multiplier ideals and modules on toric varieties a formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given variants for the multiplier module and test ideals are also treated

cross validation in survival analysis the predictive value of a statistical model is conceptually different from the explained variation in this paper we construct a measure of the predictive value of the cox proportional hazards model computed from the leave one out regression coefficients these coefficients can also be used to calculate a shrinkage factor which can be applied to improve the predictions and that can be used in r2 type measures of the proportion of explained variation our methods are illustrated by a study of chemotherapy for advanced ovarian cancer

on maximally selected chi square statistics two samples can be compared by selecting a cutpoint and then forming a 2 x 2 table of the numbers of observations above and below the cutpoint in each sample miller and siegmund 1982 biometrics 38 1011 1016 investigated asymptotic theory relating to the distribution of the standard chisquare statistic when the cutpoint is selected to maximize its value in a companion article halpern 1982 biometrics 38 1017 1023 studied the finite sample distribution of this maximally selected chi square statistic via simulation exact finite sample distribution theory derived from durbin s 1971 journal of applied probability 8 431 453 1973 distribution theory for tests based on the sample distribution function philadelphia siam combinatorial approach is presented here

the tight integral closure of a set of ideals abstract this paper celebrates the enormous contributions of david buchsbaum to commutative algebra homological algebra and representation theory with grateful appreciation for the inspiration he has provided for myself and so many others

length multiplicity and multiplier ideals let r m be an n dimensional regular local ring essentially of finite type over a field of characteristic zero given an m primary ideal a of r the relationship between the singularities of the scheme defined by a and those defined by the multiplier ideals j a c with c varying in q are quantified in this paper by showing that the samuel multiplicity of a satisfies e n n k n c n whenever j a c c m k 1 this formula generalizes an inequality on log canonical thresholds previously obtained by ein mustata and the author of this paper a refined inequality is also shown to hold for small dimensions and similar results valid for a generalization of test ideals in positive characteristics are presented

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

confidence intervals for the effect of a prognostic factor after selection of an optimal cutpoint when investigating the effects of potential prognostic or risk factors that have been measured on a quantitative scale values of these factors are often categorized into two groups sometimes an optimal cutpoint is chosen that gives the best separation in terms of a two sample test statistic it is well known that this approach leads to a serious inflation of the type i error and to an overestimation of the effect of the prognostic or risk factor in absolute terms in this paper we illustrate that the resulting confidence intervals are similarly affected we show that the application of a shrinkage procedure to correct for bias together with bootstrap resampling for estimating the variance yields confidence intervals for the effect of a potential prognostic or risk factor with the desired coverage

strong and weak f regularity are equivalent for graded rings it is shown that the tight closure of a submodule in a artinian module is the same as its finitistic tight closure when the modules are graded over a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i graded ring over a perfect field as a corollary it is deduced that for such a graded ring strong and weak f regularity are equivalent as another application the following conjecture of hochster and huneke is proved let r m be a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i graded ring over a field with unique homogeneous maximal ideal m then r is weakly f regular if and only if r m is weakly f regular

f singularities under generic linkage abstract let r k x 1 x n be a polynomial ring over a prefect field of positive characteristic let i be an equi dimensional ideal in r and let j be a generic link of i in s r u i j c r we describe the parameter test submodule of s j in terms of the test ideal of the pair r i when a reduction of i is a complete intersection or almost complete intersection as an application we deduce a criterion for when s j has f rational singularities in these cases we also compare the f pure threshold of r i and s j

practical p value adjustment for optimally selected cutpoints this paper concerns a series of simulations undertaken to examine the effects of two data features number of cutpoints and true marker prognostic effect size on three methods of p value adjustment asymptotic p acor improved bonferroni p bon and empirical permutation p emp h o rejection rates for p emp and p bon are almost indistinguishable from those for an independent validation sample p vld while those of p acor are somewhat conservative especially when the number of cutpoints is small analysis of a new breast cancer prognostic marker heat shock protein 70 illustrates the methods these results underscore many of the problems associated with data derived cutpoints in general and the need for p value adjustment

estimation in a cox regression model with a change point according to a threshold in a covariate we consider a nonregular cox model for independent and identically distributed right censored survival times with a change point according to the unknown threshold of a covariate the maximum partial likelihood estimators of the parameters and the estimator of the baseline cumulative hazard are studied we prove that the estimator of the change point is n consistent and the estimator of the regression parameters are n 1 2 consistent and we establish the asymptotic distributions of the estimators the estimators of the regression parameters and of the baseline cumulative hazard are adaptive in the sense that they do not depend on the knowledge of the change point

relative risk trees for censored survival data a method is developed for obtaining tree structured relative risk estimates for censored survival data the first step of a full likelihood estimation procedure is used in a recursive partitioning algorithm that adopts most aspects of the widely used classification and regression tree cart algorithm of breiman et al 1984 classification and regression trees belmont california wadsworth the performance of the technique is investigated through stimulation and compared to the tree structured survival methods proposed by davis and anderson 1989 statistics in medicine 8 947 961 and therneau grambsch and fleming 1990 biometrika 77 147 160

a generalization of tight closure and multiplier ideals we introduce a new variant of tight closure associated to any fixed ideal a which we call tight closure and study various properties thereof in our theory the annihilator ideal r a of all tight closure relations which is a generalization of the test ideal in the usual tight closure theory plays a particularly important role we prove the correspondence of the ideal a and the multiplier ideal associated to a or the adjoint of a in lipman s sense in normal q gorenstein rings reduced from characteristic zero to characteristic p 0 also in fixed prime characteristic we establish some properties of r a similar to those of multiplier ideals e g a briancon skoda type theorem subadditivity etc with considerably simple proofs and study the relationship between the ideal and the f rationality of rees algebras

f injective singularities are du bois in this paper we prove that singularities of f injective type are du bois this extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of frobenius in positive characteristic

frobenius and cartier algebras of stanley reisner rings we study the generation of the frobenius algebra of the injective hull of a complete stanley reisner ring over a field with positive characteristic in particular by extending the ideas used by m katzman to give a counterexample to a question raised by g lyubeznik and k e smith about the finite generation of frobenius algebras we prove that the frobenius algebra of the injective hull of a complete stanley reisner ring can be only principally generated or infinitely generated also by using our explicit description of the generators of such algebra and applying the recent work by m blickle about cartier algebras and generalized test ideals we are able to show that the set of f jumping numbers of generalized test ideals associated to complete stanley reisner rings form a discrete subset inside the non negative real numbers

survival trees by goodness of split abstract a tree based method for censored survival data is developed based on maximizing the difference in survival between groups of patients represented by nodes in a binary tree the method includes a pruning algorithm with optimal properties analogous to the classification and regression tree cart pruning algorithm uniform convergence of the estimates of the conditional cumulative hazard and survival functions is discussed and an example is given to show the utility of the algorithm for developing prognostic classifications for patients

tree stability diagnostics and some remedies for instability stability aspects of recursive partitioning procedures are investigated using resampling techniques diagnostic tools to assess single split stability and overall tree stability are introduced to correct for the procedure s preference for covariates with many unique realizations corrected p values are used in the factor selection component of the algorithm finally methods to stabilize tree based predictors are discussed

approximating the distribution of maximally selected mcnemar s statistics summary it is common in epidemiologic analyses to summarize continuous outcomes as falling above or below a threshold with paired data and with a threshold chosen without reference to the outcomes mcnemar s test of marginal homogeneity may be applied to the resulting dichotomous pairs when testing for equality of the marginal distributions of the underlying continuous outcomes if the threshold is chosen to maximize the test statistic however referring the resulting test statistic to the nominal x2 distribution is incorrect instead the p value must be adjusted for the multiple comparisons here the distribution of a maximally selected mcnemar s statistic is derived and it is shown that an approximation due to durbin 1985 journal of applied probability22 99 122 may be used to estimate approximate p values the methodology is illustrated by an application to measurements of insulin like growth factor i igf i in matched prostate cancer cases and controls from the physicians health study the results of simulation experiments that assess the accuracy of the approximation in moderate sample sizes are reported

on a non vanishing conjecture of kawamata and the core of an ideal we show under suitable hypothesis which are sharp in a certain sense that the core of an m primary ideal in a regular local ring of dimension d is equal to the adjoint or multiplier ideal of its d th power this generalizes the fundamental formula for the core of an integrally closed ideal in a two dimensional regular local ring due to huneke and swanson we also find a generalization of this result to singular non regular settings which we show to be intimately related to the problem of finding non zero sections of ample line bundles on projective varieties in particular we show that a graded analog of our formula for core would imply a remarkable conjecture of kawamata predicting that every adjoint ample line bundle on a smooth variety admits a non zero section

maximally selected chi square statistics two samples can be compared by selecting a cut point and then forming a 2 x 2 table of the numbers of observations above and below the cut point in each sample when the cut point is selected so as to maximize the standard chi square statistic the chi square percentile points are inappropriate actual significance levels are computed for large samples and correct percentile poillts are tabulated

correlations in multifractals we present an analysis of the implicit assumptions that lie at the basis of the scaling relations for fractal measures proposed by cates et al 1 furthermore the scaling relations which have been shown to hold for trivial multifractals like the generalized cantor set 2 are tested numerically for critical wave functions of an incommensurate system

drops in joint hilbert kunz multiplicities and projective equivalence abstract given two m primary ideals a and i in a noetherian local or graded ring we introduce the level ideals of i with respect to a as a measure of the asymptotic growth of the drop in the joint hilbert kunz multiplicity e hk a t i e hk a t i x we study the properties that must hold for two ideals a b to determine the same level ideals for all i and we show that this is related to projective equivalence

two point statistics of multifractal measures abstract the relationship between the function of a multifractal and the spatial correlations of its singularity strengths is examined an expression to compute the probability of observing two different singularities and within a distance r is derived for measures arising from isotropic random multiplicative processes the correlation of s is shown to decay logarithmically with distance possible applications to turbulence models are discussed and the results are illustrated for a binomial measure where the scaling of two point correlation functions is shown to exhibit a phase transition

f regularity does not deform we show that the property of f regularity does not deform and thereby settle a long standing open question in the theory of tight closure specifically we construct a three dimensional graded domain r which is not f regular or even f pure but has a quotient r tr which is f regular examples are constructed over fields of characteristic p 0 as well as over fields of characteristic zero 1 introduction throughout this paper all rings are commutative noethe rian and have an identity element the theory of tight closure was developed by melvin hochster and craig huneke in hh2 and draws attention to rings which have the property that all their ideals are tightly closed called weakly f regular rings the term f regular is reserved for rings all of whose localizations are weakly f regular a natural question that arose with the development of the the ory was whether the property of f regularity deforms i e if r m k is a local ring such that r tr is f regular for some nonzerodivisor t m m ustr be f regular see the epilogue of ho hochster and huneke showed that this is indeed true if the ring r is gorenstein hh3 and their work has been followed by various attempts at extending this result see akm si sm3 our pri mary goal here is to settle this question by constructing a family of examples to show that f regularity does not deform we shall throughout be considering graded rings but local examples can be obtained in all cases by localizing at the homogeneous maximal ideals our main result is theorem 1 1 there exists an graded ring r of dimension three finitely generated over a field r 0 k of characteristic p 2 which is not f pure but has an f regular quotient r tr where t m is a homogeneous nonzerodivisor specifically for positive integers m and n satisfying m m n 2 consider the ring r k a b c d t i where i is generated by the size two minors of the

maximally selected x2 statistics for kx 2 tables summary it is common in epidemiologic analyses to summarize continuous outcomes as falling above or below a threshold with such a dichotomized outcome the usual x2 statistics for association or trend can be used to test for equality of proportions across strata of the study population however if the threshold is chosen to maximize the test statistic the nominal x2 reference distributions are incorrect in this paper the asymptotic distributions of maximally selected x2 statistics for association and for trend for the k 2 table are derived the methodology is illustrated with data from an aids clinical trial the results of simulation experiments that assess the accuracy of the asymptotic distributions in moderate sample sizes are also reported

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

discreteness and rationality of f jumping numbers on singular varieties we prove that the f jumping numbers of the test ideal tau x delta mathfrak a t are discrete and rational under the assumptions that x is a normal and f finite scheme over a field of positive characteristic p kx is mathbb q cartier of index not divisible p and either x is essentially of finite type over a field or the sheaf of ideals mathfrak a is locally principal this is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero

on the discreteness and rationality of f jumping coefficients abstract we prove that the f jumping coefficients of a principal ideal of an excellent regular local ring of characteristic p 0 are all rational and form a discrete subset of r

weak f regularity deforms in q gorenstein rings weakly f regular inother words does the property of weak f regularity deform it has beenshown by singh that in general weak f regularity does not deform si2 wxit is known that f rationality deforms hh2 4 2 and the proof iswxstraightforward however the same difficulties that arise when trying to

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

formulas for multiplier ideals on singular varieties we prove a generalization of demailly ein lazarsfeld s subadditivity formula and musta s summation formula for multiplier ideals to the case of singular varieties using characteristic p methods as an application of our formula we improve hochster huneke s result on the growth of symbolic powers of ideals in singular affine algebras

f signature of pairs and the asymptotic behavior of frobenius splittings abstract we generalize f signature to pairs r d where d is a cartier subalgebra on r as defined by the first two authors in particular we show the existence and positivity of the f signature for any strongly f regular pair in one application we answer an open question of aberbach and enescu by showing that the f splitting ratio of an arbitrary f pure local ring is strictly positive furthermore we derive effective methods for computing the f signature and the f splitting ratio in the spirit of the work of r fedder

the arc space of a toric variety abstract the nash problem on arc families is affirmatively answered for a toric variety by ishii and kollar s paper which also shows the negative answer for general case the nash problem is one of questions about the relation between arc families and valuations in this paper the relation is described clearly for a toric variety the arc space of a toric variety admits an action of the group scheme determined by the torus each orbit on the arc space corresponds to a lattice point in the cone and therefore corresponds to a toric valuation the dominant relation among the orbits is described in terms of the lattice points as a corollary we obtain the answer to the embedded version of the nash problem for an invariant ideal on a toric variety

adjoint ideals along closed subvarieties of higher codimension in this paper we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using char acteristic p methods when x is a normal gorenstein closed subvariety of a smooth complex variety a we formulate a restriction property of the adjoint ideal sheaf adj x a of a along x involving the l c i ideal sheaf dx of x the proof relies on a modification of generalized test ideals of hara and yoshida 11

f singularities of pairs and inversion of adjunction of arbitrary codimension we generalize the notions of f regular and f pure rings to pairs r mathfrak a t of rings r and ideals mathfrak a subset r with real exponent t 0 and investigate these properties these f singularities of pairs correspond to singularities of pairs of arbitrary codimension in birational geometry via this correspondence we prove a sort of inversion of adjunction of arbitrary codimension which states that for a pair x y of a smooth variety x and a closed subscheme y subsetneq x if the restriction z y z to a normal gorenstein closed subvariety z subsetneq x is klt resp lc then the pair x y z is plt resp lc near z

castelnuovo mumford regularity and the discreteness of f jumping coefficients in graded rings in this paper we show that the sets of f jumping coefficients of ideals form discrete sets in certain graded f finite rings we do so by giving a criterion based on linear bounds for the growth of the castelnuovo mumford regularity of certain ideals we further show that these linear bounds exists for one dimensional rings and for ideals of most two dimensional domains we conclude by applying our technique to prove that all sets of f jumping coefficients of all ideals in the determinantal ring given as t he quotient by 2 2 minors in a 2 3 matrix of indeterminates form discrete sets

on the constancy regions for mixed test ideals abstract in this note we study the partition of r 0 n given by the regions where the mixed test ideals a 1 t 1 a n t n are constant we show that each region can be described as the preimage of a natural number under a p fractal function r 0 n n in addition we give some examples illustrating that these regions do not need to be composed of finitely many rational polytopes

the f pure threshold of a calabi yau hypersurface we compute the f pure threshold of the affine cone over a calabi yau hypersurface and relate it to the order of vanishing of the hasse invariant on the versal deformation space of the hypersurface

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

maximally selected rank statistics a common statistical problem is the assessment of the predictive power of a quantitative variable for some dependent variable a maximally selected rank statistic regarding the quantitative variable provides a test and implicitly an estimate of a cutpoint as a simple classification rule restricting the selection to an arbitrary given inner part of the support of the quantitative variable we show that the asymptotic null distribution of the maximally selected rank statistic is the distribution of the supremum of the absolute value of a standardized gaussian process on an interval the asymptotic argument holds also in the case of tied or censored observations we compare monte carlo results with an approximation of the asymptotic distribution under the null hypothesis in addition we investigate the behaviour of the test procedure and of the familiar spearman rank test for independence under some alternatives moreover we discuss some aspects of the problem of estimating an underlying cutpoint

globalizing f invariants abstract in this paper we define and study the global hilbert kunz multiplicity and the global f signature of prime characteristic rings which are not necessarily local our techniques are made meaningful by extending many known theorems about hilbert kunz multiplicity and f signature to the non local case

when does the subadditivity theorem for multiplier ideals hold demailly ein and lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals we prove that in the two dimensional case the subadditivity theorem holds on log terminal singularities however in the higher dimensional case we have several counterexamples we consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three dimensional toric ring

on f pure thresholds using the frobenius map we introduce a new invariant for a pair r a of a ring r and an ideal a r which we call the f pure threshold c a of a and study its properties we see that the f pure threshold characterizes several ring theoretic properties by virtue of hara and yoshida s result trans amer math soc 355 2003 3143 the f pure threshold c a in characteristic zero corresponds to the log canonical threshold lc a which is an important invariant in birational geometry using the f pure threshold we prove some ring theoretic properties of three dimensional terminal singularities of characteristic zero also in fixed prime characteristic we establish several properties of the f pure threshold similar to those of the log canonical threshold with quite simple proofs

d modules bernstein sato polynomials and f invariants of direct summands abstract we study the structure of d modules over a ring r which is a direct summand of a polynomial or a power series ring s with coefficients over a field we relate properties of d modules over r to d modules over s we show that the localization r f and the local cohomology module h i i r have finite length as d modules over r furthermore we show the existence of the bernstein sato polynomial for elements in r in positive characteristic we use this relation between d modules over r and s to show that the set of f jumping numbers of an ideal i r is contained in the set of f jumping numbers of its extension in s as a consequence the f jumping numbers of i in r form a discrete set of rational numbers we also relate the bernstein sato polynomial in r with the f thresholds and the f jumping numbers in r

depths of multiplier ideals and integral closure in this note we study how the depths of multiplier ideals behave under restriction we also study possible values of the depths of multiplier ideals in the filtrations induced from maximal ideal sheaves we then use it to give a sufficient condition for the integral closedness of the product of a multiplier ideal and a power of maximal ideal sheaf in the spirit of huneke

f singularities via alterations we give characterizations of test ideals and f rational singularities via regular alterations formally the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions lastly we establish nadel type vanishing theorems up to finite maps for test ideals and further demonstrate how these vanishing theorems may be used to extend sections

rees algebras of f regular type abstract we study the f regularity of rees algebras r i a it in terms of the global f regularity of the blowing up x proj r i of spec a as it reads global f regularity is a global analog of strong f regularity defined via splitting of frobenius maps in prime characteristic and these notions are extended to characteristic zero by reduction modulo p 0 we study in detail the case where a m is a two dimensional local ring and i is an m primary ideal in characteristic zero the condition for r i to have f regular type is described in terms of the dual graph of a resolution x on which i o x is invertible we also prove some miscellaneous results concerning singularities of rees algebras and extended rees algebras of higher dimension

f signature exists suppose r is a d dimensional reduced f finite noetherian local ring with prime characteristic p 0 and perfect residue field let r 1 p e be the ring of p e th roots of elements of r for e and let a e denote the maximal rank of a free r module appearing in a direct sum decomposition of r 1 p e we show the existence of the limit s r lim e to infty frac a e p ed called the f signature of r this invariant which can be extended to all local rings in prime characteristic was first formally defined by c huneke and g leuschke in math ann 324 2 391 404 2002 and has previously been shown to exist only in special cases the proof of our main result is based on the development of certain uniform hilbert kunz estimates of independent interest additionally we analyze the behavior of the f signature under finite ring extensions and recover explicit formulae for the f signatures of arbitrary finite quotient singularities

predictive value of statistical models a review is given of different ways of estimating the error rate of a prediction rule based on a statistical model a distinction is drawn between apparent optimum and actual error rates moreover it is shown how cross validation can be used to obtain an adjusted predictor with smaller error rate a detailed discussion is given for ordinary least squares logistic regression and cox regression in survival analysis finally the splitsample approach is discussed and demonstrated on two data sets

on the existence of f thresholds and related limits we show the existence of f thresholds in full generality in addition we study properties of standard graded algebras over a field for which f pure threshold and f threshold at the irrelevant maxi

some results on multifractal correlations abstract we have performed a numerical and analytical study of multifractal correlations in a two scale random cantor set at large distances for not very large we find qualitative agreement between canonical simulation results also counting results and the formula recently proposed by meneveau and chhabra following work of cates and deutsch for large a new correlation scaling function which is linear in log r is obtained due to a transition of the moment scaling function from cates and deutsch behavior to a differing behavior for some part of its domain this linear is consistent with numerical results

a canonical linear system associated to adjoint divisors in characteristic p 0 suppose that x is a projective variety over an algebraically closed field of characteristic p 0 further suppose that l is an ample or more generally in some sense positive divisor we study a natural linear system in h 0 x ox kx l we further generalize this to incorporate a boundary divisor we show that these subsystems behave like the global sections associated to multiplier ideals h 0 x j x l in characteristic zero in particular we show that these systems are in many cases base point free while the original proof utilized kawamata viehweg vanishing and variants of multiplier ideals our proof uses test ideals

test multiplier and invariant ideals abstract this paper gives an explicit formula for the multiplier ideals and consequently for the log canonical thresholds of any gl v gl w invariant ideal in s sym v w where v and w are vector spaces over a field of characteristic 0 this characterization is done in terms of a polytope constructed from the set of young diagrams corresponding to the schur modules generating the ideal our approach consists in computing the test ideals of some invariant ideals of s in positive characteristic namely we compute the test ideals and so the f pure thresholds of any sum of products of determinantal ideals not all the invariant ideals are as the latter not even in characteristic 0 but they are up to integral closure and this is enough to reach our goals the results concerning the test ideals are obtained as a consequence of general results holding true in a special situation within such framework fall determinantal objects of a generic matrix as well as of a symmetric matrix and of a skew symmetric one similar results are thus deduced for the gl v invariant ideals in sym sym 2 v and in sym 2 v monomial ideals also fall in this framework thus we recover howald s formula for their multiplier ideals and more generally hara yoshida s formula for their test ideals in the proof we introduce the notion of floating test ideals a property that in a sense is satisfied by ideals defining schemes with the nicest possible singularities as will be shown products of determinantal ideals and by passing to characteristic 0 ideals generated by a single schur module have this property

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

big tight closure test elements for some non reduced excellent rings abstract this paper is concerned with existence of big tight closure test elements for a commutative noetherian ring r of prime characteristic p let r denote the complement in r of the union of the minimal prime ideals of r a big test element for r is an element of r which can be used in every tight closure membership test for every r module and not just the finitely generated ones the main results of the paper are that if r is excellent and satisfies condition r 0 and c r is such that r c is gorenstein and weakly f regular then some power of c is a big test element for r if i r is a homomorphic image of an excellent regular ring of characteristic p for which the frobenius homomorphism is intersection flat or ii r is f pure or iii r is local the gamma construction is not used

f thresholds of hypersurfaces for every 0 that is f finite i e such that the frobenius morphism f r r is finite if a is an ideal in r and if is a nonnegative real number then the corresponding test ideal is denoted by a in this context we say that is an f jumping exponent or an f threshold if a 6 a for every the following is our main result about f jumping exponents in positive characteristic theorem 1 1 if r is an f finite regular ring and if a f is a principal ideal then the f jumping exponents of a are rational and they form a discrete set the discreteness and the rationality of f jumping numbers has been proved in bms for every ideal when the ring r is essentially of finite type over an f finite field we mention also that for r k x y with k a finite field the above result has been proved in ha using a completely different approach we stress that the difficulty in attacking this result does notcome from the fact that there is no available resolution of singularities in positive characteristic even

bernstein sato polynomials in positive characteristic abstract in characteristic zero the bernstein sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an euler operator on a suitable d module we consider analogous d modules in positive characteristic and use them to define a sequence of bernstein sato polynomials corresponding to the fact that we need to consider also divided powers euler operators we show that the information contained in these polynomials is equivalent to that given by the f jumping exponents of the hypersurface in the sense of hara and yoshida n hara k i yoshida a generalization of tight closure and multiplier ideals trans amer math soc 355 2003 3143 3174

f rationality of rees algebras in this paper using the notion of the tight integral closure we will give a criterion for f rationality of rees algebras of m primary ideals in a cohen macaulay local ring as its application we prove the following results 1 in dimension two if a is f rational and i is integrally closed then the rees algebra r i is f rational on the other hand in higher dimensions we construct many examples of cohen macaulay normal rees algebras which are not f rational 2 if both a and r i are f rational then so is the extended rees algebra r i 3 if r i is f rational and a g i 1 then a is f rational on the other hand using resolution of singularities we will prove that a two dimensional rational singularity always admits f rational rees algebras in particular this theorem gives another way than that devised by watanabe 1997 j pure appl algebra122 323 328 to construct counterexamples to the boutot type theorem for f rational rings