RELATIVE DIHEDRAL GROUP ACTIONS ON RATIONAL ELLIPTIC SURFACES

In this article we examine conditions for the appearance or nonappearance of the twoextra-special 2-groups of order 32 as Galois groups over a fieldFof characteristic not 2. The groups in question are the central productsDDof two dihedral groups of order 8, andDQof a dihedral group with the quaternion group, obtained by identifying the central elements of order 2 in each factor group. It is shown that the realizability of each of these groups as Galois groups overFimplies the realizability of other 2-groups (which are not their quotient groups), and in turn that realizability of certain other 2-groups implies the realizability ofDDandDQ. We conclude by providing an explicit construction of field extensions with Galois groupDD.

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.

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.

The classification of automorphism groups of rational elliptic surfaces with section

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.

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.

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL2(/3), an extension of SL2(/3) by /2, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.

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.

Blocked triangular patterns and their symmetry groups

Perfect Isometries for Principal Blocks with Abelian Defect Groups and Elementary Abelian 2-Inertial Quotients

On some 10-arcs for deriving the minimum order for complete arcs in small projective planes

A group $G$ is a CI-group with respect to graphs if two Cayley graphs of $G$ are isomorphic if and only if they are isomorphic by a group automorphism of $G$. We show that an infinite family of groups which include $D_n\times F_{3p}$ are not CI-groups with respect to graphs, where $p$ is prime, $n\not = 10$ is relatively prime to $3p$, $D_n$ is the dihedral group of order $n$, and $F_{3p}$ is the nonabelian group of order $3p$.