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A fine moduli space is constructed, for cyclic-by-$mathsf{p}$ covers of an affine curve over an algebraically closed field $k$ of characteristic $mathsf{p}>0$. An intersection of finitely many fine moduli spaces for cyclic-by-$mathsf{p}$ covers of affine curves gives a moduli space for $mathsf{p}$-by-$mathsf{p}$ covers of an affine curve. A local moduli space is also constructed, for cyclic-by-$mathsf{p}$ covers of $Spec(k((x)))$, which is the same as the global moduli space for cyclic-by-$mathsf{p}$ covers of $mathbb{P}^1-{0}$ tamely ramified over $infty$ with the same Galois group. Then it is shown that a restriction morphism is finite with degrees on connected components $mathsf{p}$ powers: There are finitely many deleted points of an affine curve from its smooth completion. A cyclic-by-$mathsf{p}$ cover of an affine curve gives a product of local covers with the same Galois group of the punctured infinitesimal neighbourhoods of the deleted points. So there is a restriction morphism from the global moduli space to a product of local moduli spaces.
It is well known that the Prym variety of an etale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny is an isomorp
We compute the number of moduli of all irreducible components of the moduli space of smooth curves on Enriques surfaces. In most cases, the moduli maps to the moduli space of Prym curves are generically injective or dominant. Exceptional behaviour is
This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We improve the Arakelov inequality for the direct ima
This article accompanies my lecture at the 2015 AMS summer institute in algebraic geometry in Salt Lake City. I survey the recent advances in the study of tautological classes on the moduli spaces of curves. After discussing the Faber-Zagier relation
We introduce the moduli space of hybrid curves as the hybrid compactification of the moduli space of curves thereby refining the one obtained by Deligne and Mumford. As the main theorem of this paper we then show that the universal family of canonica