Tomographic and Lie algebraic significance of generalized symmetric informationally complete measurements

Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank one. They are interesting originally because of their connection with rank-one SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal IC measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs are optimal among minimal IC measurements with given average purity of measurement outcomes. Besides its significance to the current study, this result may help understand tomographic efficiencies of minimal IC measurements under the influence of noise. When minimal IC measurements are taken as bases for the Lie algebra of the unitary group, generalized SICs are uniquely characterized by the antisymmetry of the associated structure constants.Comment: 6.2 pages, comments and suggestions welcom

Categoricity from one successor cardinal in Tame Abstract Elementary Classes

Let K be an abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties. Theorem 1. Suppose K is \chi-tame. If K is categorical in some \lambda^+ >LS(K) then it is categorical in all \mu\geq (\lambda+\chi)^+. Theorem 2. If K is LS(K)-tame and is categorical both in LS(K) and in LS(K)^+ then K is categorical in all \mu\geq LS(K).Comment: 20 page

On comparison between relative log de Rham-Witt cohomology and relative log crystalline cohomology

In this article, we prove the comparison theorem between the relative log de Rham-Witt cohomology and the relative log crystalline cohomology for a log smooth saturated morphism of fs log schemes satisfying certain condition. Our result covers the case where the base fs log scheme is etale locally log smooth over a scheme with trivial log structure or the case where the base fs log scheme is hollow, and so it generalizes the previously known results of Matsuue. In Appendix, we prove that our relative log de Rham-Witt complex and our comparison map are compatible with those of Hyodo-Kato.Comment: 54 pages, Appendix adde

Shokurov's boundary property

For a birational analogue of minimal elliptic surfaces X/Y, the singularities of the fibers define a log structure in codimension one on Y. Via base change, we have a log structure in codimension one on Y', for any birational model Y' of Y. We show that these codimension one log structures glue to a unique log structure, defined on some birational model of Y (Shokurov's BP Conjecture). We have three applications: inverse of adjunction for the above mentioned fiber spaces, the invariance of Shokurov's FGA-algebras under restriction to exceptional lc centers, and a remark on the moduli part of parabolic fiber spaces.Comment: LaTex, 26 page

Thin groups of fractions

A number of properties of spherical Artin groups extend to Garside groups, defined as the groups of fractions of monoids where least common multiples exist, there is no nontrivial unit, and some additional finiteness conditions are satisfied \cite{Dgk}. Here we investigate a wider class of groups of fractions, called {\it thin}, which are those associated with monoids where minimal common multiples exist, but they are not necessarily unique. Also, we allow units in the involved monoids. The main results are that all thin groups of fractions satisfy a quadratic isoperimetric inequality, and that, under some additional hypotheses, they admit an automatic structure

On one question of Shemetkov about composition formations

In this paper one construction of composition formations was introduced. This construction contains formations of quasinilpotent groups, $c$-supersoluble groups, groups defined by ranks of chief factors and some new classes of groups. A partial answer on a question of L.\,A. Shemetkov about the intersection of $\mathfrak{F}$-maximal subgroups and the $\mathfrak{F}$-hypercenter was given for these composition formations.Comment: arXiv admin note: substantial text overlap with arXiv:1711.0168

The Miyaoka-Yau inequality and uniformisation of canonical models

We establish the Miyaoka-Yau inequality in terms of orbifold Chern classes for the tangent sheaf of any complex projective variety of general type with klt singularities and nef canonical divisor. In case equality is attained for a variety with at worst terminal singularities, we prove that the associated canonical model is the quotient of the unit ball by a discrete group action.Comment: v3: final version, added section on "Further directions"; accepted for publication by Annales scientifiques de l'Ecole normale sup\'erieur

Equivalence of two notions of log moduli stacks

We show the equivalence between two notions of log moduli stacks which appear in literatures. In particular, we generalize M.Olsson's theorem of representation of log algebraic stacks and answer a question posted by him (\cite{Ol4} 3.5.3). As an application, we obtain several fundamental results of algebraic log stacks which resemble to those in algebraic stacks.Comment: 27 page

Computing Quot schemes via marked bases over quasi-stable modules

Let $\Bbbk$ be a field of arbitrary characteristic, $A$ a Noetherian $\Bbbk$-algebra and consider the polynomial ring $A[\mathbf x]=A[x_0,\dots,x_n]$. We consider homogeneous submodules of $A[\mathbf x]^m$ having a special set of generators: a marked basis over a quasi-stable module. Such a marked basis inherits several good properties of a Gr\"obner basis, including a Noetherian reduction relation. The set of submodules of $A[\mathbf x]^m$ having a marked basis over a given quasi-stable module has an affine scheme structure that we are able to exhibit. Furthermore, the syzygies of a module generated by such a marked basis are generated by a marked basis, too (over a suitable quasi-stable module in $\oplus^{m'}_{i=1} A[\mathbf x](-d_i)$). We apply the construction of marked bases and related properties to the investigation of Quot functors (and schemes). More precisely, for a given Hilbert polynomial, we can explicitely construct (up to the action of a general linear group) an open cover of the corresponding Quot functor made up of open functors represented by affine schemes. This gives a new proof that the Quot functor is the functor of points of a scheme. We also exhibit a procedure to obtain the equations defining a given Quot scheme as a subscheme of a suitable Grassmannian. Thanks to the good behaviour of marked bases with respect to Castelnuovo-Mumford regularity, we can adapt our methods in order to study the locus of the Quot scheme given by an upper bound on the regularity of its points.Comment: 28 pages, exposition improved. This version contains the results of the previous one, and also the application to Quot scheme

Virtual Resolutions for a Product of Projective Spaces

Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In this paper, we construct some much shorter free complexes that better encode the geometry.Comment: 22 pages, 1 figur