Archive ouverte UNIGE | last documentshttps://archive-ouverte.unige.ch/Latest objects deposited in the Archive ouverte UNIGEengRelative cohomology of polynomial mappingshttps://archive-ouverte.unige.ch/unige:12295https://archive-ouverte.unige.ch/unige:12295Let $F$ be a polynomial mappping from $mathbb{C}^n$ to $mathbb{C}^q$ with $n>q$. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre $F^{-1}(infty)$ "at infinity" and its cohomology. Let us fix a weighted homogeneous degree on $mathbb{C}[x_1,...,x_n]$ with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of $F$. We introduce the cohomology groups $H^k(F^{-1}(infty))$ of $F$ at infinity. These groups enable us to compute all the other cohomology groups of $F$. For instance, if the fibre at infinity has an isolated singularity at the origin, we prove that every weighted homogeneous basis of $H^{n-q}(F^{-1}(infty))$ is a basis of all the groups $H^{n-q}(F^{-1}(y))$ and also a basis a the $(n-q)^{th}$ relative cohomology group of $F$. Moreover the dimension of $H^{n-q}(F^{-1}(infty))$ is given by a global Milnor number of $F$, which only depends on the leading terms of the coordinate functions of $F$.Tue, 02 Nov 2010 14:43:13 +0100Conjugacy classes of affine automorphisms of Kn and linear automorphisms of Pn in the Cremona groupshttps://archive-ouverte.unige.ch/unige:12114https://archive-ouverte.unige.ch/unige:12114We describe the conjugacy classes of affine automorphisms in the group Aut(n,K) (respectively Bir(Kn)) of automorphisms (respectively of birational maps) of Kn. From this we deduce also the classification of conjugacy classes of automorphisms of Pn in the Cremona group Bir(Kn).Fri, 15 Oct 2010 15:30:47 +0200Rational classes and divisors on curves of genus 2https://archive-ouverte.unige.ch/unige:12095https://archive-ouverte.unige.ch/unige:12095We describe amethod of looking for rational divisor classes on a curve of genus 2. We have an algorithm to decide if a given class of divisors of degree 3 contains a rational divisor. It is known that the shape of the kernel of Cassel's morphism (X − T ) is related to the existence of rational classes of degree 1. Our key tool is the dual Kummer surface.Thu, 14 Oct 2010 11:49:22 +0200Minimal length of two intersecting simple closed geodesicshttps://archive-ouverte.unige.ch/unige:9810https://archive-ouverte.unige.ch/unige:9810On a hyperbolic Riemann surface, given two simple closed geodesics that intersect n times, we address the question of a sharp lower bound Ln on the length attained by the longest of the two geodesics. We show the existence of a surface Sn on which there exists two simple closed geodesics of length Ln intersecting n times and explicitly find Ln for n ≤ 3.Fri, 30 Jul 2010 13:49:29 +0200Double covers of Kummer surfaceshttps://archive-ouverte.unige.ch/unige:9792https://archive-ouverte.unige.ch/unige:9792Besides its construction as a quotient of an abelian surface, a Kummer surface can be obtained as the quotient of a K3 surface by a Z/2Z-action. In this paper, we classify all such K3 surfaces. Our classification is expressed in terms of period lattices and extends Morrison's criterion of K3 surfaces with a Shioda–Inose structure. Moreover, we list all the K3 surfaces associated to a general Kummer surface and provide very geometrical examples of this phenomenon.Thu, 29 Jul 2010 16:38:59 +0200