where Πg(C) is the geometric (profinite) fundamental group of C×Spec(Ks) (i.e. Πg(C) is equal to the Galois group of the maximal unramified extension of F (C)⊗Ks). This sequence induces a homomorphism ρC from GK to Out(Πg(C)) which is the group of automorphisms modulo inner automorphisms of Πg(C). It is well known that ρC is an important tool for studying C. For instance, it determines C up to ...

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Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a cross-section is adde...

Let AutKK(x) be the Galois group of the transcendental degree one pure field extension K ⊆ K(x). In this paper we describe polynomial time algorithms for computing the field Fix(H) fixed by a subgroup H ⊆ AutKK(x) and for computing the fixing group Gf of a rational function f ∈ K(x).

The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame method in differential geometry. The generating set of rational invariants appears as the coefficients of a Gröbner basis, reduction with respect to which allo...

Let G be a finite group. We prove that every rational G-connected Hopf G-space with two nontrivial homotopy group systems is G-homotopy equivalent to an infinite loop G-space.

Relations between the global structure of the gauge group in elliptic F-theory compactifications, fractional null string junctions, and the Mordell-Weil lattice of rational sections are discussed. We extend results in the literature, which pertain primarily to rational elliptic surfaces and obtain π(G̃) where G̃ is the semi-simple part of the gauge group. We show how to obtain the full global str...