Soient $X$ un espace analytique affinoïde réduit sur un corps $K$ complet pour une valeur absolue non archimédienne, $r:X\to \widehat{X}$ sa réduction canonique et $p\in \widehat{X}$ un point de la variété algébrique affine $\widehat{X}$. Ce travail décrit la singularité du point $p$ à l’aide d’objets associés à l’espace $X$: la localisation formelle ${𝒪}_{X,\left(p\right)}$ qui est une $K$-algèbre noethérienne de spectre maximal $r^{-1}\left(p\right)$ et dont la réduction est ${𝒪}_{\widehat{X},\left(p\right)}$ ; un complété formel ${𝒪}_{X,\left(p\right)}$ qui est une $K$-algèbre noethérienne dont la réduction est ${𝒪}_{\widehat{X},\left(p\right)}$. Les résultats essentiels sont obtenus...

Si calcola esplicitamente con l'aiuto di un computer l'espressione di ogni germe di biolomorfismo in un punto di una iperquadrica reale $Q$ in ${\mathbb{C}}^3$, che porti $Q$ in $Q$. Tale germe risulta ovviamente una trasformazione lineare fratta, che lascia $Q$ invariante.

In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate.In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity $2$ and logarithmic Kodaira dimension $2$, any...

There is a well known relation between simple algebraic groups and simple singularities, cf. [5], [28]. The simple singularities appear as the generic singularity in codimension two of the unipotent variety of simple algebraic groups. Furthermore, the semi-universal deformation and the simultaneous resolution of the singularity can be constructed in terms of the algebraic group. The aim of these notes is to extend this kind of relation to loop groups and simple elliptic singularities. It is the...

For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical $k$-jets$J_{\mathsf{\text{vert}}}^k\left(𝒳\right)$ of the universal hypersurface$$𝒳\subset {\mathbb{P}}^{n+1}\times {\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}$$parametrizing all projective hypersurfaces $X\subset {\mathbb{P}}^{n+1}\left(ℂ\right)$ of degree $d$. In 2004, for $k=n$, Siu announced that there exist two constants $c_n\ge 1$ and $c_n^{\prime }\ge 1$ such that the twisted tangent bundle$$T_{J_{\mathsf{\text{vert}}}^n\left(𝒳\right)}\otimes {𝒪}_{{\mathbb{P}}^{n+1}}\left(c_n\right)\otimes {𝒪}_{{\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}}\left(c_n^{\prime }\right)$$is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset $\Sigma \subset J_{\mathsf{\text{vert}}}^n\left(𝒳\right)$ defined by the vanishing of certain Wronskians,...

For 1 < p < 2 we obtain sharp lower bounds for the uniform norm of products of homogeneous polynomials on $L_p\left(\mu \right)$, whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in infinite-dimensional settings). The result also holds for the Schatten classes ${}_p$. For p > 2 we present some estimates on the constants involved.