The Hurwitz scheme was originally conceived as a parameter space for simply branched covers of the projective line. A variant of this is a parameter spaec for simply branched covers of the projective line, up to automorphisms of P the so called unparametrized Hurwitz scheme. Other variants involve fixing branching types which are not necessarily simple (see [D-D-H]). A rigorous algebraic defini...

A permutation rational function f ∈ Fq(x) is a rational function that induces a bijection on Fq, that is, for all y ∈ Fq there exists exactly one x ∈ Fq such that f(x) = y. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show ho...

In this paper, we compute the number of covers of the projective line with given ramification in two related families of examples. We are able to conclude that for a particular choice of degeneration, all covers in these families degenerate to separable (admissible) covers. Starting from a good understanding of the complex case, the proof is centered on the theory of stable reduction of Galois ...

let $m$ be a right module over a ring $r$, $tau_m$ a preradical on $sigma[m]$, and$ninsigma[m]$. in this note we show that if $n_1, n_2in sigma[m]$ are two$tau_m$-lifting modules such that $n_i$ is $n_j$-projective ($i,j=1,2$), then $n=n_1oplusn_2$ is $tau_m$-lifting. we investigate when homomorphic image of a $tau_m$-lifting moduleis $tau_m$-lifting.