MathSciDoc: An Archive for Mathematician ∫

We associate to a finite digraph D a lattice polytope PD whose vertices are the rows of the Laplacian matrix of D. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of PD equals the complexity of D, and PD contains the origin in its relative interior if and only if D is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, the h∗-polynomial, and the integer decomposition property of PD in these cases. We extend Braun and Meyer’s study of cycles by considering cycle digraphs. In this setting, we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.

Let X⊂Pr be an integral and non-degenerate variety. Set n:=dim(X). We prove that if the (k+n−1)-secant variety of X has (the expected) dimension (k+n−1)(n+1)−1<r and X is not uniruled by lines, then X is not k-weakly defective and hence the k-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique S⊂X with ♯(S)=k. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures G1,d. If X is the Segre embedding of a multiprojective space we prove identifiability for the k-secant variety (assuming that the (k+n−1)-secant variety has dimension (k+n−1)(n+1)−1<r, this is a known result in many cases), beating several bounds on the identifiability of tensors.

On considère une variété riemannienne (M,g) non compacte, complète, à géométrie bornée et courbure de Ricci parallèle. Nous montrons que certains opérateurs “affines” en la courbure de Ricci sont localement inversibles, dans des espaces de Sobolev classiques, au voisinage de g.

We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree d, resp. (d,d′), onto smooth elliptic curves, with particular attention to the case d prime.

Soit G un groupe réductif défini sur un corps algébriquement clos de caractéristique positive. Nous montrons que le foncteur contraction par Frobenius de la catégorie des G-modules est adjoint à droite de celui de tensorisation deux fois par le module de Steinberg du tordu de Frobenius du G-module de départ. Il s’ensuit en particulier que le foncteur de contraction par Frobenius préserve le caractère injectif et l’existence de bonne filtration mais pas la semi-simplicité. (For a reductive group G defined over an algebraically closed field of positive characteristic, we show that the Frobenius contraction functor of $Gs-modules is right adjoint to the Frobenius twist of the modules tensored with the Steinberg module twice. It follows that the Frobenius contraction functor preserves injectivity, good filtrations, but not semi-simplicity.)

We show that sparse and Carleson coefficients are equivalent for every countable collection of Borel sets and hence, in particular, for dyadic rectangles, the case relevant to the theory of bi-parameter singular integrals. The key observation is that a dual refomulation by I. E. Verbitsky for Carleson coefficients over dyadic cubes holds also for Carleson coefficients over general sets.

In this paper, we describe all (2,3)-torus structures of a highly symmetric 39-cuspidal degree 12 curve. A direct computer-aided determination of these torus structures seems to be out of reach. We use various quotients by automorphisms to find torus structures. We use a height pairing argument to show that there are no further structures.

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal to 1. Consequently, we obtain an elementary proof that, in general, the integral simplicial volume of (triangulated) manifolds is not computable.

We construct positive loops of Legendrian submanifolds in several instances. In particular, we partially recover G. Liu’s result stating that any loose Legendrian admits a positive loop, under some mild topological assumptions on the Legendrian. Moreover, we show contractibility of the constructed loops under an extra topological assumption.

It is proved that dicritical singularities of real analytic Levi-flat sets coincide with the set of Segre degenerate points.