A Computational View on the Non-degeneracy Invariant for Enriques Surfaces

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 .

Nikulin [On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2 reflections, J. Soviet. Math. 22 (1983) 1401–1476; Surfaces of type [Formula: see text] with a finite automorphism group and a Picard group of rank three, Proc. Steklov Institute of Math. (3) (1985) 131–155] and Vinberg [Classification of 2-reflective hyperbolic lattices of rank 4, Trans. Moscow Math. Soc. (2007) 39–66] proved that there are only a finite number of lattices of rank [Formula: see text] that are the Néron–Severi lattice of projective [Formula: see text] surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such [Formula: see text] surfaces [Formula: see text], when these surfaces have moreover no elliptic fibrations. In that case, we show that such [Formula: see text] surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study the converse, i.e. if the geometric description we obtained characterizes these surfaces. In four cases, the description is sufficient, in each of the four other cases, there is exactly another one possibility which we study. We obtain that at least five moduli spaces of [Formula: see text] surfaces (among the eight we study) are unirational.

We classify Enriques surfaces with smooth K3 cover and finite automorphism group in arbitrary positive characteristic. The classification is the same as over the complex numbers except that some types are missing in small characteristics. Moreover, we give a complete description of the moduli of these surfaces. Finally, we realize all types of Enriques surfaces with finite automorphism group over the prime fields $\mathbb{F}_p$ and $\mathbb{Q}$ whenever they exist.

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:

Complex Enriques surfaces with a finite group of automorphisms are classified into seven types. In this paper, we determine which types of such Enriques surfaces exist in characteristic 2. In particular we give a 1-dimensional family of classical and supersingular Enriques surfaces with the automorphism group A u t ( X ) \mathrm {Aut}(X) isomorphic to the symmetric group S 5 \mathfrak {S}_5 of degree 5.

We classify supersingular and classical Enriques surfaces with finite automorphism group in characteristic 2 into 8 types according to their dual graphs of all $(-2)$-curves (nonsigular rational curves). We give examples of these Enriques surfaces together with their canonical coverings. It follows that the classification of all Enriques surfaces with finite automorphism group in any characteristics has been finished.

Torsion-free groups having finite automorphism groups. I

This paper consists mainly of a review and applications of our old results relating to the title. We discuss how many elliptic fibrations and elliptic fibrations with infinite automorphism groups (or Mordell–Weil groups) an algebraic K3 surface over an algebraically closed field can have. As examples of applications of the same ideas, we also consider K3 surfaces with exotic structures: with a finite number of non-singular rational curves, with a finite number of Enriques involutions, and with naturally arithmetic automorphism groups.

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 elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic 3, which is supersingular. The specialization induces an embedding of the Neron–Severi lattices. Using this embedding, we determine the automorphism group of this K3 surface over a discrete valuation ring of mixed characteristic whose residue field is of characteristic 3. The elliptic modular surface of level 4 has a fixed-point-free involution that gives rise to the Enriques surface of type IV in Nikulin–Kondo–Martin’s classification of Enriques surfaces with finite automorphism group. We investigate the specialization of this involution to characteristic 3.

We classify all complex surfaces with quotient singularities that do not\ncontain any smooth rational curves, under the assumption that the canonical\ndivisor of the surface is not pseudo-effective. As a corollary we show that if\n$X$ is a log del Pezzo surface such that for every closed point $p\\in X$, there\nis a smooth curve (locally analytically) passing through $p$, then $X$ contains\nat least one smooth rational curve.\n

K3 Surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers

in this note we consider smooth rational curves C of degree n in threedimensional projective space IP 3 (over a closed field of characteristic 0). To avoid trivial exceptions we shall always assume that n ~ 4 (this does not hold however for certain auxiliary curves we shall consider). Let N = N c be the normal bundle of C in IP 3. Since degel(IP3)=4, and d e g c l ( l P 0 = 2 , we have that d e g c l ( N ) = 4 n 2 . By a well-known theorem of Grothendieck the bundle N is a direct sum of two line bundles. Hence N ~ O c ( 2 n l a ) G O c ( 2 n 1 +a) for some non-negative a=a(C), which is uniquely determined by C. The question we would like to answer is an obvious one: which values of a occur? We shall show (Theorem 4 below) that a value a occurs if and only if 0_ =0, therefore Hi(C, N)=O. It follows [K, p. 150] that C represents a smooth point on the Chow variety Ch(3, 1, n) of effective cycles of dimension 1 and degree n in IP 3. Since the set of all smooth rational curves with a fixed degree is obviously connected, we see that the smooth C's represent a smooth, irreducible, 4n-dimensional (Zariski-)open subset S of Ch(3, 1, n). In a for thcoming paper [ E V ] we shall prove the following

Large finite groups have large automorphism groups [4]; infinite groups may, like the infinite cyclic group, have finite automorphism groups, but their endomorphism semigroups are infinite (see Baer [1, p. 530] or [2, p. 68]). We show in this paper that the corresponding propositions for semigroups are false.