small ball probabilities for the slepian gaussian fields the d dimensional slepian gaussian random field s t t r d is a mean zero gaussian process with covariance function es s s t i d i i max 0 a i s i t i for a i 0 and t t i t d r d small ball probabilities for s t are obtained under the l 2 norm on 0 1 d and under the sup norm on 0 1 2 which implies talagrand s result for the brownian sheet the method of proof for the sup norm case is purely probabilistic and analytic and thus avoids ingenious combinatoric arguments of using decreasing mathematical induction in particular riesz product techniques are new ingredients in our arguments

dichotomy results for the l1 norm of the discrepancy function abstract it is a well known conjecture in the theory of irregularities of distribution that the l 1 norm of the discrepancy function of an n point set satisfies the same asymptotic lower bounds as its l 2 norm in dimension d 2 this fact has been established by halasz while in higher dimensions the problem is wide open in this note we establish a series of dichotomy type results which state that if the l 1 norm of the discrepancy function is too small smaller than the conjectural bound then the discrepancy function has to be large in some other function space

a vanishing theorem for log canonical pairs using inversion of adjunction we deduce from nadel s theorem a vanishing property for ideals sheaves on projective varieties a special case of which recovers a result due to bertram ein lazarsfeld this enables us to generalize to a large class of projective schemes certain bounds on castelnuovo mumford regularity previously obtained by bertram ein lazarsfeld in the smooth case and by chardin ulrich for locally complete intersection varieties with rational singularities our results are tested on several examples

manifolds covered by lines and the hartshorne conjecture for quadratic manifolds small codimensional embedded manifolds defined by equations of small degree are fano and covered by lines they are complete intersections exactly when the variety of lines through a general point is so and has the right codimension this allows us to prove the hartshorne conjecture for manifolds defined by quadratic equations and to obtain the list of such hartshorne manifolds

smooth threefolds in bbb p 5 without apparent triple or quadruple points and a quadruple point formula for a projective variety x of codimension 2 in bbb p n 2 defined over the complex number field bbb c it is traditionally said that x has no apparent k 1 ple points if the k 1 secant lines of x do not fill up the ambient projective space bbb p n 2 equivalently the locus of k 1 ple points of a generic projection of x to bbb p n 1 is empty we show that a smooth threefold in bbb p 5 has no apparent triple points if and only if it is contained in a quadric hypersurface we also obtain an enumerative formula counting the quadrisecant lines of x passing through a general point of bbb p 5 and give necessary cohomological conditions for smooth threefolds in bbb p 5 without apparent quadruple points this work is intended to generalize the work of f severi fse and a aure au where it was shown that a smooth surface in bbb p 4 has no triple points if and only if it is either a quintic elliptic scroll or contained in a hyperquadric furthermore we give open questions along these lines

varieties with quadratic entry locus i we introduce and study l qel manifolds x subset mathbb p n of type a class of projective varieties whose extrinsic and intrinsic geometry is very rich especially when 0 we prove a strong divisibility property for lqel manifolds of type 3 allowing the classification of those of type delta geq frac dim x 2 in particular we obtain a new and very short proof that severi varieties have dimension 2 4 8 or 16 and also an almost self contained proof of their classification due to zak we also provide the classification of special cremona transformations of type 2 3 and 2 5

a vanishing theorem and asymptotic regularity of powers of ideal sheaves abstract let i be an ideal sheaf on p n in the first part of this paper we bound the asymptotic regularity of powers of i as s p reg i p s p e where e is a constant and s is the s invariant of i we also give the same upper bound for the asymptotic regularity of symbolic powers of i under some conditions in the second part by using multiplier ideal sheaves we give a vanishing theorem of powers of i when it defines a local complete intersection subvariety with log canonical singularities

regularity on abelian varieties i we introduce the notion of mukai regularity m regularity for coherent sheaves on abelian varieties the definition is based on the fourier mukai transform and in a special case depending on the choice of a polarization it parallels and strengthens the usual castelnuovo mumford regularity mukai regularity has a large number of applications ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties to higher dimensional type statements and to a study of special classes of vector bundles some of these applications are explained here while others are the subject of upcoming sequels

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

fat points on rational normal curves abstract in this paper we study 0 dimensional schemes z made of fat points in p n n 2 whose support lies on a rational normal curve we conjecture that the hilbert function of z does not depend on the choice of the points and we show this under some numerical hypotheses we also study the hilbert function of the infinitesimal neighborhoods of the rational normal curve and we find the value where it coincides with the hilbert polynomial

a geometric approach to the minimum sensitivity design problem in this paper a riemannian geometric framework is given for the minimum sensitivity design problem and its solution using a natural optimization criterion the theory is then applied to the case of linear systems to generate a class of minimum sensitivity realizations related to the so called balanced realizations in particular conditions are given which are applicable in the cases of fixed point and floating point implementations

generic projections the equations defining projective varieties and castelnuovo regularity for a reduced irreducible projective variety x of degree d and codimension e in mathbb p n the castelnuovo mumford regularity rm reg x is defined as the least k such that x is k regular i e h i mathbb p n mathcal i x k i 0 for i ge1 where mathcal i x subset cal o mathbb p n is the sheaf of ideals of x there is a long standing conjecture about k regularity see 5 rm reg x le d e 1 here we show that rm reg x le d e 1 10 for any smooth fivefold and rm reg x le d e 1 20 for any smooth sixfold by extending methods used in 10 furthermore we give a bound for the regularity of a reduced connected and equidimensional locally cohen macaulay curve or surface in terms of degree d codimension e and an arithmetic genus rho a see theorem 4 1

regularity and normality of the secant variety to a projective curve abstract for a smooth curve of genus g embedded by a line bundle of degree at least 2 g 3 we show that the ideal sheaf of the secant variety is 5 regular this bound is sharp with respect to both the degree of the embedding and the bound on the regularity further we show that the secant variety is projectively normal for the generic embedding of degree at least 2 g 3

bundle reduction and the alignment distance on spaces of state space lti systems this paper introduces a large class of differential geometric distances between finite dimensional linear dynamical systems collectively called the alignment distance contrary to the existing distances the alignment distance is based on the state space description of dynamical systems and is defined on the manifolds of systems of fixed order and fixed input output dimension under a matrix rank constraint e g minimality controllability or observability while the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory distances natural to this structure have not been studied the starting point for defining such a distance is to identify a linear system of order n with its equivalence class of state space realizations all related by the so called similarity action i e state space change of basis under gl n the lie group of nonsingular n times n matrices the main idea of the alignment distance is to first find the best state space change of basis that brings a realization of a system as close as possible to a realization of another system the alignment step and then compare the aligned realizations a direct implementation of this idea due to noncompactness of gl n is complicated however using the notion of reduction of the structure group of a principal bundle we show that the change of basis can be restricted to an orthogonal change of basis provided one uses realizations in a reduced subbundle this key observation brings about significant computational benefits as a technical contribution possibly of independent interest we show that several forms of realization balancing available in the control literature have differential geometric significance and are indeed examples of reducing the structure group from gl n to its subgroup of orthogonal matrices o n the alignment distance can be defined for stable and unstable systems discrete or continuous time and stochastic systems

filtering cohomology and lifting vector bundles for a module m over a local cohen macaulay ring r we develop a finite sequence of presentations of m which facilitates the study of invariants arising from the cohomology modules of m as an application we use this data in case r is regular and m represents a vector bundle on the punctured spectrum of r with a vanishing cohomology module to obtain bounds on how far m can be lifted as a vector bundle in an earlier article 5 we used the dual of a filtration of auslander and bridger 1 to obtain some results for nontrivial kth syzygies of rank k in this article we establish a different method for filtering a module over a local cohen macaulay ring or more precisely a method for filtering a free presentation of a module see discussion following theorem 1 8 this filtration is accomplished through a finite sequence of presentations to be specific let r be a local cohen macaulay ring and let m be a finitely generated r module such that depth ext m r i for all i less than or equal to the dimension of r in theorem 1 2 we establish an exact sequence which we term a q th presentation of m

multiparameter operators and sharp weighted inequalities this article is concerned with the operators from harmonic analysis which are naturally associated to a multiple parameter family of dilations we are especially interested here in dealing with questions from the theory of such operators whose answers cannot be obtained by a reduction to the case of product operators we also introduce a new tool in order to carry out this multiparameter analysis which comes from the classical theory this is the concept of sharp weighted inequalities where for operators such as the hilbert transform or classical square function one asks for hilbert space inequalities which imply the correct estimates of the norm on lp of these operators as p becomes large 0 introduction in this article our main goal is to study classes of oper ators associated with multiparameter groups of dilations on rn in some sense the consideration of these operators is a natural next step or simplest case after those of the classical calderon zygmund theory and the product space theory it turns out that in order to obtain estimates for the multiparameter operators we shall be led to the following type of extremely classical question how does one obtain l2 estimates on r1 for the hilbert transform or other singular integrals which yield sharp estimates on the lp rl operator norm of h as p approaches 1 or oc these sharp weighted inequalities are of some independent interest and their study will be seen to be essentially equivalent to the deep work of chang wilson and wolff 3 on the exponential square integrability of functions whose area integral is bounded they also have other applications for example they provide an immediate extension of the chang wilson wolff estimates to product spaces as given by pipher in 12 at this point let us be more specific about the exact nature of the multipa rameter operators which we shall consider consider in r3 the dilations given by ps t x y z sx ty9 stz for s91 0 then there is a maximal operator first considered by zygmund ml and singular integrals tl introduced by ricci and stein 13 naturally associated to these dilations the maximal operator af3 is defined by

minimal free resolutions and asymptotic behavior of multigraded regularity let s be a standard nk graded polynomial ring over a field k let i be a multigraded homogeneous ideal of s and let m be a finitely generated zk graded s module we prove that the resolution regularity a multigraded variant of castelnuovo mumford regularity of inm is asymptotically a linear function this shows that the well known z graded phenomenon carries to the multigraded situation

the small ball problem for the brownian sheet we show that the logarithm of the probability that the brownian sheet has a supremum at most e over 0 1 2 is of order e 2 log 1 e 3

cohomology of projective schemes from annihilators to vanishing we provide bounds on the castelnuovo mumford regularity in terms of defining equations by using elements that annihilates some cohomology modules inspired by works of miyazaki nagel schenzel and vogel the elements in these annihilators are provided either by liaison or by tight closure theories our results hold in any characteristic

higher linear syzygies of inner projections let x pr be a smooth projective variety embedded by the complete linear system associated to a very ample line bundle l on x in this paper we prove that if x satisfies property np for p 1 then every isomorphic inner projection x pr 1 of x satisfies np 1 we also give some applications of our results to various examples

effectivity of brauer manin obstructions we study brauer manin obstructions to the hasse principle and to weak approximation with special regard to effectivity questions

asymptotic behaviour of arithmetically cohen macaulay blow ups this paper addresses problems related to the existence of arithmetic macaulayfications of projective schemes let y be the blow up of a projective scheme x projr along the ideal sheaf of i r it is known that there are embeddings y projk i e c for c d i e 1 where d i denotes the maximal generating degree of i and that there exists a cohen macaulay ring of the form k i e c if and only if h 0 y oy k h i y oy 0 for i 1 dimy 1 y is equidimensional and cohen macaulay cutkosky and herzog asked when there is a linear bound on c and e ensuring that k i e c is a cohen macaulay ring we obtain a surprising compelte answer to this question namely that under the above conditions there are well determined invariants and e0 such that k i e c is cohen macaulay for all c d i e and e e0 our approach is based on recent results on the asymptotic linearity of the castelnuovo mumford regularity of ideal powers we also investigate the existence of a cohen macaulay rees algebra of the form r i e ct which provides an arithmetic macaulayfication for x if r has negative a invariant we prove that such a cohen macaulay rees algebra exists if and only if oy ox r i oy 0 for i 0 y is equidimensional and cohen macaulay especially these conditions imply the cohen macaulayness of r i e ct for all c d i e and e e0 the above results can be applied to obtain several new classes of cohen macaulay algebras

generic initial ideals of singular curves in graded lexicographic order abstract in this paper we are interested in the generic initial ideals of singular projective curves with respect to the graded lexicographic order let c be a singular irreducible projective curve of degree d 5 with the arithmetic genus a c in p r where r 3 if m i c is the regularity of the lexicographic generic initial ideal of i c in a polynomial ring k x 0 x r then we prove that m i c is 1 d 1 2 a c which is obtained from the monomial x r 3 x r 1 d 1 2 a c provided that dim tan p c 2 for every singular point p c this number is equal to one plus the number of secant lines through the center of general projection into p 2 our result generalizes the work of j ahn 2008 1 for smooth projective curves and that of a conca and j sidman 2005 9 for smooth complete intersection curves in p 3 the case of singular curves was motivated by a conca and j sidman 2005 9 example 4 3 we also provide some illuminating examples of our results via calculations done with macaulay 2 and singular decker et al 2011 10 grayson and stillman 16

probability estimates for multiparameter brownian processes soit f une fonction de repartition sur 0 1 d et w f le processus gaussien qui est la limite faible du processus empirique determine par f si g est une fonction sur 0 1 d on obtient des bornes superieures et inferieures pour p sup t 0 1 d w f t g t e

graded mapping cone theorem multisecants and syzygies abstract let x be a reduced closed subscheme in p n as a slight generalization of property n p due to green lazarsfeld one says that x satisfies property n 2 p scheme theoretically if there is an ideal i generating the ideal sheaf i x p n such that i is generated by quadrics and there are only linear syzygies up to p th step cf eisenbud et al 2005 8 vermeire 2001 20 recently many algebraic and geometric results have been proved for projective varieties satisfying property n 2 p cf choi kwak and park 2008 6 eisenbud et al 2005 8 kwak and park 2005 15 in this case the castelnuovo regularity and normality can be obtained by the blowing up method as reg x e 1 where e is the codimension of a smooth variety x cf bertram ein and lazarsfeld 2003 3 on the other hand projection methods have been very useful and powerful in bounding castelnuovo regularity normality and other classical invariants in geometry cf beheshti and eisenbud 2010 2 kwak 1998 14 kwak and park 2005 15 lazarsfeld 1987 16 we first prove the graded mapping cone theorem on partial eliminations as a general algebraic tool to study syzygies of the non complete embedding of x for applications we give an optimal bound on the length of zero dimensional intersections of x and a linear space l in terms of graded betti numbers we also deduce several theorems about the relationship between x and its projections with respect to the geometry and syzygies for a projective scheme x satisfying property n 2 p scheme theoretically in addition we give not only interesting information on the regularity of fibers of the projection for the case of n d p d 2 but also geometric structures for projections according to moving the center

on the small ball inequality in all dimensions abstract let h r denote an l normalized haar function adapted to a dyadic rectangle r 0 1 d we show that for choices of coefficients r we have the following lower bound on the l norms of the sums of such functions where the sum is over rectangles of a fixed volume n d 1 2 r 2 n r h r x l 0 1 d 2 n r 2 n r for some 0 1 2 the point of interest is the dependence upon the logarithm of the volume of the rectangles with n d 1 2 on the left above the inequality is trivial while it is conjectured that the inequality holds with n d 2 2 this is known in the case of d 2 michel talagrand the small ball problem for the brownian sheet ann probab 22 3 1994 1331 1354 mr 95k 60049 and a recent paper of two of the authors dmitriy bilyk michael t lacey on the small ball inequality in three dimensions duke math j 2006 in press arxiv math ca 0609815 proves a partial result towards the conjecture in three dimensions in this paper we show that the argument of dmitriy bilyk michael t lacey on the small ball inequality in three dimensions duke math j 2006 in press arxiv math ca 0609815 can be extended to arbitrary dimension we also prove related results in the subjects of the irregularity of distribution and approximation theory the authors are unaware of any prior results on these questions in any dimension d 4

asymptotic linear bounds for the castelnuovo mumford regularity we prove asymptotic linear bounds for the castelnuovo mumford regularity of certain filtrations of homogeneous ideals whose rees algebras need not be noetherian

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

metrical results on the discrepancy of halton kronecker sequences by a halton kronecker sequence we mean a sequence in the s t dimensional unit cube which is the combination of an s dimensional halton sequence and a t dimensional kronecker sequence the distribution of such sequences was studied for the first time quite recently by niederreiter in this paper we obtain metrical results for the discrepancy of halton kronecker sequences which are similar to results for the pure kronecker sequences obtained by khintchine and by w m schmidt

vanishing theorems a theorem of severi and the equations defining projective varieties in 1903 severi s proved that if x x c p are smooth surfaces whose union is the complete intersection of r 2 hypersurfaces of degrees di dr 2 then hypersurfaces of degree k z di r cut out a complete linear series on x cf sr xiii 9 8 the purpose of this paper is first of all to show that elementary arguments using the kodaira vanishing theorem lead to a simple variant of severi s statement theorem 7 below which extends it in serval directions more importantly we hope to convince the reader that this result has a surprising number of applications to questions involving the equations defining projective varieties consider to begin with a smooth complex projective variety x c pr of dimension n and codimension e r n in this setting our theorem asserts the vanishing of the higher cohomology of suitable twists of the ath power of the ideal sheaf yx of x in apr proposition 1 assume that x is cut out scheme theoretically in p by hypersurfaces of degrees d1 d2 d then h p r k 0 for i 1 provided k ad d2 de r note that only the degrees of the first e codim x pr defining equations come into play here when n 2 and a 1 this is a consequence of severi s

orlicz spaces bounds for special classes of hyperbolic sums abstract let r r 1 r d denote a dyadic rectangle in the unit cube 0 1 d d 3 let h r be the l normalized haar function supported on r in 10 the conjectured signed small ball inequality r 2 n r h r n d 2 where r 1 was proven under the additional assumption that the coefficients also satisfy the splitting property r r 1 r 2 r d with r 1 r 2 r d 1 we give another proof of this result using a duality argument based on this approach we also show r 2 n r h r exp l a n d 2 1 a 2 a for any integer n 1 and any choice of coefficients r 1 1 which satisfy the splitting property the above inequality has been conjectured for general coefficients r 1 1 in d 3 these bounds are investigated further for more general coefficients r 1 1 the proof of the sharpness of the l lower bound of hyperbolic sums with coefficients satisfying the splitting property is also provided

the power of online thinning in reducing discrepancy consider an infinite sequence of independent uniformly chosen points from 0 1 d after looking at each point in the sequence an overseer is allowed to either keep it or reject it and this choice may depend on the locations of all previously kept points however the overseer must keep at least one of every two consecutive points we call a sequence generated in this fashion a two thinning sequence here the purpose of the overseer is to control the discrepancy of the empirical distribution of points that is after selecting n points to reduce the maximal deviation of the number of points inside any axis parallel hyper rectangle of volume a from na our main result is an explicit low complexity two thinning strategy which guarantees discrepancy of o log 2d 1 n for all n with high probability compare with theta sqrt n log log n without thinning the case d 1 of this result answers a question of benjamini we also extend the construction to achieve the same asymptotic bound for 1 beta thinning a set up in which rejecting is only allowed with probability beta independently for each point in addition we suggest an improved and simplified strategy which we conjecture to guarantee discrepancy of o log d 1 n compare with theta log d n the best known construction of a low discrepancy sequence finally we provide theoretical and empirical evidence for our conjecture and provide simulations supporting the viability of our construction for applications

regularity of prime ideals we answer several natural questions which arise from a recent paper of mccullough and peeva providing counterexamples to the eisenbud goto regularity conjecture we give counterexamples using rees algebras and also construct counterexamples that do not rely on the mayr meyer construction furthermore examples of prime ideals for which the difference between the maximal degree of a minimal generator and the maximal degree of a minimal first syzygy can be made arbitrarily large are given using a result of ananyan hochster we show that there exists an upper bound on regularity of prime ideals in terms of the multiplicity alone

metric entropy and the small ball problem for gaussian measures abstract we establish a precise link between the small ball problem for a gaussian measure on a separable banach space and the metric entropy of the unit ball of the hubert space h generating this link allows us to compute small ball probabilities from metric entropy results and vice versa

a sharp bound for the castelnuovo mumford regularity of subspace arrangements abstract we show that the ideal of an arrangement of d linear subspaces of projective space is d regular in the sense of castelnuovo and mumford answering a question of b sturmfels in particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d

singularities of generic linkage of algebraic varieties let y be a generic link of a subvariety x of a nonsingular variety a we give a description of the grauert riemenschneider canonical sheaf of y in terms of the multiplier ideal sheaves associ ated to x and use it to study the singularities of y as the first application we give a criterion when y has rational singularities and show that log canonical threshold increases and log canonical pairs are preserved in generic linkage as another application we give a quick and simple liaison method to generalize the results of de fernex ein and chardin ulrich on the castelnuovo mumford regularity bound for a projective variety

small ball probabilities of gaussian fields lower bounds on the small ball probability are given for brownian sheet type gaussian fields as well as for general gaussian fields with stationary increments in d in particular a sharp bound is found for the fractional levy brownian fields

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

balanced canonical forms for system identification parameterizations of linear state space systems by balanced canonical forms are considered the structure in particular the topological properties of these parameterizations are investigated based on these results an algorithm for identification obtained by optimizing a likelihood type criterion function is proposed and analyzed in the algorithm both the estimation of real valued parameters and of integer parameters defining dynamic specification is incorporated the analysis of the algorithm emphasizes consistency results

finiteness of hilbert functions and bounds for castelnuovo mumford regularity of initial ideals bounds for the castelnuovo mumford regularity and hilbert coefficients are given in terms of the arithmetic degree if the ring is reduced or in terms of the defining degrees from this it follows that there exists only a finite number of hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension a good bound is also given for the castelnuovo mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field

asymptotic linear bounds of castelnuovo mumford regularity in multigraded modules abstract let a be a noetherian standard n graded algebra over an artinian local ring a 0 let i 1 i t be homogeneous ideals of a and m a finitely generated n graded a module we prove that there exist two integers k k such that reg i 1 n 1 i t n t m n 1 n t k k for all n 1 n t n

on projective varieties of codimension 2 one of the important developments of the early 1970 s in algebraic geometry was the discovery by barth 1 subsequently enlarged on by many others that submanifolds x of projective space whose codimension is small compared with their dimension look homologically like complete intersections this focused attention on the fact that no such x were known other than complete intersections and led hartshorne 4 to conjecture that x must in fact be a complete intersection if 2 codim x dim x hartshorne and independently barth and van de ven 3 proved that if the degree hence also the codimension of x are much smaller than its dimension then x is a complete intersection the purpose of this paper is to contribute to the program of proving that projective varieties of small codimension are complete intersections by proving the following

syzygies and singularities of tensor product surfaces of bidegree 2 1 let u h op1 p1 2 1 be a basepoint free four dimensional vector space the sections corresponding to u determine a regular map u p1 p1 p3 we study the associated bigraded ideal iu k s t u v from the standpoint of commutative algebra proving that there are exactly six numerical types of possible bigraded minimal free resolution these resolutions play a key role in determining the implicit equation for u p1 p1 via work of buse jouanolou 5 buse chardin 6 botbol 2 and botbol dickensteindohm 3 on the approximation complex z in four of the six cases iu has a linear first syzygy remarkably from this we obtain all differentials in the minimal free resolution in particular this allows us to explicitly describe the implicit equation and singular locus of the image

a bound for castelnuovo mumford regularity by double point divisors abstract let x p r be a non degenerate smooth projective variety of dimension n codimension e and degree d defined over an algebraically closed field of characteristic zero in this paper we first show that reg o x d e and classify the extremal and the next to extremal cases our result reduces the eisenbud goto regularity conjecture for the smooth case to the problem finding a castelnuovo type bound for normality it is worth noting that mccullough peeva recently constructed counterexamples to the regularity conjecture by showing that reg o x is not even bounded above by any polynomial function of d when x is not smooth for a normality bound in the smooth case we establish that reg x n d 2 1 which improves previous results obtained by mumford bertram ein lazarsfeld and noma finally by generalizing mumford s method on double point divisors we prove that reg x d 1 m where m is an invariant arising from double point divisors associated to outer general projections using double point divisors associated to inner projections we also obtain a slightly better bound for reg x under suitable assumptions

sharp square function inequalities for conditionally symmetric martingales let t be a conditionally symmetric martingale taking values in a hilbert space 1ei and let s i be its square function if vp is the smallest positive zero of the confluent hypergeometric function and ap is the largest positive zero of the parabolic cylinder function of parameter p then the following inequalities are sharp 3 vp iis llp 2 moreover the constants vp and ap for the cases mentioned above are also best possible for the marcinkiewicz paley inequalities for haar functions