faces of the cone of euclidean distance matrices characterizations structure and induced geometry in this paper we obtain new characterizations of the faces of the cone of euclidean distance matrices in one case the characterization is based on a special subspace associated with each distance matrix in other case on linear restrictions of coordinate matrices we also relate the faces to the supporting hyperplanes of the cone and we show how the supporting hyperplanes induce certain geometry on the configurations that belong to that face

the best euclidian fit to a given distance matrix in prescribed dimensions abstract in this paper we determine the euclidean distance matrix which is closest to a given not necessarily euclidian one we admit a wide class of appropriate matrix norms as measures of closeness majorization results on sums of eigenvalues of symmetric matrices serve as basic tools

algebraic hyperbolicity of the very general quintic surface in p3 abstract we prove that a curve of degree dk on a very general surface of degree d 5 in p 3 has geometric genus at least d k d 5 k 2 1 this gives a substantial improvement on the celebrated genus bounds of geng xu as a corollary we deduce the algebraic hyperbolicity of a very general quintic surface in p 3 resolving a long standing conjecture of demailly this completely determines which very general hypersurfaces in p 3 are algebraically hyperbolic

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

twisted genus bounds for subvarieties of generic hypersurfaces we prove some lower bounds on certain twists of the canonical bundle of a subvariety of a generic hypersurface in projective space in particular we prove that the generic sextic threefold contains no rational or elliptic curves and no nondegenerate curves of genus 2

on the uniqueness of euclidean distance matrix completions the euclidean distance matrix completion problem edmcp is the problem of determining whether or not a given partial matrix can be completed into a euclidean distance matrix in this paper we present necessary and sufficient conditions for a given solution of the edmcp to be unique

kontsevich spaces of rational curves on fano hypersurfaces we investigate the spaces of rational curves on a general hypersurface in particular we show that for a general degree d hypersurface in pn with n d 2 the space m0 0 x e of degree e kontsevich stable maps from a rational curve to x is an irreducible local complete intersection stack of dimension e n d 1 n 4 this resolves all but one case of a conjecture of coskun harris and starr

on the use of digital elevation model data for hortonian and fractal analyses of channel networks a common method of channel network extraction from digital elevation model dem data is based on specification of a threshold area ath that is the minimum support area required to drain to a point for a channel to form current efforts to predict ath from dem data are inconclusive and usually an arbitrary constant ath value is chosen for channel network extraction in this paper we study the effects of threshold area selection on the morphometric properties such as drainage density length of drainage paths and external and internal links and scaling properties such as horton s laws and fractal dimension of a channel network we also study the related problem of dem data resolution and its effect on estimation of scaling properties the results indicate that morphometric properties vary considerably with ath and thus values reported without their associated ath are meaningless and should be used in hydrologic analysis with caution also the completeness of a channel network in terms of having the outlet stream flowing directly into a higher order stream is found to depend on ath in a random unpredictable way even if only the complete channel networks are used in the analysis the statistical variability of scaling properties estimates due to ath selection is significant and can be of comparable size to the variability due to dem resolution and variability between estimates of different river networks our analysis highlights the need to carefully study the problem of network source representation or channel initiation scale from dems which will point to an appropriate ath for channel network extraction and estimation of morphometric properties

subvarieties of general type on a general projective hypersurface we study subvarieties of a general projective degree d hypersurface x d subset mathbf p n our main theorem which improves previous results of l ein and c voisin implies in particular the following sharp corollary any subvariety of a general hypersurface x d subset mathbf p n for n geq 6 and d geq 2n 2 is of general type

subvarieties of generic hypersurfaces in a nonsingular projective toric variety we investigate the subvarieties contained in generic hypersurfaces of projective toric varieties and prove two main theorems the first generalizes clemens famous theorem on the genus of curves in hypersurfaces of projective spaces to curves in hypersurfaces of toric varieties and the second improves the bound in the special case of toric varieties in a theorem of ein on the positivity of subvarieties contained in sufficiently ample generic hypersurfaces of projective varieties both depend on a hypothesis which deals with the surjectivity of multiplication maps of sections of line bundles on the toric variety we also obtain an infinitesimal torelli theorem for hypersurfaces of toric varieties

a variance inequality ensuring that a pre distance matrix is euclidean we derive a variance inequality on the squares of the pre distances between n objects ensuring that the corresponding pre distance matrix is euclidean the result is discussed in relation with some transformations making a pre distance matrix euclidean

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

the cone of distance matrices abstract the geometry of the cone of euclidean distance matrices edms is analyzed using a new characterization of an edm the facial structure and the angle that edms of embedding dimension one make with the center ray are found this result follows from a complete analysis of the critical points of the distance function in frobenius norm from the matrix e consisting of zero diagonal and ones elsewhere to the edms of embedding dimension one