face counting on an acyclic birkhoff polytope in this paper we present some algorithms allowing an exhaustive account on the number of edges and faces of the acyclic birkhoff polytope

on a class of special linear systems of mathbb p 3 in this paper we deal with linear systems of p 3 through fat points we consider the behavior of these systems under a cubic cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones

motivic decomposition of a compactification of a merkurjev suslin variety we provide a motivic decomposition of a twisted form of a smooth hyperplane section of gr 3 6 this twisted form is a norm variety corresponding to a symbol in the milnor k theory k3 3 as an application we construct a torsion element in the chow group of this variety msc2000 20g15 14f43

fractional perfect b matching polytopes i general theory the fractional perfect b matching polytope of an undirected graph g is the polytope of all assignments of nonnegative real numbers to the edges of g such that the sum of the numbers over all edges incident to any vertex v is a prescribed nonnegative number b v general theorems which provide conditions for nonemptiness give a formula for the dimension and characterize the vertices edges and face lattices of such polytopes are obtained many of these results are expressed in terms of certain spanning subgraphs of g which are associated with subsets or elements of the polytope for example it is shown that an element u of the fractional perfect b matching polytope of g is a vertex of the polytope if and only if each component of the graph of u either is acyclic or else contains exactly one cycle with that cycle having odd length where the graph of u is defined to be the spanning subgraph of g whose edges are those at which u is positive

on a notion of speciality of linear systems in p n 2014 american mathematical society given a linear system in pn with assigned multiple general points we compute the cohomology groups of its strict transforms via the blow up of its linear base locus this leads us to give a new definition of expected dimension of a linear system which takes into account the contribution of the linear base locus and thus to introduce the notion of linear speciality we investigate such a notion giving sufficient conditions for a linear system to be linearly non special for an arbitrary number of points and necessary conditions for a small number of points

quadratic form theory over preordered von neumann regular rings abstract to each pair r t consisting of a unitary commutative von neumann regular ring r where 2 is a unit and t is a preorder on r we associate a reduced special group g t r which faithfully reflects quadratic form theory modulo t over free r modules and then show using the representation of r as the ring of global sections of its affine scheme together with results from m dickmann f miraglia on quadratic forms whose total signature is zero mod 2 n solution to a problem of m marshall invent math 133 1998 243 278 m dickmann f miraglia lam s conjecture algebra colloq 10 2003 149 176 m dickmann f miraglia algebraic k theory of special groups j pure appl algebra 204 2006 195 234 that g t r satisfies a powerful k theoretic property the smc property from this we conclude that quadratic form theory modulo t over free r modules verifies marshall s signature conjecture lam s conjecture as well as a reduced version of milnor s witt ring conjecture

motivic rigidity of severi brauer varieties let d be a central division algebra over a field f we study in this note the rigidity of the motivic decompositions of the severi brauer varieties of d with respect to the ring of coefficients and to the base field we first show that if the ring of coefficient is a field these decompositions only depend on its characteristic in a second part we show that if d remains division over a field extension e f the motivic decompositions of several severi brauer varieties of d remain the same when extending the scalars to e

a variety associated to an algebra with involution abstract a variety iv a is canonically associated to a central simple algebra a with orthogonal involution of the first kind the properties of this variety are the main topic of this paper the paper is divided into two parts we first define this variety to be the subvariety of the grassmannian of the n dimensional subspaces of the n2 dimensional vector space a consisting of all the n dimensional right ideals i of a such that i i 0 the geometric properties of iv a are investigated in particular the generic property of this variety is identified further the kernel of the scalar extension map br f br k where k is the function field of iv a is also computed lastly a criterion on the embeddability of the field k into the function field of the brauer severi variety of a is given in the second part the quillen k theory of iv a is computed the k theory result then leads to an index reduction formula for the function field k

the index of a brauer class on a brauer severi variety let d and e be central division algebras over k let k be the generic splitting field of e we show that the index of d k k is the minimum of the indices of d e i as i varies we use this to calculate the index of d under related central extensions and to construct division algebras with special properties

the cox ring of overline _ 0 6 we prove that the cox ring of the moduli space v 0 6 of stable rational curves with 6 marked points is finitely generated by sections corresponding to the boundary divisors and divisors which are pull backs of the hyperelliptic locus in m 3 via morphisms p m 0 6 m 3 that send a 6 pointed rational curve to a curve with 3 nodes by identifying 3 pairs of points in particular this gives a self contained proof of hassett and tschinkel s result about the effective cone of m 0 6 being generated by the above mentioned divisors

positivity of divisors on blown up projective spaces ii we construct log resolutions of pairs on the blow up of the projective space in an arbitrary number of general points and we discuss the semi ampleness of the strict transforms as an application we give an explicit proof that the abundance conjecture holds for an infinite family of such pairs for n 2 points these strict transforms are f nef divisors on the moduli space m0 n 3 in a kapranov s model we show that all of them are nef

thin algebras of embedding dimension three let r r0 r1 r2 be a finitely generated commutative graded k algebra with hilbert series hr n 0dimk rn n suppose we require that r have a presentation as r k x1 xg 1 r where g and r and the degrees d1 dr of the homogeneous polynomials 1 r are specified in advance no one knows precisely which hubert series such an r may have a lower bound is the coefficient wise inequality hr 1 g i 1r 1 di absolute value symbols denoting the initial non negative segment of a power series can this lower bound always be attained this deceptively simple sounding question has been answered affirmatively only when g 2 r froberg report no 37 department of mathematics university of stockholm stockholm sweden 1982 a iarrobino memoir no 188 amer math soc providence r i 1977 when r g and when r g 1 and char k 0 a iarrobino compressed algebras trans amer math soc 285 1984 337 378 this paper considers the case g 3 settling it affirmatively whenever k is an infinite field

on linear systems of p 3 through multiple points abstract we prove a conjecture about the dimension of linear systems of surfaces of degree d in p 3 through at most eight multiple points in general position

the geometric interpretation of froberg iarrobino conjectures on infinitesimal neighbourhoods of points in projective space abstract the study of infinitesimal deformations of a variety embedded in projective space requires at ground level that of deformation of a collection of points as specified by a zero dimensional scheme further basic problems in infinitesimal interpolation correspond directly to the analysis of such schemes an optimal hilbert function of a collection of infinitesimal neighbourhoods of points in projective space is suggested by algebraic conjectures of r froberg and a iarrobino we discuss these conjectures from a geometric point of view they give for each such collection a function based on dimension number of points and order of each neighbourhood which should serve as an upper bound to its hilbert function weak conjecture the strong conjecture predicts when the upper bound is sharp in the case of equal order throughout in general we refer to the equality of the hilbert function of a collection of infinitesimal neighbourhoods with that of the corresponding conjectural function as the strong hypothesis we interpret these conjectures and hypotheses as accounting for the infinitesimal neighbourhoods of projective subspaces naturally occurring in the base locus of a linear system with prescribed singularities at fixed points we develop techniques and insight toward the conjectures verification and refinement the main result gives an upper bound on the hilbert function of a collection of infinitesimal neighbourhoods in p n based on hilbert functions of certain such subschemes of p n 1 further equality occurs exactly when the scheme has only the expected linear obstructions to the linear system at hand it follows that an infinitesimal neighbourhood scheme obeys the weak conjecture provided that the schemes identified in codimension one satisfy the strong hypothesis this observation is then applied to show that the weak conjecture does hold valid in p n for n 3 the main feature here is that the result is obtained although the strong hypothesis is not known to hold generally in p 2 and further p 2 presents special exceptional cases consequences of the main result in higher dimension are then examined we note then that the full weight of the strong conjecture and validity of the strong hypothesis are not necessary toward using the main theorem in the next dimension we end with the observation of how our viewpoint on the strong hypothesis pertains to extra algebraic information namely on the structure of the minimal free resolution of an ideal generated by linear forms

eigenvalues of tridiagonal pseudo toeplitz matrices abstract in this article we determine the eigenvalues of sequences of tridiagonal matrices that contain a toeplitz matrix in the upper left block

some remarks about acyclic and tridiagonal birkhoff polytopes abstract for a given tree t we consider the facial structure of the acyclic birkhoff polytope t we also determine the f vector of the polytope n t consisting of all tridiagonal doubly stochastic matrices of order n finally we count the number of combinatorially distinct faces of n t in each dimension

postulation of general quintuple fat point schemes in p 3 we study the postulation of a general union y of double triple quartuple and quintuple points of p 3 in characteristic 0 we prove that y has good postulation in degree d 11 the proof is based on the combination of the horace differential lemma with a computer assisted proof we also classify the exceptions in degree 9 and 10

vanishing theorems for linearly obstructed divisors we study divisors on the blow up of pn at points in general position that are non special with respect to the notion of linear speciality introduced in 6 we describe the cohomology groups of their strict transforms via the blow up of the space along their linear base locus we extend the result to non effective divisors that sit in a small region outside the effective cone as an application we describe linear systems of divisors in pn blown up at points in star configuration and their strict transforms via the blow up of the linear base locus

alternating sign matrices related 0 1 matrices and the smith normal form abstract we investigate the smith normal form snf of alternating sign matrices asms and related matrices of 0 s and 1 s 0 1 matrices we identify certain classes of asms and 0 1 matrices whose snfs are 0 1 matrices we relate some of our work to various ranks in particular the 0 1 rank of a 0 1 matrix that is the bipartite partition number of a bipartite graph

powers of the fundamental ideal in the witt ring abstract let k be a field of characteristic different from 2 in the algebraic theory of quadratic forms one studies the witt ring w k of equivalence classes of non degenerate quadratic forms the witt ring has a filtration given by the powers in k of the fundamental ideal i k of even dimensional forms the ideal in k is generated by the set pn k of n fold pfister forms a1 an ni 1 1 ai where ai k k 0 many questions about this filtration and its quotients have arisen in the study of w k the spectacular work of voevodsky together with his collaborative work with orlov and vishik allows one to answer many old questions the purpose of this paper is to indicate some of these solutions

the skeleton of acyclic birkhoff polytopes abstract for a fixed tree t with n vertices the corresponding acyclic birkhoff polytope n t consists of doubly stochastic matrices having support in positions specified by t this is a face of the birkhoff polytope n which consists of all n n doubly stochastic matrices the skeleton of n t is the graph where vertices and edges correspond to those of n t and we investigate some properties of this graph in particular we characterize adjacency of pairs of vertices compute the minimum degree of a vertex and show some properties of the maximum degree of a vertex in the skeleton we also determine the maximum degree for certain classes of trees including paths stars and caterpillars

l rays of permutation matrices and doubly stochastic matrices abstract let m n be the space of real n n matrices we investigate a linear transformation m n r n called an l ray motivated by x ray which is defined in terms of sums of the entries in the blocks of a certain l shaped partition of the positions of a matrix a m n we find descriptions of the image of the classes of permutation matrices and doubly stochastic matrices under this map and show connections to majorization theory

inequalities for the spectra of symmetric doubly stochastic matrices we prove majorization relations for symmetric doubly stochastic matrices whose nonzero off diagonal entries lie in an edge transitive graph these results are then used to solve the fastest mixing markov chain problem on edge transitive graphs

tridiagonal doubly stochastic matrices abstract we study the facial structure of the polytope t n in r n n consisting of the tridiagonal doubly stochastic matrices of order n we also discuss some subclasses of t n with focus on spectral properties and rank formulas finally we discuss a connection to majorization

an estimate for the nonstochastic eigenvalues of doubly stochastic matrices abstract we introduce a new measure of irreducibility of a doubly stochastic matrix and find the best estimate from below for the distance of nonstochastic eigenvalues from one the estimate is used to obtain a cheeger type inequality for the second laplacian eigenvalue of a weighted graph

bounds for the representation of quadratic forms abstract we prove first that for fixed integers n m 1 there is a uniform bound on the number of pfister forms of degree n over any pythagorean field f necessary to represent in the witt ring of f any form of dimension m as a linear combination of such forms with non zero coefficients in f uniform means that the bound does not depend either on the form or on the field f it is given by a recursive function f of n and m similar results hold for the reduced special groups arising from preordered fields and from fields whose pythagoras number is bounded by a fixed integer we single out a large class of pythagorean fields and more generally of reduced special groups cf 4 for which f has a simply exponential bound of the form cmn 1 c a constant such a class is closed under certain possibly infinitary operations which preserve marshall s signature conjecture in the case of groups of finite stability index s we obtain an upper bound for f which is quadratic on m 2n where c depends on s

the diameter of the acyclic birkhoff polytope abstract in this work we give an interpretation of vertices and edges of the acyclic birkhoff polytope t n n t where t is a tree with n vertices in terms of graph theory we generalize a recent result relatively to the diameter of the graph g t n

linear systems on the blow up of p1 n abstract in this note we study linear systems on the blow up of p 1 n at r points in very general position we prove that the fibers of the projections p 1 n p 1 s 1 s n 1 can give contribution to the speciality of the linear system this allows us to give a new definition of expected dimension of a linear system in p 1 n which we call fiber dimension finally we state a conjecture about linear systems on p 1 3

on the dimensions of anisotropic quadratic forms in i4 let wf denote the witt ring of a field f of characteristic 2 and let i n f denote the n th power of the ideal if of even dimensional forms in wf the arason pfister hauptsatz states that if 0 i n f is anisotropic then dim 2 n pfister also showed that if i 3 f is anisotropic and dim 8 then dim 12 we extend this result to i 4 f and show that if i 4 f is anisotropic and dim 16 then dim 24 and we provide some results on anisotropic 24 dimensional forms in i 4 f

collisions of fat points and applications to interpolation theory abstract we address the problem to determine the flat limit of the collision of fat points in p n we give a description of the limit scheme in many cases in particular in low dimension and multiplicities the problem turns out to be closely related with interpolation theory and as an application we exploit collisions to prove new cases of laface ugaglia conjecture

some new examples in the theory of quadratic forms we construct a 6 dimensional anisotropic quadratic form phi and a 4 dimensional quadratic form psi over some fieldf such that phi becomes isotropic over the function field f psi but every proper subform of phi is still anisotropic over f psi it is an example of non standard isotropy with respect to some standard conditions of isotropy for 6 dimensional forms over function fields of quadrics known previously besides of that we produce an 8 dimensional quadratic form phi with trivial determinant such that the index of the clifford invariant of phi is 4 but phi can not be represented as a sum of two 4 dimensional forms with trivial determinants using this we find a 14 dimensional quadratic form with trivial discriminant and clifford invariant which is not similar to a difference of two 3 fold pfister forms the proofs are based on computations of the topological filtration on the grothendieck group of certain projective homogeneous varieties to do these computations we develop several methods covering a wide class of varieties and being to our mind of independent interest

the kernels of the incidence matrices of graphs revisited in this paper we study the structure of some special bases for the null space of the incidence matrix of a graph recently it was shown that if g is a graph with no cut vertex then g has a 1 0 1 basis we generalize this result showing that the statement remains valid for every graph with no cut edge for the null space of any bipartite graph we construct a 1 0 1 basis for any bipartite graph we obtain the support sizes of all elements in the null space of its incidence matrix among other things we prove that for a graph g there exists a 1 1 vector for the null space of g if and only if the degree of any vertex of g is even and g has an even number of edges

isotropy of six dimensional quadratic forms over function fields of quadrics abstract letfbe a field of characteristic different from 2 and let be an anisotropic six dimensional quadratic form overf we study the last open cases in the problem of describing the quadratic forms such that becomes isotropic over the function fieldf

integral motives relative krull schmidt principle and maranda type theorems in the present article we investigate properties of the category of integral grothendieck chow motives over a field we discuss the krull schmidt principle for integral motives provide a complete list of generalized severi brauer varieties with indecomposable integral motive and exploit a relation between the category of motives of twisted flag varieties and integral p adic representations of finite groups

faces of faces of the tridiagonal birkhoff polytope abstract the tridiagonal birkhoff polytope n t is the set of real square matrices with nonnegative entries and all rows and columns sums equal to 1 that are tridiagonal this polytope arises in many problems of enumerative combinatorics statistics combinatorial optimization etc in this paper for a given a p face of n t we determine the number of faces of lower dimension that are contained in it and we discuss its nature in fact a 2 face of n t is a triangle or a quadrilateral and the cells can only be tetrahedrons pentahedrons or hexahedrons

around 16 dimensional quadratic forms in i 3_q we determine the indexes of all orthogonal grassmannians of a generic 16 dimensional quadratic form in i 3 q this is applied to show that the 3 pfister number of the form is ge 4 other consequences are a new and characteristic free proof of a recent result by chernousov merkurjev on proper subforms in i 2 q originally available in characteristic 0 as well as a new and characteristic free proof of an old result by hoffmann tignol and izhboldin karpenko on 14 dimensional quadratic forms in i 3 q originally available in characteristic ne 2 we also suggest an extension of the method based on investigation of the topological filtration on the grothendieck ring of a maximal orthogonal grassmanian which applies to quadratic forms of dimension higher than 16