rational curves of castelnuovo mumford regularity d r 1 abstract the purpose here is to show that for r 4 and d r 4 a nondegenerate rational curve of degree d in p r has a unique d r 1 secant line if and only if x fails to be d r regular this is a next case to a result of gruson lazarsfeld and peskine which asserts that for r 3 and d r 2 a nondegenerate curve x of degree d in p r has a d r 2 secant line if and only if x fails to be d r 1 regular

coverage probability bias objective bayes and the likelihood principle summary we review objective bayes procedures based on both parametric and predictive coverage probability bias and explore the extent to which such procedures contravene the likelihood principle in the case of a scalar parameter the discussion encompasses choice of objective priors objective posterior probability statements and objective predictive probability statements we conclude with some remarks concerning the future development and implementation of objective priors based on small coverage probability bias

castelnuovo mumford regularity bounds for singular surfaces we prove the regularity conjecture namely eisenbud goto conjecture for a normal surface with rational gorenstein elliptic and log canonical singularities along the way we bound the regularity for a dimension zero scheme by its loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree

linear systems cut out by quadrics on projections of varieties let x be a variety embedded in p where r is an algebraically closed field let l x k denote the linear system cut out on x by the hypersurfaces of degree k then l x k is complete for large enough k the question arises given a particular embedding of x how large does k have to be before l x k is complete in the case c is a curve the problem was studied by castelnuovo 2 who showed in particular that if cc p3 then l c k is complete if k deg c 2 see szpiro 12 for a modern proof due to gruson and more recently by lazarfeld et al 5 who have shown that if c a nondegenerate curve in p then l c k is complete for f deg c n 1 in this paper we study the following special case let x be a variety in p with ideal sheaf 7 to say that l x k is complete is equivalent to saying that the sequence

multigraded castelnuovo mumford regularity we develop a multigraded variant of castelnuovo mumford regularity motivated by toric geometry we work with modules over a polynomial ring graded by a finitely generated abelian group as in the standard graded case our definition of multigraded regularity involves the vanishing of graded components of local cohomology we establish the key properties of regularity its connection with the minimal generators of a module and its behavior in exact sequences for an ideal sheaf on a simplicial toric variety x we prove that its multigraded regularity bounds the equations that cut out the associated subvariety we also provide a criterion for testing if an ample line bundle on x gives a projectively normal embedding

on the derived category of the classical godeaux surface abstract we construct an exceptional sequence of length 11 on the classical godeaux surface x which is the z 5 z quotient of the fermat quintic surface in p 3 this is the maximal possible length of such a sequence on this surface which has grothendieck group z 11 z 5 z in particular the result answers kuznetsov s nonvanishing conjecture which concerns hochschild homology of an admissible subcategory in the negative the sequence carries a symmetry when interpreted in terms of the root lattice of the simple lie algebra of type e 8 we also produce explicit nonzero objects in the right orthogonal to the exceptional sequence

probability matching priors for non regular cases we study probability matching priors in non regular cases both one parameter and multi parameter cases are considered the resulting priors are shown to satisfy certain differential equations several examples are discussed it is observed that the reference prior in the sense of bernardo is in many cases also probability matching as a necessary tool for finding such priors asymptotic expansion of posterior distributions for multi parameter non regular cases is also obtained

noether resolutions in dimension 2 let r k x 1 ldots x n be a polynomial ring over an infinite field k and let i subset r be a homogeneous ideal with respectto a weight vector omega omega 1 ldots omega n in mathbb z n such that dim r i d in this paper we study the minimal graded free resolution of r i as a module that we call the noether resolution of r i whenever a k x n d 1 ldots x n is a noether normalization of r i when d 2 and i is saturated we give an algorithm for obtaining this resolution that involves the computation of a minimalgr obner basis of i with respect to the weighted degree reverse lexicographic order in the particular case when r i is a 2 dimensional semigroup ring we also describe the multigraded version of this resolution in terms of the underlying semigroup whenever we have the noether resolution of r i or its multigraded version we obtain formulas for the corresponding hilbert series of r i and when i is homogeneous we obtain a formula for the castelnuovo mumford regularity of r i moreover in the more general setting that r i is a simplicial semigroup ring of any dimension we provide its macaulayfication as an application of the results for 2 dimensional semigroup rings we provide a new upper bound for the castelnuovo mumford regularity of the coordinate ring of a projective monomial curve finally we describe the multigraded noether resolution and the macaulayfication of either the coordinate ring of a projective monomial curve mathcal c l subseteq mathbb p k n associated to an arithmetic sequence or the coordinate ring of any canonical projection pi r mathcal c of mathcal c to mathbb p k n 1

small schemes and varieties of minimal degree we prove that if x p r is any 2 regular scheme in the sense of castelnuovo mumford then x is small this means that if l is a linear space and y l x is finite then y is linearly independent in the sense that the dimension of the linear span of y is deg y 1 the converse is true and well known for finite schemes but false in general the main result of this paper is that the converse small implies 2 regular is also true for reduced schemes algebraic sets this is proven by means of a delicate geometric analysis leading to a complete classification we show that the components of a small algebraic set are varieties of minimal degree meeting in a particularly simple way from the classification one can show that if x p r is 2 regular then so is xred and so also is the projection of x from any point of x our results extend the del pezzo bertini classification of varieties of minimal degree the charac terization of these as the varieties of regularity 2 by eisenbud goto and the construction of 2 regular square free monomial ideals by froberg 0 introduction throughout this paper we will work with projective schemes x p r over an algebraically closed field k the castelnuovo mumford regularity of x p r is a basic homological measure of the complexity of x and its embedding in p r that gives a bound for the degrees of the generators of the defining ideal ix of x and for many other invariants the only schemes of regular ity 1 are the linear spaces but no classification is known for projective schemes of regularity 2 in this paper we prove a structure theorem for reduced 2 regular schemes showing that their irreducible components are varieties of minimal degree and characterizing how these components can meet we also show that the reduced structure on any 2 regular scheme is 2 regular and thus we obtain a complete description of the reduced structures on 2 regular schemes since a high veronese re embedding of any zero dimensional scheme is 2 regular one cannot hope to characterize the isomorphism types of all 2 regular nonreduced schemes before stating our results we review some basic notions for any subscheme x p r we write span x for the smallest linear subspace of p r containing x recall that every variety reduced irreducible scheme x p r satisfies the condition deg x 1 codim x span x

on the invariance of noninformative priors various noninformative priors have been proposed in the literature this article explores the invariance or lack thereof of some of these noninformative priors including the reference prior of berger and bernardo the reverse reference prior of j k ghosh and the probabilitymatching prior of peers and stein under reparameterization berger and bernardo s m group ordered reference prior is shown to remain invariant under a special type of reparameterization the reverse reference prior of j k ghosh is shown not to remain invariant under reparameterization however the probability matching prior is shown to remain invariant under any reparameterization also for spherically symmetric distributions certain noninformative priors are derived using the principle of group invariance

likelihood estimation and inference in a class of nonregular econometric models we study inference in structural models with a jump in the conditional density where location and size of the jump are described by regression curves two prominent examples are auction models where the bid density jumps from zero to a positive value at the lowest cost and equilibrium job search models where the wage density jumps from one positive level to another at the reservation wage general inference in such models remained a long standing unresolved problem primarily due to nonregularities and computational difficulties caused by discontinuous likelihood functions this paper develops likelihood based estimation and inference methods for these models focusing on optimal bayes and maximum likelihood procedures we derive convergence rates and distribution theory and develop bayes and wald inference we show that bayes estimators and confidence intervals are attractive both theoretically and computationally and that bayes confidence intervals based on posterior quantiles provide a valid large sample inference method

a combinatorial proof of the eisenbud goto conjecture for monomial curves and some simplicial semigroup rings abstract we will give a pure combinatorial proof of the eisenbud goto conjecture for arbitrary monomial curves moreover we will show that the conjecture holds for certain simplicial affine semigroup rings

frequentist validity of posterior quantiles for a two parameter exponential family summary berger bernardo s 1989 reference priors and tibshirani 1989 priors are derived for distributions in bar lev reiser s 1982 two parameter exponential family when either the location or the scale parameter is of interest the reference prior is shown to be a tibshirani prior furthermore conditions under which a tibshirani prior is a higherorder matching prior are investigated when both the parameters are of interest it is also shown that the reference prior agrees with the prior matching the posterior and frequentist expansions based on the signed square root of log likelihood ratio the normal inverse gaussian and gamma distributions are used to illustrate the results

on priors providing frequentist validity of bayesian inference for multiple parametric functions summary we characterise priors which match up to o n 1 the posterior joint cumulative distribution function of multiple parametric functions with the corresponding frequentist cumulative distribution function this work extends and unifies the work of ghosh mukerjee 1993 and datta ghosh 1995a on the topic of probability matching priors a set of necessary and sufficient conditions is obtained for the above characterisation some of these conditions depend only on the parametric functions and not on the prior examples are given where the joint probability matching is possible and where it is not possible

some remarks on noninformative priors abstract this article focuses primarily on a comparison between the reference priors of berger and bernardo and the reverse reference priors suggested by j k ghosh sufficient conditions are given that provide agreement between the two classes of priors several examples are given showing the agreement or disagreement between the two in addition these priors are compared under a criterion that requires the frequentist coverage probability of the posterior region of a real valued parametric function to match a nominal level with a remainder of o n 1 where n denotes the sample size the latter priors first introduced by welch and peers are obtained by solving a differential equation due to peers finally in the presence of several parameters of interest a general class of priors that satisfies the matching criterion separately for each parameter is constructed and examples are given to illustrate how reference or reverse reference priors fit within this class of priors

a bound on the castelnuovo mumford regularity for curves recall that a projective curve in bbb p r with ideal sheaf cal i is said to be n regular if h i bbb p r cal i otimes cal o bbb p r n i 0 for every integer i 0 and that in this case it is cut out scheme theoretically by equations of degree at most n the purpose here is to show that an irreducible reduced projective curve of degree d and large arithmetic genus p a satisfies a smaller regularity bound than the optimal one d r 2 for example if p a geq r 2 then a curve is d 2r 4 regular unless it is embedded by a complete linear system of degree 2p a 2

curves of degree r 2 in pr cohomological geometric and homological aspects abstract let c p rk be a non degenerate projective curve of degree r 2 where r 3 by means of the hartshorne rao module of c we distinguish 4 different possible cases and an exceptional case which only appears if r 3 in any case c is obtained either by means of an embedding of an arbitrary smooth curve c 0 of genus 2 or by projecting an elliptic normal curve from a point or by projecting a rational normal curve from a line finally the possible minimal free resolutions of the homogeneous coordinate ring of c are studied

a castelnuovo mumford regularity bound for scrolls abstract let x p r be a scroll of codimension e and degree d over a smooth projective curve of genus g the purpose of this paper is to prove a linear castelnuovo mumford regularity bound that reg x d e 1 g e 1 this bound works over an algebraically closed field of arbitrary characteristic

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

on the complexity of the integral closure the computation of the integral closure of an affine ring has been the focus of several modern algorithms we will treat here one related problem the number of generators the integral closure of an affine ring may require this number and the degrees of the generators in the graded case are major measures of cost of the computation we prove several polynomial type bounds for various kinds of algebras and establish in characteristic zero an exponential type bound for homogeneous algebras with a small singular locus

simple modifications for signed roots of likelihood ratio statistics summary in the context of inference about a scalar parameter in the presence of nuisance parameters some simple modifications for the signed root of the log likelihood ratio statistic r are developed that reduce the order of error in the standard normal approximation to the distribution of r from o n 1 2 to o n 1 barndorff nielsen has introduced a variable u such that the error in the standard normal approximation to the distribution of r r r log u r is of order o n 3 2 but calculation of urequires the specification of an exact or approximate ancillary statistic a this paper proposes an alternative variable to u denoted by t that is available without knowledge of a and satisfies t u op n 1 in general thus the standard normal approximation to the distribution of r r log t r has error of order o n 1 and it can be used to construct approximate confidence limits having coverage error of order o n 1 in certain cases however tand uare identical the derivation of tinvolves the bayesian approach to constructing confidence limits considered by welch and peers and peers similar modifications for the signed root of the conditional likelihood ratio statistic are also developed and these modifications are seen to be useful when a large number of nuisance parameters are present several examples are presented including inference for natural parameters in exponential models and inference about location scale models with type ii censoring in each case the relationship between tand u is discussed numerical examples are also given including inference for regression models inference about the means of log normal distributions and inference for exponential lifetime models with type i censoring where barndorff nielsen s variable u is not available

noninformative priors for one parameter of many summary we consider the problem of constructing a prior that is noninformative for a single parameter in the presence of nuisance parameters our approach is to require that the resulting marginal posterior intervals have accurate frequentist coverage stein 1985 derived nonrigorously a sufficient condition for such a prior through the use of orthogonal parameters we give a general form for the class of priors satisfying stein s condition the priors are proportional to the square root of the information element for the parameter of interest times an arbitrary function of the nuisance parameters this is in contrast to jeffreys 1946 invariant prior for the overall parameter which is proportional to the square root of the determinant of the information matrix several examples are given and comparisons are made to the reference priors of bernardo 1979

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

hypersurfaces cutting out a projective variety let x be a nondegenerate projective variety of degree d and codimension e in a projective space p n defined over an algebraically closed field we study the following two problems is the length of the intersection of x and a line l in p n at most d e 1 if l g x is the scheme theoretic intersection of all hypersurfaces of degree at most d e 1 containing x equal to x to study the second problem we look at the locus of points from which x is projected nonbirationally

natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method this paper presents a semi analytical approach to solve the eigenproblem of an acoustic cavity with multiple elliptical boundaries to satisfy the helmholtz equation in the elliptical coordinate system the multipole expansion for the acoustic pressure is formulated in terms of angular and radial mathieu functions the boundary conditions are satisfied by uniformly collocating points on the boundaries the acoustic pressure at each point is directly calculated in each elliptical coordinate system in different coordinate systems the normal derivative of the acoustic pressure is calculated by using the appropriate directional derivative an alternative to the addition theorem by truncating the multipole expansion a finite linear algebraic system is derived the direct searching approach is employed to determine the natural frequencies by using the singular value decomposition svd numerical results are widely discussed for several examples including an elliptical cavity a confocal elliptical annulus cavity and an elliptical cavity with two elliptical cylinders the accuracy and numerical convergence of the presented method is validated by comparison with available results from the analytical method and the commercial finite element code abaqus no spurious eigensolutions are found in the proposed formulation excellent accuracy and fast rate of convergence are the key features of the present method thanks to its semi analytical feature

rational points of bounded height on projective surfaces we give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

some effects of property n p on the higher normality and defining equations of nonlinearly normal varieties for a smooth projective variety x p embedded by the complete linear system property np has been studied for a long time 5 11 12 10 etc on the other hand castelnuovo mumford regularity conjecture and related problems have been focused for a projective variety which is not necessarily linearly normal 2 13 15 17 20 etc this paper aims to explain the influence of property np on higher normality and defining equations of a smooth variety embedded by sub linear system also we prove a claim about property np of surface scrolls which is a generalization of green s work in 11 about property np for curves

bayesian intervals with good frequentist behaviour in the presence of nuisance parameters given a random sample from a distribution with density function that depends on an unknown parameter 1 p we are concerned with the problem of setting confidence intervals for a particular component of it say 1 treating the remaining components as a nuisance parameter adopting an objective bayesian approach we show that the bayes intervals with a certain conditional prior density on the parameter of interest 1 are confidence intervals as well having nearly the correct frequency of coverage the frequentist performance of the proposed intervals is tested in a simulation study for gamma mean and shape parameters

estimating a product of means bayesian analysis with reference priors abstract suppose that we observe x n 1 and independently y n 1 and are concerned with inference mainly estimation and confidence statements about the product of means this problem arises most obviously in situations of determining area based on measurements of length and width it also arises in other practical contexts however for instance in gypsy moth studies the hatching rate of larvae per unit area can be estimated as the product of the mean of egg masses per unit area times the mean number of larvae hatching per egg mass approximately independent samples can be obtained for each mean see southwood 1978 noninformative prior bayesian approaches to the problem are considered in particular the reference prior approach of bernardo 1979 an appropriate reference prior for the problem is developed and relatively easily implementable formulas for posterior moments e g the posterior mean and variance and credible sets are derived comparisons with alternative noninf

on priors providing frequentist validity for bayesian inference summary we derive the differential equation that a prior must satisfy if the posterior probability of a one sided credibility interval for a parametric function and its frequentist probability agree up to o n 1 this equation turns out to be identical with stein s equation for a slightly different problem for which also our method provides a rigorous justification our method is different in details from stein s but similar in spirit to dawid 1991 and bickel ghosh 1990 some examples are provided