The classification of automorphism groups of rational elliptic surfaces with section

It is well-known that a rational elliptic surface with global sections is a nine-pointsblowing-up of p which may be infinitely near and can be obtained from the pencil of cubic curves in P. This object has been studied by Demauzuer-Pinkham [1], Looijenga [4] and Naruki [5] [6]. In this paper, we shall study the structure of rational elliptic surfaces with multiple fibers. A rational elliptic surface with multiple fibers of multiplicity m can be obtained from a rational elliptic surface with global sections by performing logarithmic transformations at one point on the base curve P. It can also be blown down to P. In this case, if the nine points pl5 p2, ..., Pg on P 2 are mutually distinct and the multiple fiber is of type m/0, it is obtained from the pencil generated by m-fold cubic which passes through Pj's and a curve of degree 3m which has an ordinary singularity of multiplicity m at each Pt(l <i<9) and is non-singular outside them. The method of this paper is to study birational geometry of curves in P through the logarithmic transformations. Our main result is as follows.

We constructed several families of elliptic K3 surfaces with Mordell-Weil groups of ranks from 1 to 4. We studied F-theory compactifications on these elliptic K3 surfaces times a K3 surface.Gluing pairs of identical rational elliptic surfaces with nonzero Mordell-Weil ranks yields elliptic K3 surfaces, the Mordell-Weil groups of which have nonzero ranks. The sum of the ranks of the singularity type and the Mordell-Weil group of any rational elliptic surface with a global section is 8. By utilizing this property, families of rational elliptic surfaces with various nonzero Mordell-Weil ranks can be obtained by choosing appropriate singularity types. Gluing pairs of these rational elliptic surfaces yields families of elliptic K3 surfaces with various nonzero Mordell-Weil ranks.We also determined the global structures of the gauge groups that arise in F-theory compactifications on the resulting K3 surfaces times a K3 surface. U(1) gauge fields arise in these compactifications.

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

Our main result, Theorem (0.1), classifies multiple fibers on rational elliptic surfaces over algebraically closed fields of arbitrary characteristic. One result of this is the existence in positive characteristics of tame multiple fibers of additive type for several of the Kodaira fiber-types for which no examples were previously known.

Twisted first homology groups of the automorphism group of a free group

Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index at most 2 or rational elliptic surfaces, and construct the only family of each of the three extremal cases where X^g contains 10 (maximum possible) curves. We also classify rational log Enriques surfaces of index 2. Our approach is more geometrical rather than lattice-theoretical (see Nikulin's paper for the latter approach).

We present some rational elliptic surfaces over a certain field such that certain open subschemes do not satisfy the surjectivity of the boundary map on K 2 arising from the localization sequence. We consider two cases that the base field is transcendental over its prime subfield and that the base field is characteristic zero.

Simple graphs containing induced subgraphs whose automorphism groups are isomorphic to subgroups of a given finite group

Color algebras and affine connections on S6

We consider twisted conjugacy classes and the R ∞-property for classical linear groups. In particular, it is stated that the general linear group GL n (K) and the special linear group SL n (K), where $$ n \geqslant 3 $$ , possess the R ∞-property if either K is an infinite integral domain with trivial automorphism group or K is an integral domain containing a subring of integers, whose automorphism group Aut(K) is finite. By an integral domain we mean a commutative ring with identity which has no zero divisors.

For a finitely generated group G, we study deformation spaces of metric G-trees, which are analogues of the Teichmuller spaces of surfaces for group actions on trees. Deformation spaces of metric G-trees generalize Culler-Vogtmann’s Outer space, the deformation space of free actions on trees, which has proven to be immensely useful in the study of Out(Fn), the outer automorphism group of the free group of rank n ≥ 2. Let D be a deformation space of metric G-trees. The group of positive real numbers R>0 acts on D by scaling the metrics on the trees and we define the projectivized deformation space as the quotient PD = D/R>0. The outer auto- morphism group Out(G) contains a certain subgroup OutD(G) that acts on D and PD by precomposing the G-actions on the trees. In Chapter 1, we present a complete argument that under certain assumptions the projectivized deformation space PD is a model for the classifying space of OutD(G) for a family of subgroups. In Chapter 2, we introduce an asymmetric pseudometric on PD that generalizes the asymmetric Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmuller space. Making use of the Lipschitz metric on PD, we prove existence of train track representatives for irreducible automorphisms of virtually free groups and nonelementary generalized Baumslag-Solitar groups that contain no solvable Baumslag-Solitar group BS(1, n) with n ≥ 2. In Chapter 3, we define the higher holomorphs Aut(G, k), k ∈ N, which are “higher-pointed” variants of the automorphism group Aut(G). Following the construction of the spine of Outer space, we construct a family of simplicial complexes S(PD, k), k ∈ N on which certain subgroups AutD(G, k) ≤ Aut(G, k) act and we show that these complexes are always contractible.

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibrations on the K3 surfaces explicitly and to study their specializations.

Admissible subcategories are building blocks of semiorthogonal decompositions. Many examples of them are known, but few general properties have been proved, even for admissible subcategories in the derived categories of coherent sheaves on basic varieties such as projective spaces. We use a relation between admissible subcategories and anticanonical divisors to study admissible subcategories of del Pezzo surfaces. We show that any admissible subcategory of the projective plane has a full exceptional collection, and since all exceptional objects and collections for the projective plane are known, this provides a classification result for admissible subcategories. We also show that del Pezzo surfaces of degree at least three do not contain so-called phantom subcategories. These are the first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms.

We give a short proof of the Torelli theorem for $ALH^*$ gravitational instantons using the authors’ previous construction of mirror special Lagrangian fibrations in del Pezzo surfaces and rational elliptic surfaces together with recent work of Sun-Zhang. In particular, this includes an identification of 10 diffeomorphism types of $ALH^*_b$ gravitational instantons.

In this paper we give an explicit method for constructing every rational elliptic surface that has a four-torsion section. We use this construction to study the moduli space of birational equivalence classes of Galois covers that arise from rational elliptic surfaces with a relative dihedral group action. As a conclusion we find that the moduli space for such Galois covers whose Galois groups are the dihedral group of order eight is a nodal rational curve.