In this paper, we use the perspective of linear series, and in particular results following from the degeneration tools of limit linear series, to give a number of new results on existence and non-existence of tamely branched covers of the projective line in positive characteristic. Our results are both in terms of ramification indices and the sharper invariant of monodromy cycles, and the firs...

We study branched covering spaces in several contexts, proving that under suitable circumstances the cover satisfies the same upper curvature bounds as the base space. The first context is of a branched cover of an arbitrary metric space that satisfies Alexandrov's curvature condition CAT(κ), over an arbitrary complete convex subset. The second context is of a certain sort of branched cover of ...

For every fixed $h \geq 1$, we construct an infinite family of simply connected symplectic 4-manifolds $X'_{g,h}[i]$, for all $g > h$ and $0 \leq i < 2p - where $p = \lfloor \frac{g+1}{h+1} \rfloor$. Each manifold $X'_{g,h}[i]$ is the total space a genus $g$ Lefschetz pencil constructed by explicit monodromy factorization. We then show that each diffeomorphic to complex surface fiber sum formed...

To a branched cover f:Σ ˜→Σ between closed surfaces one can associate combinatorial datum given by the topological types of Σ ˜ and Σ, degree d f, number n branching points partitions local degrees f at preimages points. This must satisfy Riemann-Hurwitz condition plus some extra ones if either or both are non-orientable. A very old question posed Hurwitz [14] in 1891 asks whether satisfying th...

A number of recent results are presented which bear on the question of what geometric information can be gleaned from the representation of a three-manifold as a branched cover over a fixed universal link. Results about Seifert-fibered manifolds, graph manifolds and hyperbolic manifolds are discussed. Section 0 Introduction. Closed, orientable three-manifolds admit a variety of universal constr...

In this paper we study the reduction of Galois covers of curves, from characteristic zero to positive characteristic. The starting point is a recent result of Raynaud, which gives a criterion for good reduction for covers of the projective line branched at three points. We use the ideas of Raynaud to study the case of covers of the projective line branched at four points. Under some condition o...