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Factorization of point configurations, cyclic covers, and conformal blocks

Michele Bolognesi, Noah Giansiracusa (2015)

Journal of the European Mathematical Society

We describe a relation between the invariants of n ordered points in projective d -space and of points contained in a union of two linear subspaces. This yields an attaching map for GIT quotients parameterizing point configurations in these spaces, and we show that it respects the Segre product of the natural GIT polarizations. Associated to a configuration supported on a rational normal curve is a cyclic cover, and we show that if the branch points are weighted by the GIT linearization and the rational...

Families of curves and alterations

A. Johan De Jong (1997)

Annales de l'institut Fourier

In this article it is shown that any family of curves can be altered into a semi-stable family. This implies that if S is an excellent scheme of dimension at most 2 and X is a separated integral scheme of finite type over S , then X can be altered into a regular scheme. This result is stronger then the results of [ Smoothness, semi-stability and alterations to appear in Publ. Math. IHES]. In addition we deal with situations where a finite group acts.

Families of elliptic curves with genus 2 covers of degree 2.

Claus Diem (2006)

Collectanea Mathematica

We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism).A classical theorem says that a genus 2 cover of an elliptic curve of degree 2 over a field of characteristic ≠ 2 is birational to a product of two elliptic curves over the projective...

Fragmented deformations of primitive multiple curves

Jean-Marc Drézet (2013)

Open Mathematics

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that Y red is smooth. We study the deformations of Y to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity n of Y). We are particularly interested in deformations to n disjoint smooth irreducible components, which are called fragmented deformations. We describe them completely. We give also a characterization...

Galois Covers and the Hilbert-Grunwald Property

Pierre Dèbes, Nour Ghazi (2012)

Annales de l’institut Fourier

Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a p -adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over ℚ . The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.

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