desingularization of quasi excellent schemes in characteristic zero grothendieck proved in ega iv that if any integral scheme of finite type over a locally noetherian scheme x admits a desingularization then x is quasi excellent and conjectured that the converse is probably true we prove this conjecture for noetherian schemes of characteristic zero namely starting with the resolution of singularities for algebraic varieties of characteristic zero we prove the resolution of singularities for noetherian quasi excellent q schemes

uniformization of analytic spaces in this article we give an explicit elementary proof of a local version of resolution of singularities in characteristic zero local means that the centres of blowing up are chosen locally so that a finite number of finite sequences of local blowings up may be required to cover a neighbourhood of a given point hence we use the term uniformization rather than local desingularization although uniformization in our sense is considerably stronger than the original idea of zariski 16 in the last two decades several mathematicians notably abhyankar 1 hironaka and spivakovsky 12 and youssin 14 15 have proposed simpler or more explicit versions of the inductive procedure in hironaka s proof of his great theorem 8 these approaches would seem to lead to some form of uniformization though full details of none of them have yet appeared the idea of seeking an explicit local procedure to determine the centres of blowing up is that a sufficiently good local choice should globalize automatically our method for example gives global resolution of singularities of surfaces in any codimension throughout this paper k denotes either the field of real numbers r or the field of complex numbers c we will work in the category of analytic spaces over k although our techniques apply as well to algebraic spaces over a field

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

some irreducible hilbert schemes number thus h a parametrizes subschemes of if r consisting of finitely many points there are some examples of irreducible hilbert schemes of curves of low genus and degree cp 5 but the only classes of irreducible hilbert schemes which are mentioned in the literature seem to be those of plane curves embedded in some p and of hypersurfaces cp 1 thm 1 4 and 7 prop 3 the aim of the paper is to find a class of examples which in some sense is a mixture of these two classes in order to state the result we have to recall a theorem of f s macaulay according to which the hilbert scheme h e is non empty if and only if

sklyanin algebras and hilbert schemes of points we construct projective moduli spaces for torsion free sheaves on noncommutative projective planes these moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual commutative projective plane p2 the generic noncommutative plane corresponds to the sklyanin algebra s skl e constructed from an automorphism of infinite order on an elliptic curve e p2 in this case the fine moduli space of line bundles over s with first chern class zero and euler characteristic 1 n provides a symplectic variety that is a deformation of the hilbert scheme of n points on p2 e

hilbert series of modules of gk dimension two over elliptic algebras abstract we characterize the hilbert functions and minimal resolutions of critical cohen macaulay graded right modules of gelfand kirillov dimension two over generic quadratic and cubic three dimensional artin schelter regular algebras see also y berest g wilson ideal classes of the weyl algebra and noncommutative projective geometry with an appendix by michel van den bergh int math res not 2002 26 2002 1347 1396 l le bruyn moduli spaces for right ideals of the weyl algebra j algebra 172 1995 32 48 t a nevins j t stafford sklyanin algebras and hilbert schemes of points math ag 0310045 2003 8 14 15

standard bases of perfect homogeneous polynomial ideals of height 2 this paper concerns minimal systems of homogeneous generators i e standard bases of homogeneous ideals z in a commutative polynomial ring k with coefficients in a field k a basic numerical question about a standard basis b of z is to determine its cardinality or more generally the number of the elements in b of degree t for each t 0 this number depends only on z and not on the choice of b and will be denoted by v z in this paper we obtain a sharp estimate for the numbers v p when p is a perfect homogeneous ideal of height 2 in k precisely in 2 1 we provide upper and lower bounds for v p in terms of the hilbert function h h k p of k p and of an integer i which ranges in an interval defined by h then in 3 3 we show the sharpness of these bounds in fact we prove a more general result by exhibiting examples of ideals p with given h and i having for each degree t one of the possible values v p expected from 2 1 finally in 4 1 we modify the preceding examples and obtain under certain hypotheses radical ideals with the same properties the preceding results offer refined versions of well known theorems in fact from 2 1 we obtain cf 2 3 a more complete form of dubreil s inequalities cf 2 11 moreover 3 3 and 4 1 improve in the particular hypotheses under consideration a classic result of macaulay cf lo 143 and its reduced version cf 6 11 131 finally particular cases of our estimates in 2 1 can be found in 7 8 5 121 for points in generic position in the projective plane in 1 for hilbert function completeintersections and in 4 for monomial ideals of height 2

technical allocative and overall efficiency estimation and inference abstract nonparametric data envelopment analysis and free disposal hull estimators are frequently used to estimate cost revenue and profit efficiency as well as the corresponding allocative efficiencies papers in the literature often report sample means of such estimates along with sample standard deviations inviting readers to make inference about means of these efficiencies using classical methods based on the standard lindeberg feller central limit theorem clt a number of papers explicitly make inference using the classical methods however the statistical properties of these estimators are until now unknown this paper establishes rates of convergence and existence of limiting distributions for the various estimators these properties are needed in order to make inference about individual producers using subsampling methods in addition properties of the first two moments of the estimators are derived and these results are subsequently used to establish new clts for the estimators providing formal justification for inference making the results reveal that the classical clts and methods do not provide valid inference when fdh estimators are used and provide valid inference when dea estimators only in a few restrictive special cases

the general hyperplane section of a curve in this paper we discuss some necessary and sufficient conditions for a curve to be arithmetically cohen macaulay in terms of its general hyperplane section we obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non arithmetically cohen macaulay curve of p 3 we prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non arithmetically cohen macaulay curve of p 3 arise also as degree matrices of some smooth integral non arithmetically cohen macaulay curve and we characterize the exceptions we give a necessary condition on the graded betti numbers of the general plane section of an arithmetically buchsbaum non arithmetically cohen macaulay curve in p for curves in p 3 we show that any set of betti numbers that satisfies that condition can be realized as the betti numbers of the general plane section of an arithmetically buchsbaum non arithmetically cohen macaulay curve we also show that the matrices that arise as a degree matrix of the general plane section of an arithmetically buchsbaum integral smooth non arithmetically cohen macaulay space curve are exactly those that arise as a degree matrix of the general plane section of an arithmetically buchsbaum non arithmetically cohen macaulay space curve and have positive subdiagonal we also prove some bounds on the dimension of the deficiency module of an arithmetically buchsbaum space curve in terms of the degree matrix of the general plane section of the curve and we prove that they are sharp

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

hypersurface singularities in positive characteristic we present results on multiplicity theory differential operators on smooth schemes play a central role in the study of the multiplicity of an embedded hypersurface at a point this follows from the fact that the multiplicity is defined by the taylor development of the defining equation at such point and the taylor development involves higher order differentials on the other hand the multiplicity of a hypersurface can be expressed in terms of general projections defined at etale neighborhoods of such point given a local embedding of a d 1 dimensional hypersurface in a smooth d dimensional scheme a general projection on a smooth d 1 dimensional scheme is a finite map on the hypersurface and the multiplicity at the point is the degree of this finite map we relate both approaches in fact we study invariants of embedded hypersurfaces defined in terms of differential operators which express properties of the ramification of the morphism of particular interest is the case of hypersurfaces over fields of positive characteristic a central result in multiplicity theory of hypersurfaces is a form of elimination of one variable in the description of highest multiplicity locus this form of elimination known over fields of characteristic zero is achieved with the notion of tschirnhausen polynomial introduced by abhyankar this is a key point in the proof of embedded desingularization in this paper we provide a characteristic free approach to this form of elimination our alternative approach is based on projections on smooth d 1 dimensional schemes the properties of this new form of elimination remain weaker in positive characteristic than it does in characteristic zero when it comes to compatibility of elimination with permissible monoidal transformation this opens the way to new questions on resolution problems we also discuss the behavior of invariants attached to a singularity at a point with this form of elimination

parametrizations of ideals in k x y and k x y z abstract we parametrize the affine space of artinian affine ideals of k x y which have a given initial ideal with respect to the degree reverse lexicographic term order the fact that the term order is degree compatible allows us to extend the parametrization to the projective case namely zero dimensional subschemes of p 2 with some extra assumption and to determine the betti strata of these grobner cells this allows us to prove a formula due to a iarrobino for the codimension of the betti strata of codimension two punctual schemes in p 2

algebras associated to elliptic curves this paper completes the classification of artin schelter regular algebras of global dimension three for algebras generated by elements of degree one this has been achieved by artin schelter tate and van den bergh we are therefore concerned with algebras which are not generated in degree one we show that there exist some exceptional algebras each of which has geometric data consisting of an elliptic curve together with an automorphism just as in the case where the algebras are assumed to be generated in degree one in particular we study the elliptic algebras a a and a a where a p2 which were first defined in an earlier paper we omit a set s p2 consisting of 11 specified points where the algebras a a become too degenerate to be regular theorem let a represent a a or a a where a p2 s then a is an artin schelter regular algebra of global dimension three moreover a is a noetherian domain with the same hilbert series as the appropriately graded commutative polynomial ring in three variables this combined with our earlier results completes the classification

good points and constructive resolution of singluarities in this paper we present a new algorithm of resolution of singularities over fields of characteristic zero making use of invariants that come from abhyankar s good point theory abl we also prove new properties on constructive or algorithmic desingularization let us explain what we mean by an algorithm of resolution consider a pair x w where w is a regular variety over a base field k of characteristic zero not necessarily irreducible i e w smooth over k and xc w is a closed non empty subscheme call c the class of all such pairs over different base fields k natural maps i x w1 x w arise within this class for instance if w1 w is a smooth map over a fixed field k or if w1 w arises from an arbitrary change of base field fix now a totally ordered set i and suppose assigned for each pair 7 x w of u a function p x i which is upper semi continuous and takes only finitely many values say 1 c r c i let max bp be the biggest ai and set

non commutative mori contractions and p1 bundles abstract we give a method for constructing maps from a non commutative scheme to a commutative projective curve with the aid of artin zhang s abstract hilbert schemes this is used to construct analogues of the extremal contraction of a k negative curve with self intersection zero on a smooth projective surface this result will hopefully be useful in studying artin s conjecture on the birational classification of non commutative surfaces as a non trivial example of the theory developed we look at non commutative ruled surfaces and more generally at non commutative p 1 bundles we show in particular that non commutative p 1 bundles are smooth have well behaved hilbert schemes and we compute its serre functor we then show that non commutative ruled surfaces give examples of the aforementioned non commutative mori contractions

global resolution of singularities subordinated to a 1 dimensional foliation abstract let m be an analytic manifold over r or c a 1 dimensional log canonical resp monomial singular distribution and i a coherent ideal sheaf defined on m we prove the existence of a resolution of singularities for i that preserves the log canonicity resp monomiality of the singularities of furthermore we apply this result to provide a resolution of a family of ideal sheaves when the dimension of the parameter space is equal to the dimension of the ambient space minus one

linear equations over noncommutative graded rings abstract we call a graded connected algebra r effectively coherent if for every linear equation over r with homogeneous coefficients of degrees at most d the degrees of generators of its module of solutions are bounded by some function d d for commutative polynomial rings this property has been established by hermann in 1926 we establish the same property for several classes of noncommutative algebras including the most common class of rings in noncommutative projective geometry that is strongly noetherian rings which includes noetherian pi algebras and sklyanin algebras we extensively study so called universally coherent algebras that is such that the function d d is bounded by 2d for d 0 for example finitely presented monomial algebras belong to this class as well as many algebras with finite groebner basis of relations

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

resolution except for minimal singularities ii the case of four variables in this sequel to resolution except for minimal singularities i we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings the main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities

betti strata of height two ideals abstract let r k x y denote the polynomial ring in two variables over an infinite field k we study the betti strata of the family g h parametrizing graded artinian quotients of r k x y having given hilbert function h the betti stratum g h parametrizes all quotients a of having the graded betti numbers determined by h and the minimal relation degrees with i dim k tor 1 r i k i we recover that the betti strata are irreducible and we calculate their codimension in the family g h theorem the codimension of g h in g h satisfies letting i generators of degree i and i relations of degree i cod g h i i i here min i h i i 1 the initial degree of the ideals when k is algebraically closed we also show that the closure of a betti stratum is the union of more special strata and is cohen macaulay theorem 2 18 our method is to identify g h as a product of determinantal varieties ours is a more direct argument than that of a iarrobino v kanev power sums gorenstein algebras and determinantal loci lecture notes in math vol 1721 springer heidelberg 1999 345 xxvii pp theorem 5 63 which obtains the codimension of betti substrata of a postulation stratum for punctual subschemes of p 2 there we relied on a result of m boij determining the codimension of betti strata of height three gorenstein algebras m boij betti number strata of the space of codimension three gorenstein artin algebras preprint 2001 key tools throughout include properties of an invariant v the number of generators of the ancestor ideal v r and previous results concerning the projective variety g h in a iarrobino punctual hilbert schemes mem amer math soc 10 188 1977 we adapt a method of m boij that reduces the calculation of the codimension of the betti strata to showing that the most special stratum has the right codimension as applications we determine which hilbert functions are possible for artinian quotients of r given the socle type theorem 3 2 and we also determine the hilbert function of the intersection of t general enough level ideals in r each having a specified hilbert function theorem 3 6

equivalence and resolution of singularities abstract this article provides a simple presentation of an algorithm to resolve singularities of algebraic varieties over fields of characteristic zero by means of a sequence of blowing ups with smooth centers contained in the set of points of maximum multiplicity the algorithm uses primarily multiplicity rather than the hilbert samuel function to control the resolution process and it does not involve a local embedding into a smooth variety the paper introduces a generalization of the usual notion of equivalence in the theory of resolution of singularities which is important to justify an essential step in the construction of the algorithm

ideals of cubic algebras and an invariant ring of the weyl algebra hasselt univ dept wni b 3590 diepenbeek belgium univ antwerp dept wiskunde informat b 2020 antwerp belgium de naeghel k hasselt univ dept wni agoralaan gebouw d b 3590 diepenbeek belgium koendenaeghel hotmail com nicolas marconnet ua ac be

the gotzmann coefficients of hilbert functions abstract in this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the hilbert function of the general hyperplane section of a variety green s hyperplane restriction theorem these geometric consequences improve some results in this direction first given by green and extend others by bigatti geramita and migliore other applications of our detailed investigation of how the hilbert polynomial is written as a sum of binomials are to conditions that must be satisfied by a polynomial if it is to be the hilbert polynomial of a non degenerate integral subscheme of p n a problem posed by r p stanley we also give some new restrictions on the hilbert function of a zero dimensional reduced scheme with the uniform position property

families of artinian and one dimensional algebras abstract the purpose of this paper is to study families of artinian or one dimensional quotients of a polynomial ring r with a special look to level algebras let gradalg h r be the scheme parametrizing graded quotients of r with hilbert function h let b a be any graded surjection of quotients of r with hilbert function h b 1 h 1 h j and h a respectively if dim a 0 respectively dim a depth a 1 and a is a truncation of b in the sense that h a 1 h 1 h j 1 0 0 respectively h a 1 h 1 h j 1 for some h j then we show there is a close relationship between gradalg h a r and gradalg h b r concerning e g smoothness and dimension at the points a and b respectively provided b is a complete intersection or provided the castelnuovo mumford regularity of a is at least 3 sometimes 2 larger than the regularity of b in the complete intersection case we generalize this relationship to non truncated artinian algebras a which are compressed or close to being compressed for more general artinian algebras we describe the dual of the tangent and obstruction space of graded deformations in a manageable form which we make rather explicit for level algebras of cohen macaulay type 2 this description and a linkage theorem for families allow us to prove a conjecture of iarrobino on the existence of at least two irreducible components of gradalg h r h 1 3 6 10 14 10 6 2 whose general elements are artinian level algebras of type 2

on incidence between strata of the hilbert scheme of points on p 2 the hilbert scheme of n points in the projective plane has a natural stratification obtained from the associated hilbert series in general the precise inclusion relation between the closures of the strata is still unknown guerimand ph d thesis universite de nice 2002 studied this problem for strata whose hilbert series are as close as possible preimposing a certain technical condition he obtained necessary and sufficient conditions for the incidence of such strata in this paper we present a new approach based on deformation theory to guerimand s result this allows us to show that the technical condition is not necessary

pairs of commuting nilpotent matrices and hilbert function abstract let k be an infinite field there has been recent study of the family h n k of pairs of commuting nilpotent n n matrices relating this family to the fibre h n of the punctual hilbert scheme a n hilb n a 2 over the point np of the symmetric product sym n a 2 where p is a point of the affine plane a 2 v baranovsky the variety of pairs of commuting nilpotent matrices is irreducible transform groups 6 1 2001 3 8 r basili on the irreducibility of commuting varieties of nilpotent matrices j algebra 268 1 2003 56 80 a premet nilpotent commuting varieties of reductive lie algebras invent math 154 3 2003 653 683 in this study a pair of commuting nilpotent matrices a b is related to an artinian algebra k a b there has also been substantial study of the stratification of the local punctual hilbert scheme h n by the hilbert function as j briancon description de hilb n c x y invent math 41 1 1977 45 89 and others however these studies have been hitherto separate we first determine the stable partitions i e those for which p itself is the partition q p of a generic nilpotent element of the centralizer of the jordan nilpotent matrix j p we then explore the relation between h n k and its stratification by the hilbert function of k a b suppose that dim k k a b n and that k is algebraically closed of characteristic 0 or large enough p we show that a generic element of the pencil a b k has jordan partition the maximum partition p h whose diagonal lengths are the hilbert function h of k a b

an elementary coordinate dependent local resolution of singularities and applications abstract an elementary classical analysis resolution of singularities method is developed extensively using explicit coordinate systems the algorithm is designed to be applicable to subjects such as oscillatory integrals and critical integrability exponents as one might expect the trade off for such an elementary method is a weaker theorem than hironaka s work h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero i ann of math 2 79 1964 109 203 h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero ii ann of math 2 79 1964 205 326 or its subsequent simplications and extensions such as e bierstone p milman canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant invent math 128 2 1997 207 302 s encinas o villamayor good points and constructive resolution of singularities acta math 181 1 1998 109 158 j kollar resolution of singularities seattle lectures preprint a n varchenko newton polyhedra and estimates of oscillatory integrals funct anal appl 18 3 1976 175 196 nonetheless the methods of this paper can be used to prove a variety of theorems of interest in analysis as illustration two consequences are given first and most notably a general theorem regarding the existence of critical integrability exponents are established secondly a quick proof of a well known inequality of lojasiewicz s lojasiewicz ensembles semi analytiques inst hautes etudes sci bures sur yvette 1964 is given the arguments here are substantially different from the general algorithms such as h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero i ann of math 2 79 1964 109 203 h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero ii ann of math 2 79 1964 205 326 or the elementary arguments of e bierstone p milman semianalytic and subanalytic sets inst hautes etudes sci publ math 67 1988 5 42 and h sussman real analytic desingularization and subanalytic sets an elementary approach trans amer math soc 317 2 1990 417 461 the methods here have as antecedents the earlier work of the author m greenblatt a direct resolution of singularities for functions of two variables with applications to analysis j anal math 92 2004 233 257 phong and stein d h phong e m stein the newton polyhedron and oscillatory integral operators acta math 179 1997 107 152 and varchenko a n varchenko newton polyhedra and estimates of oscillatory integrals funct anal appl 18 3 1976 175 196

on the group of automorphisms of a quasi affine variety let mathbb k be an algebraically closed field of characteristic zero we show that if the automorphisms group of a quasi affine variety x over mathbb k is infinite then x is uniruled

nonparametric tests of regularity farrell efficiency and goodness of fit abstract the goal of this paper is to reveal the links among several seemingly unrelated literatures 1 1 the nonparametric regularity test literature see afriat 1972 diewert and parkan 1983 hanoch and rothschild 1972 varian 1984 2 2 the efficiency measurement literature farrell 1957 is the classic reference but also included is what has been dubbed data envelopment analysis and 3 3 the goodness of fit approach associated with varian 1990 the unifying threads are the shephard type distance function and the mahler inequality mahler 1939 our synthesis yields regularity tests of cost minimization and technical change which are nonparametric yet provide information as to goodness of fit based on farrell type efficiency measures

functorial desingularization of quasi excellent schemes in characteristic zero the nonembedded case we prove that any reduced noetherian quasi excellent scheme of characteristic zero admits a strong desingularization which is functorial with respect to all regular morphisms as a main application we deduce that any reduced formal variety of characteristic zero admits a strong functorial desin gularization also we show that as an easy formal consequence of our main result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi excellent stacks and formal schemes and complex or non archimedean analytic spaces moreover these functors easily generalize to non compact setting by use of generalized convergent blow up sequences with regular centers

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

techniques for the study of singularities with applications to resolution of 2 dimensional schemes we give an overview of invariants of algebraic singularities over perfect fields we then show how they lead to a synthetic proof of embedded resolution of singularities of 2 dimensional schemes

resolution except for minimal singularities i abstract the philosophy of this article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities the idea is used in two related problems 1 we give a proof of resolution of singularities of a variety or a divisor except for simple normal crossings i e which avoids blowing up simple normal crossings and ends up with a variety or a divisor having only simple normal crossings singularities 2 for more general normal crossings in a local analytic or formal sense such a result does not hold we find the smallest class of singularities in low dimension or low codimension with which we necessarily end up if we avoid blowing up normal crossings singularities several of the questions studied were raised by kollar

positivity of mixed multiplicities let r u v n2r u v be a standard bigraded algebra over an artinian local ring k r 0 0 standard means r is generated over k by a finite number of elements of degree 1 0 and 0 1 since the length r u v of r u v is finite we can consider r u v as a function in two variables u and v this function was first studied by van der waerden w and bhattacharya b who proved that there is a polynomial pr u v of total degree dimr 2 such that r u v pr u v for u and v large enough katz mandal andverma kmv found out that the degree of pr u v is equal to rdimr 2 where rdimr is the relevant dimension of r defined as follows let r denote the ideal generated by the homogeneous elements of degree u v with u 1 v 1 let projr be the set of all homogeneous prime ideals r of r then rdimr max dimr projr if projr and rdimr can be any negative integer if projr if we write

functional equations for zeta functions of groups and rings we introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings the specific form of these formulae enables us to deduce local functional equations more precisely we prove local functional equations for the subring zeta functions associated to rings the subgroup conjugacy and representation zeta functions of finitely generated torsion free nilpotent or t groups and the normal zeta functions of t groups of class 2 in particular we solve the two problems posed in 9 section 5 we deduce our theorems from a blueprint result on certain p adic integrals which generalises work of denef and others on igusa s local zeta function the malcev correspondence and a kirillov type theory developed by howe are used to linearise the problems of counting subgroups and representations in t groups respectively

dea with efficiency classification preserving conditional convexity abstract we propose to relax the standard convexity property used in data envelopment analysis dea by imposing additional qualifications for feasibility of convex combinations we specifically focus on a condition that preserves the koopmans efficiency classification this yields an efficiency classification preserving conditional convexity property which is implied by both monotonicity and convexity but not conversely substituting convexity by conditional convexity we construct various empirical dea approximations as the minimal sets that contain all dmus and are consistent with the imposed production assumptions imposing an additional disjunctive constraint to standard convex dea formulations can enforce conditional convexity computation of efficiency measures relative to conditionally convex production set can be performed through disjunctive programming dp

a stratification of the hilbert scheme of points in the projective plane this result is analogous to a theorem of iarrobino and brian on on the stratification of the punctual hilbert scheme where a different hilbert function defined by initial forms instead is used however cp i l thm 2 9 p 26 the proof essentially makes use of the smoothness of h a cp f p 517 and a general result of bialynicki birula on the action of the multiplicative group on a smooth projective scheme as already was done in a study of the homology of h a by ellingsrud and stromme this method is applied here to a study of h a around points which are invariant under the borel group in the appendix a method for computing the dimension of h is represented which rests on iarrobino s description of the tangent space to certain points of h a and the theorem of bialynicki birula finally i would like to mention that hirschowitz has outlined a general program for studying stratifications including those defined by hilbert functions cp h p 175 but the result obtained here can at best be regarded only as the most simple perhaps non trivial example

the hilbert function of generic plane sections of curves of ℙ3 a characteristic condition is given on a zero dimensional differentiable 0 sequenceh hi i 0 h1 3 in order to be the hilbert function of a generic plane section of a reduced irreducible curve of 3 hence of points of 2 with the uniform position property in this way an answer is given to some question stated by harris in ha2

ideal classes of three dimensional artin schelter regular algebras abstract we determine the possible hilbert functions of graded rank one torsion free modules over three dimensional artin schelter regular algebras it turns out that as in the commutative case they are related to castelnuovo functions from this we obtain an intrinsic proof that the space of torsion free rank one modules on a non commutative p 2 is connected a different proof of this fact based on deformation theoretic methods and the known commutative case has recently been given by nevins and stafford sklyanin algebras and hilbert schemes of points math ag 0310045 for the weyl algebra it was proved by wilson invent math 133 1 1998 1 41

simple hironaka resolution in characteristic zero building upon work of villamayor and bierstone milman we give a proof of the canonical hironaka principalization and desingularization the idea of homogenized ideals introduced in the paper gives a priori the canonicity of algorithm and radically simplifies the proof

the non lefschetz locus abstract we study the weak lefschetz property of artinian gorenstein algebras and in particular of artinian complete intersections in codimension four and higher it is an open problem whether all complete intersections have the weak lefschetz property for a given artinian gorenstein algebra a we ask what linear forms are lefschetz elements for this particular algebra i e which linear forms l give maximal rank for all the multiplication maps l a i a i 1 this is a zariski open set and its complement is the non lefschetz locus for monomial complete intersections we completely describe the non lefschetz locus for general complete intersections of codimension three and four we prove that the non lefschetz locus has the expected codimension which in particular means that it is empty in a large family of examples for general gorenstein algebras of codimension three with a given hilbert function we prove that the non lefschetz locus has the expected codimension if the first difference of the hilbert function is of decreasing type for completeness we also give a full description of the non lefschetz locus for artinian algebras of codimension two

a game for the resolution of singularities we propose a combinatorial game on finite graphs called stratify t hat is played by two protagonists a and b the game captures the logical structure of a proof of the resolution of singularities in each round the graph of the game is modified by the moves of the players when it assumes a final configuration a has won otherwise the game goes on forever and nobody wins in particular b cannot win the game but can only prevent a from winning we show that a always possesses a winning strategy regardless of the initial shape of the graph and of the moves of b this implies translating the game back to algebraic geometry that there is a choice of centres for the blowup of singular varieties in characteristic zero which eventually leads to their resolution the algebra needed for this implication is elementary the transcription from varieties to graphs and from blowups to modifications of the graph thus axiomatizes the proof of the resolution of singularities in principle the same logic could also work in positive characteristic once an appropriate descent in dimension is settled

building blocks of etale endomorphisms of complex projective manifolds etale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true they are the endomorphisms of abelian varieties and the nearly etale rational endomorphisms of weak calabi yau varieties

resolution of singularities of arithmetical threefolds ii abstract we prove grothendieck s conjecture on resolution of singularities for quasi excellent schemes x of dimension three and of arbitrary characteristic this applies in particular to x spec a a a reduced complete noetherian local ring of dimension three and to algebraic or arithmetical varieties of dimension three similarly if f is a number field a complete discretely valued field or more generally the quotient field of any excellent dedekind domain o any regular projective surface x f has a proper and flat model x over o which is everywhere regular

on the use of naturality in algorithmic resolution abstract this paper provides a simplified presentation of a known algorithm for resolution of singularities in characteristic zero it works in the context of marked ideals and uses naturality properties with respect to open restrictions and strong equivalence to solve a delicate glueing problem that arises when induction on the dimension of the objects considered is applied

subspaces of non commutative spaces this paper concerns the closed points closed subspaces open subspaces weakly closed and weakly open subspaces and effective divisors on a non commutative space

non parametric tests of productive efficiency with errors in variables we develop a non parametric test of productive efficiency that accounts for errors in variables following the approach of varian 1985 nonparametric analysis of optimizing behavior with measurement error journal of econometrics 30 1 2 445 458 the test is based on the general pareto koopmans notion of efficiency and does not require price data statistical inference is based on the sampling distribution of the l norm of errors the test statistic can be computed using a simple enumeration algorithm the finite sample properties of the test are analyzed by means of a monte carlo simulation using real world data of large eu commercial banks

zeta functions for analytic mappings log principalization of ideals and newton polyhedra in this paper we provide a geometric description of the possi ble poles of the igusa local zeta function z s f associated to an analytic mapping f f1 fl u k n k l and a locally constant function with support in u in terms of a log principalizaton of the k x ideal if f1 fl typically our new method provides a much shorter list of possible poles compared with the previous methods we determine the largest real part of the poles of the igusa zeta function and then as a corollary we ob tain an asymptotic estimation for the number of solutions of an arbitrary sys tem of polynomial congruences in terms of the log canonical threshold of the subscheme given by if we associate to an analytic mapping f f1 fl a newton polyhedron f and a new notion of non degeneracy with respect to f the novelty of this notion resides in the fact that it depends on one newton polyhedron and khovanskii s non degeneracy notion depends on the newton polyhedra of f1 fl by constructing a log principalization we give an explicit list for the possible poles of z s f l 1 in the case in which f is non degenerate with respect to f

uniform position properties and hilbert functions for points on a smooth quadric abstract if x is a set of points of p3 with the uniform position property we study the structure of the hilbert function of x with special care in the case when x lies on a smooth quadric q in this particular situation we also investigate the behaviour of the hilbert matrix of x when the uniform position property is enlarged to the divisors of q

the jacobian module the milnor fiber and the d module generated by f s for a germ f on a complex manifold x we introduce a complex derived from the liouville form acting on logarithmic differential forms and give an exactness criterion we use this liouville complex to connect properties of the d module generated by f s to homological data of the jacobian ideal specifically we show that for a large class of germs the annihilator of f s is generated by derivations through local cohomology we connect the cohomology of the milnor fiber to the jacobian module via logarithmic differentials in particular we consider not necessarily reduced hyperplane arrangements we prove a conjecture of terao on the annihilator of 1 f we confirm in many cases a corresponding conjecture on the annihilator of f s but we disprove it in general we show that the bernstein sato polynomial of an arrangement is not determined by its intersection lattice we prove that arrangements for which the annihilator of f s is generated by derivations fulfill the strong monodromy conjecture and that this includes as very special cases all arrangements of coxeter and of crystallographic type and all multi arrangements in dimension 3

zeta functions and oscillatory integrals for meromorphic functions abstract in the 70 s igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero in the present article this theory is extended to the case of rational functions or more generally meromorphic functions f g with coefficients in a local field of characteristic zero this generalization is far from being straightforward due to the fact that several new geometric phenomena appear also the oscillatory integrals have two different asymptotic expansions the usual one when the norm of the parameter tends to infinity and another one when the norm of the parameter tends to zero the first asymptotic expansion is controlled by the poles with negative real parts of all the twisted local zeta functions associated to the meromorphic functions f g c for certain special values c the second expansion is controlled by the poles with positive real parts of all the twisted local zeta functions associated to f g

resolution of singularities for 3 folds in positive characteristic a concise complete proof of resolution of singularities of 3 folds in positive characteristic 5 is given abhyankar first proved this theorem in 1966

weights on cohomology invariants of singularities and dual complexes we study the weight filtration on the cohomology of a proper complex algebraic variety and obtain natural upper bounds on its size when it is the exceptional divisor of a singularity we also give bounds for the cohomology of links the invariants of singularities introduced here gives rather strong information about the topology of rational and related singularities

ideal classes of three dimensional sklyanin algebras abstract in this paper we classify graded reflexive ideals up to isomorphism and shift in certain three dimensional artin schelter regular algebras this classification is similar to the classification of right ideals in the first weyl algebra a problem that was completely settled recently the situation we consider is substantially more complicated however