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Register a new userOn two extensions of Poncelet theorem
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Ciro Ciliberto
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In this note I provide two extensions of a particular case of the classical Poncelet theorem.
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A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in ${rm U(1)}$. By using the theory of indigenous bundles, we construct on a compact Riemann surface $X$ of genus $g_X geq 1$ a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in $2 pi mathbb{Z}_{>1}$, which is generically injective in the algebro-geometric sense as $g_X geq 2$. As an application, we prove the following two results about irreducible metrics: $bullet$ as $g_X geq 2$ and $d$ is even and greater than $12g_X - 7$, the effective divisors of degree $d$ which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension $geq 2(d+3-3g_X)$ in ${rm Sym}^d(X)$; $bullet$ as $g_X geq 1$, for almost every effective divisor $D$ of degree odd and greater than $2g_X-2$ on $X$, there exist finitely many cone spherical metrics representing $D$.
The main purpose of this paper is to make Nakayamas theorem more accessible. We give a proof of Nakayamas theorem based on the negative definiteness of intersection matrices of exceptional curves. In this paper, we treat Nakayamas theorem on algebraic varieties over any algebraically closed field of arbitrary characteristic although Nakayamas original statement is formulated for complex analytic spaces.
We give a new proof of Bradens theorem ([Br]) about emph{hyperbolic restrictions} of constructible sheaves/D-modules. The main geometric ingredient in the proof is a 1-parameter family that degenerates a given scheme Z equipped with a G_m-action to the product of the attractor and repeller loci.
We prove the following two results 1. For a proper holomorphic function $ f : X to D$ of a complex manifold $X$ on a disc such that ${df = 0 } subset f^{-1}(0)$, we construct, in a functorial way, for each integer $p$, a geometric (a,b)-module $E^p$ associated to the (filtered) Gauss-Manin connexion of $f$. This first theorem is an existence/finiteness result which shows that geometric (a,b)-modules may be used in global situations. 2. For any regular (a,b)-module $E$ we give an integer $N(E)$, explicitely given from simple invariants of $E$, such that the isomorphism class of $Ebig/b^{N(E)}.E$ determines the isomorphism class of $E$. This second result allows to cut asymptotic expansions (in powers of $b$) of elements of $E$ without loosing any information.
Let G be the Tate module of a p-divisble group H over a perfect field k of characteristic p. A theorem of Scholze-Weinstein describes G (and therefore H itself) in terms of the Dieudonne module of H; more precisely, it describes G(C) for good semiperfect k-algebras C (which is enough to reconstruct G). In these notes we give a self-contained proof of this theorem and explain the relation with the classical descriptions of the Dieudonne functor from Dieudonne modules to p-divisible groups.
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