called local Harbourne constants, which measure the local negativity phenomenon. Moreover, we provide explicit examples of interesting configurations of rational curves in some K3 and Enriques surfaces and compute their local Harbourne constants.

We consider normal compact surfaces $Y$ obtained from a minimal class VII surface $X$ by contraction of a cycle $C$ of $r$ rational curves with $C^2<0$. Our main result states that, if the obtained cusp is smoothable, then $Y$ is globally smoothable. The proof is based on a vanishing theorem for $H^2(\Theta_Y)$. If $r<b_2(X)$ any smooth small deformation of $Y$ is rational, and if $r=b_2(X)$ (i.e. when $X$ is a half-Inoue surface) any smooth small deformation of $Y$ is an Enriques surface. The condition "the cusp is smoothable" in our main theorem can be checked in terms of the intersection numbers of the cycle, using the Looijenga conjecture (which has recently become a theorem). Therefore this is a "decidable" condition. We prove that this condition is always satisfied if $r<b_2(X)\leq 11$. Therefore the singular surface $Y$ obtained by contracting a cycle $C$ of $r$ rational curves in a minimal class VII surface $X$ with $r<b_2(X)\leq 11$ is always smoothable by rational surfaces. The statement holds even for unknown class VII surfaces.

In this chapter we consider in detail the class of K 3-surfaces and that of Enriques surfaces. We start with some notation and after that we state the main results in Sect. 2. In Chapt. IV, Sect. 3 we saw that K 3- surfaces are Kahler, a fact we use from the start. The main tool for studying moduli of K 3- surfaces is the period map and we describe these moduli spaces in terms of the corresponding period domains. This is done in Sect. 6–14 after we have proved some general facts concerning the geometry of divisors on K 3-surfaces and Kummer surfaces, collected in Sect. 3–5. The geometry of Enriques surfaces as discussed in Sect. 15–18 is then coupled with a study of the period map of their universal covers in order to arrive at a description of the moduli space in terms of certain classical bounded domains. See Sect. 19–21. We finish this chapter with more recent results for projective K 3-surfaces. First, we consider their moduli spaces. After this we discuss the construction of mirror families for K 3-surfaces. Next we present Mumford’s proof that every K 3-surface contains a (possibly singular) rational curve and a 1-dimensional algebraic family of (in general singular) elliptic curves. Then we discuss enumerative results for rational curves and we finish with an application to hyperbolic geometry (related to the Green-Griffiths and Lang conjectures).

Rational Curves on Enriques Surfaces, But Only Few

We prove that classes of rational curves on very general Enriques surfaces are always 2 2 -divisible. As a consequence, we compute the Seshadri constant of any big and nef line bundle on a very general Enriques surface, proving that it coincides with the value of the ϕ \phi -function introduced by Cossec [Math. Ann. 271 (1985), pp. 577–600].

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.

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:

For an Enriques surface S, the non-degeneracy invariant nd ( S ) retains information on the elliptic fibrations of S and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on S together with a configuration of smooth rational curves, and gives a lower bound for nd ( S ) . We provide a SageMath code that computes this combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying nd ( S ) = 10 which are not general and with infinite automorphism group. We obtain lower bounds on nd ( S ) for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes–Pardini. Finally, we recover Dolgachev and Kondō’s computation of the non-degeneracy invariant of the Enriques surfaces with finite automorphism group and provide additional information on the geometry of their elliptic fibrations.

We give a 1-dimensional family of classical and supersingular Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1. Moreover we show that there exist 30 nonsingular rational curves and ten non-effective (-2)-divisors on these Enriques surfaces whose reflection group is of finite index in the orthogonal group of the Neron-Severi lattice modulo torsion.

For an Enriques surface $S$, the non-degeneracy invariant $\mathrm{nd}(S)$ retains information on the elliptic fibrations of $S$ and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on $S$ together with a configuration of smooth rational curves, and gives a lower bound for $\mathrm{nd}(S)$. We provide a SageMath code that computes such combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying $\mathrm{nd}(S)=10$ which are not general and with infinite automorphism group. We obtain lower bounds on $\mathrm{nd}(S)$ for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes-Pardini. Finally, we recover Dolgachev and Kond\=o's computation of the non-degeneracy invariant of the Enriques surfaces with finite automorphism group and provide additional information on the geometry of their elliptic fibrations.

Brandhorst and Shimada described a large class of Enriques surfaces, called (\tau,\bar{\tau}) -generic, for which they gave generators for the automorphism groups and calculated the elliptic fibrations and the smooth rational curves up to automorphisms. In the present paper, we give lower bounds for the non-degeneracy invariant of such Enriques surfaces, we show that in most cases the invariant has generic value 10 , and we present the first known example of complex Enriques surface with infinite automorphism group and non-degeneracy invariant not equal to 10 .

We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADE-singularities and trivial canonical bundle whose Q_l-cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces).Comment: 24 pages; v3: journal version, correcting 20 root types to 19 and ruling out the remaining type 4A_2+A_1 (in new section 11)