Occurrences of a factor w in an infinite uniformly recurrent sequence u can be encoded by over finite alphabet. This is usually denoted du(w) and called the derived to u. If prefix fixed point primitive substitution φ, then Durand's result from 1998, ψ as well. For non-prefix w, only exceptionally. To study this phenomenon we introduce new notion: A set M substitutions said closed under derivat...

We construct Belyi maps having specified behavior at finitely many points. Specifically, for any curve C defined over Q, and any disjoint finite subsets S, T ⊂ C(Q), we construct a finite morphism φ : C → P such that φ ramifies at each point in S, the branch locus of φ is {0, 1,∞}, and φ(T ) ∩ {0, 1,∞} = ∅. This refines a result of Mochizuki’s. We also prove an analogous result over fields of p...

We examine the different ways of writing a cover of curves φ : C → D over a field K as a composition φ = φn ◦ φn−1 ◦ · · · ◦ φ1, where each φi is a cover of curves over K of degree at least 2 which cannot be written as the composition of two lower-degree covers. We show that if the monodromy group Mon(φ) has a transitive abelian subgroup then the sequence (deg φi)1≤i≤n is uniquely determined up...

Let χ be a linear morphism of the product of two projective spaces PG(n, F ) and PG(m,F ) into a projective space. Let γ be the Segre embedding of such a product. In this paper we give some sufficient conditions for the existence of an automorphism α of the product space and a linear morphism of projective spaces φ, such that γφ = αχ. A.M.S. classification number: 51M35.

In this note we study the fibers of a rational map from an algebraic point of view. We begin by describing two interpretations of the word ‘fiber’. Let S = k[x0, . . . , xn] be a polynomial ring over an infinite field k, I ⊂ S an ideal generated by an r+1-dimensional vector space W of forms of the same degree, and φ the associated rational map P → P = P(W ). We will use this notation throughout...

It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlap-free words was studied e. g. by Fife [8] who described all binary overlap-free infinite words and Séébold [13] who proved that the Thue-Morse word is essentially the only binary overlap-free word which is a fixed point of a morph...

We prove the following. Let R be a Noetherian commutative ring, B a finitely generated R-algebra, and A a pure R-subalgebra of B. Then A is finitely generated over R. In this paper, all rings are commutative. Let A be a ring and B an A-algebra. We say that A → B is pure, or A is a pure subring of B, if for any A-module M , the map M = M ⊗A A → M ⊗A B is injective. Considering the case M = A/I, ...

We obtain a quantitative version of the classical Chevalley-Weil theorem for curves. Let φ : C̃ → C be an unramified morphism of non-singular plane projective curves defined over a number field K. We calculate an effective upper bound for the norm of the relative discriminant of the number field K(Q) over K for any point P ∈ C(K) and Q ∈ φ(P ). 2000 MCS: 14G25, 14H25, 11G30.

Let φ : X → Y be a generically finite morphism. The purpose of this paper is to show how the OY -algebra structure on φ∗OX controls algebro-geometric aspects of X like the ring generation of graded rings associated to X and the very ampleness of line bundles on X . As the main application of this we prove some new results for certain regular surfaces X of general type. Precisely, we find the de...

We generalize the fundamental theorem for Burnside rings to the mark morphism of plus constructions defined by Boltje. The main observation is the following: If D is a restriction functor for a finite group G, then the mark morphism φ : D+ → D is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for G) after composing with a suitable isomorphism of D. As a conse...