On automorphisms of Enriques surfaces

An Enriques surface over an algebraically closed field k of characteristic 4:2 is a non-singular projective surface S with H ~ (S, d~s)= H2(S, Cs)= 0 and 2Ks = 0. The unramitied double cover defined by Ks is a K3 surface R, a non-singular projective surface with Hi(R, d~R)=0 , KR =0. Illusie has shown, [I], that the group of divisors modulo numerical equivalence is isomorphic to the Enriques lattice U ~ E 8 ( 1) where U and E s ( 1) denote, respectively, the unique even unimodular lattices of index of inertia (1,1) and (0, 8). The purpose of this note is to use this isomorphism to study the Picard group of S. We prove the existence of certain configurations of irreducible curves of arithmetic genus 0 or 1 and deduce from them the existence of certain projective models for S and R. For example, we prove the following results:

Let S be an Enriques surface over an algebraically closed field k of characteristic ^2. Then, equivalently, S is a non-singular projective surface with q(S)=pg(S) = Q and 2KS^Q. It is known (cf. Cossec [Co]) that every Enriques surface admits a morphism of degree one onto a surface of degree 10 in P with isolated rational double points, and also that every Enriques surface is birationally equivalent to a (non-normal) sextic surface in P. Then there arises the following problem:

International Journal of MathematicsVol. 10, No. 05, pp. 619-642 (1999) No AccessOPEN RATIONAL SURFACES WITH LOGARITHMIC KODAIRA DIMENSION ZEROHIDEO KOJIMAHIDEO KOJIMADepartment of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, JapanPartially supported by JSPS Research Fellowships for Young Scientists and Grant-in-Aid for Scientific Research, the Ministry of Education, Science and Culture.https://doi.org/10.1142/S0129167X99000240Cited by:4 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetailsCited By 4Eilenberg–MacLane spaces in algebraic surface theoryR. V. Gurjar, S. R. Gurjar and B. Hajra18 January 2023 | Geometriae Dedicata, Vol. 217, No. 2Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial unitsTomasz Pełka and Paweł Raźny31 July 2021 | Pacific Journal of Mathematics, Vol. 311, No. 2Correction to “Affine Lines on % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX % garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz % aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeWaeaaakeaatu % uDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0HgiuD3BaGqbbiab-Prirbaa % !4246! $ \mathbb{Q} $ -homology Planes with Logarithmic Kodaira Dimension −∞”Takashi Kishimoto and Hideo Kojima2 April 2008 | Transformation Groups, Vol. 13, No. 1Structure of affine surfaces −B with ⩽1Hideo Kojima1 Jul 2002 | Journal of Algebra, Vol. 253, No. 1 Recommended Vol. 10, No. 05 Metrics History Received 4 September 1998 PDF download

By the fundamental result of I. I. Piatetsky-Shapiro and I. R. Shafarevich (1971), the automorphism group Aut(X) of aK3 surfaceX over ℂ and its action on the Picard latticeS X are prescribed by the Picard latticeS X . We use this result and our method (1980) to show the finiteness of the set of Picard latticesS X of rank ≥ 3 such that the automorphism group Aut(X) of theK3 surfaceX has a nontrivial invariant sublatticeS 0 inS X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic latticesS 0, this has been proved by the author before (1980, 1995). Thus we extend these results to negative sublatticesS 0. We give several examples of Picard latticesS X with parabolic and negativeS 0. We also formulate the corresponding finiteness result for reflective hyperbolic lattices of hyperbolic type over purely real algebraic number fields. We give many examples of reflective hyperbolic lattices of the hyperbolic type. These results are important for the theory of Lorentzian Kac-Moody algebras and mirror symmetry.

It is shown that the fixed part of the canonical linear system of a fibre in a relatively minimal fibred surface supports at most exceptional sets of weakly elliptic singular- ities. Introduction. Let S be a non-singular projective surface and f : S → C a surjective morphism of S onto a non-singular projective curve C with connected fibres. We call f a relatively minimal fibration of genus g if a general fibre is a non-singular projective curve of genus g and there are no (−1)-curves contained in fibres. We assume that g ≥ 2 throughout the paper. Let F be afi bre off . Then the intersection form is negative semi-definite on Supp(F ) by Zariski's lemma. Furthermore, there exist a positive integer m and a 1-connected curve D such that F = mD .W henm is strictly greater than one, F is called a multiple fibre of multiplicity m and OD(D) is a torsion of order m. In (8), we considered the canonical linear system on the minimal resolution of a normal surface singularity and showed that the fixed part supports at most exceptional sets of rational singular points (cf. (1) and (2)). The present article is an extension of it to the semi-global case and we study the fixed part of the canonical linear system |KF | which we call the canonical fixed part in this paper. Recall that the canonical fixed part is closely related to the Horikawa index (see (3, p. 12)), an analytic invariant of a singular fibre germ. In fact, according to (6, Lemma 10 and Theorem 3), if g = 2, the canonical fixed part is a chain of (−2)-curves (of type A) and the Horikawa index is almost equivalent to the number of its irreducible components.

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.

Let F F be a (topologically) finitely generated free pro- p p -group, and β \beta an automorphism of F F . If p ≠ 2 p \ne 2 and the order of β \beta is 2, then there is some basis of F F such that β \beta either fixes or inverts its elements. If p p does not divide the order of β \beta , then the subgroup of F F of all elements fixed by β \beta is (topologically) infinitely generated; however this is not always the case if p p divides the order of β \beta .

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:

A central topic of interest in the theory of nonassociative algebras is the study of automorphisms and derivations of algebras. As is well known, many linear algebraic and Lie groups and their Lie algebras are obtained from the automorphism groups and derivation algebras of certain nonassociative algebras. If B is a finitedimensional algebra over a field F, then there is a natural relationship between the automorphism group Aut B and the derivation algebra Der B of B. The Lie algebra of Aut B, viewed as a linear algebraic group over F, is a subalgebra of Der B. If F is the field of real numbers or complex numbers, then Aut B is a Lie subgroup of the linear group GL(B) and Der B is the Lie algebra of Aut B [SW1].

Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

Let V be a nonsingular projective surface of Kodaira dimension K(V) > 0. Let D be a reduced, effective, nonzero divisor on V with only normal crossings. In the present article, a pair (F, D) is said to be a minimal logarithmic surface of general type, if, by definition, Kv + D is a numerically effective divisor of self intersection number (Kv + D) 2 > 0 and if Kv + D has positive intersection with every exceptional curve of the first kind on V. Here Kv is the canonical divisor of V. In the case, on the one hand, Sakai [8; Theorem 7.6] proved a Miyaoka — Yau type inequality (cf ) := (Kv + D) 2 — c2 -by making use of [8; Theorem 5.5]. In the present article, we shall prove that (cf ) > -c2 — 2 provided that the rational map &\Ky+D defined by the complete linear system \KV + D| has a surface as the image of V. Moreover, if the equality holds, then the logarithmic geometric genus pg := h°(V, Kv + D) = -(cf ) + 2 = 3, D is an elliptic curve and V is the canonical resolution in the sense of Horikawa associated with a double covering h: Y -> P. In addition, the branch locus B of h is a reduced curve of degree eight and the singular locus Sing B consists of points of multiplicity < 3 except for at most one simple quadruple point. Introduction This is a succession of the previous paper [9]. We work over an algebraically closed field fe of characteristic zero. Let V be a nonsingular projective surface defined over fe. If V is a minimal surface of general type, we have the following inequality due to M. Noether: This inequality, together with the Noether formula 12%((9V) = c±(V) 2 + c2(V Communicated by K. Saito, May 18, 1990. 1991 Mathematics Subject Classification: 14J29 Department of Mathematics, the National University of Singapore, Singapore.

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

All considered groups are torsion or do not contain infinitely generated subgroups. If such a groupGacts transitively on a connected algebraA, then all elements ofAhave the same stabilizer, so this stabilizer is a normal subgroup (it is also shown that these facts are not true for arbitrary groups). Hence the automorphism group Aut(A) is a homomorphic image ofG. In particular, if the action ofGis, in addition, faithful, thenGis isomorphic to Aut(A). By these results we first obtain that for each (not necessarily connected) algebraA, if all of its Aut(A)-orbits are connected, then Aut(A) is a subdirect product of automorphism groups of these orbits. Secondly, connected components of Aut(A)-orbits have automorphism extension property.Next, we show that ifAis a connected algebra such that Aut(A) acts transitively on it, then the group structure of Aut(A) may be transported onAsuch that the left multiplications are all automorphisms ofAand the right multiplications are all unary term operations ofA. Hence all the unary term operations ofAare bijective, so they generate a subgroup of the group of all bijections of the carrier ofA. It is shown that this group is anti-isomorphic to Aut(A). Thus the Birkhoff’s construction of an algebra with a given group of automorphisms is unique in some sense when restricted to groups we consider here.The Birkhoff’s construction can be slightly modified so as to obtain a smaller set of operations. In fact, it is enough to take the right multiplications by generators. Moreover, we show that this is the best possible lower bound for the number of unary operations in the case of groups considered here. If we admit non-unary operations, then for finite and countable groups we can reduce the number of operations to one binary operation. On the other hand, ifAis a connected algebra such that Aut(A) is torsion and acts onAtransitively, then each element generatesA. Hence ifAis such an algebra with an uncountable group Aut(A), then the cardinality of the set of operations ofAis greater or equal than the cardinality of Aut(A).

Affine surfaces X completed by an irreducible rational curve C are studied. The integer m = (C2) is an invariant of X. It is shown that the set of all such surfaces with fixed invariant m is described in terms of orbits of a group action on the space of tails; moreover, the automorphism group Aut(X) is expressed by the stabilizers of the action. Explicit formulas for generators of the group Aut(X) are given for m ≤ 5. In particular, it is shown that in zero characteristic the invariant m uniquely determines the surface X; in the general case this is not so.Bibliography: 11 titles.

Endomorphisms of undirected modifications of directed graphs