Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpecial families of curves, of Abelian varieties, and of certain minimal manifolds over curves
201
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 images of powers of the dualizing sheaf. For families of Abelian varieties we recall the characterization of Shimura curves by Arakelov equalities. For families of curves we recall the characterization of Teichmueller curves in terms of the existence of certain sub variation of Hodge structures. We sketch the proof that the moduli scheme of curves of genus g>1 can not contain compact Shimura curves, and that it only contains a non-compact Shimura curve for g=3.
rate research
Read More
Given an open subset U of a projective curve Y and a smooth family f:V-->U of curves, with semi-stable reduction over Y, we show that for a sub variation of Hodge structures of rank >2 the Arakelov inequality must be strict. For families of n-folds we prove a similar result under the assumption that the (n,0) component of the Higgs bundle defines fibrewise a birational map.
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.
Let $f:X to mathbb{P}^1$ be a non-isotrivial family of semi-stable curves of genus $ggeq 1$ defined over an algebraically closed field $k$ with $s_{nc}$ singular fibers whose Jacobians are non-compact. We prove that $s_{nc}geq 5$ if $k=mathbb C$ and $ggeq 5$; we also prove that $s_{nc}geq 4$ if ${rm char}~k>0$ and the relative Jacobian of $f$ is non-smooth.
Let $C$ be a nodal curve, and let $E$ be a union of semistable subcurves of $C$. We consider the problem of contracting the connected components of $E$ to singularities in a way that preserves the genus of $C$ and makes sense in families, so that this contraction may induce maps between moduli spaces of curves. In order to do this, we introduce the notion of mesa curve, a nodal curve $C$ with a logarithmic structure and a piecewise linear function $overline{lambda}$ on the tropicalization of $C$. This piecewise linear function determines a subcurve $E$. We then construct a contraction of $E$ inside of $C$ for families of mesa curves. Resulting singularities include the elliptic Gorenstein singularities.
These are notes of my lectures at the summer school Higher-dimensional geometry over finite fields in Goettingen, June--July 2007. We present a proof of Tates theorem on homomorphisms of abelian varieties over finite fields (including the $ell=p$ case) that is based on a quaternion trick. In fact, a a slightly stronger version of those theorems with finite coefficients is proven.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
