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

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

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

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

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

application of zhangs square root law and herding to financial markets abstract we apply an asymmetric version of kirman s herding model to volatile financial markets in the relation between returns and agent concentration we use the square root law proposed by zhang this can be derived by extending the idea of a critical mean field theory suggested by plerou et al we show that this model is equivalent to the so called 3 2 model of stochastic volatility the description of the unconditional distribution for the absolute returns is in good agreement with the dax independent whether one uses the square root or a conventional linear relation only the statistic of extreme events prefers the former the description of the autocorrelations are in much better agreement for the square root law the volatility clusters are described by a scaling law for the distribution of returns conditional to the value at the previous day in good agreement with the data

linear systems of plane curves through fixed fat points of p2 abstract given any s points p1 ps in p 2 and s positive integers m1 ms let sn be the linear system of plane curves of degree n through pi with multiplicity at least mi 1 i s we give numerical bounds for the regularity of sn in the following cases a the points pi are non singular points of an integral curve of degree d b the pi s are in general position c the pi s are in uniform position d the pi s are generic points of p 2 we also study the sharpness of such bounds

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

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

projective curves of degree codimension 2 in this article we study nondegenerate projective curves x subset mathbb p d 1 of degree d which are not arithmetically cohen macaulay note that x pi p widetilde x for a rational normal curve widetilde x subset mathbb p d and a point p in mathbb p d setminus widetilde x 2 our main result is about the relation between the geometric properties of x and the position of p with respect to widetilde x we show that the graded betti numbers of x are uniquely determined by the rank hbox rk widetilde x p of p with respect to widetilde x in particular x satisfies property n 2 p if and only if p leq quad rk widetilde x p 3 therefore property n 2 p of x is controlled by quad rk widetilde x p and conversely quad rk widetilde x p can be read off from the minimal free resolution of x this result provides a non linearly normal example for which the converse to theorem 1 1 in eisenbud et al compositio math 141 1460 1478 2005 holds also our result implies that for nondegenerate projective curves x subset mathbb p d 1 of degree d which are not arithmetically cohen macaulay there are exactly lfloor frac d 2 2 rfloor distinct betti tables

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

stochastic multiplicative processes for financial markets abstract we study a stochastic multiplicative system composed of finite asynchronous elements to describe the wealth evolution in financial markets we find that the wealth fluctuations or returns of this system can be described by a walk with correlated step sizes obeying truncated levy like distribution and the cross correlation between relative updated wealths is the origin of the nontrivial properties of returns including the power law distribution with exponent outside the stable levy regime and the long range persistence of volatility correlations

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

automorphism groups of some ag codes in this correspondence we show that in many cases the automorphism group of a curve and the permutation automorphism group of a corresponding ag code are the same this generalizes a result of wesemeyer beyond the case of planar curves

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

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

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

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

closed form solutions for the levy stable distribution the levy stable distribution is the attractor of distributions which hold power laws with infinite variance this distribution has been used in a variety of research areas for example in economics it is used to model financial market fluctuations and in statistical mechanics it is used as a numerical solution of fractional kinetic equations of anomalous transport this function does not have an explicit expression and no uniform solution has been proposed yet this paper presents a uniform analytical approximation for the levy stable distribution based on matching power series expansions for this solution the trans stable function is defined as an auxiliary function which removes the numerical issues of the calculations of the levy stable distribution then the uniform solution is proposed as a result of an asymptotic matching between two types of approximations called the inner solution and the outer solution finally the results of analytical approximation are compared to the numerical results of the levy stable distribution function making this uniform solution valid to be applied as an analytical approximation

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

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

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

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

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

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

volatility cluster and herding stock markets can be characterized by fat tails in the volatility distribution clustering of volatilities and slow decay of their time correlations for an explanation models with several mechanisms and consequently many parameters as the lux marchesi model have been used we show that a simple herding model with only four parameters leads to a quantitative description of the data as a new type of data we describe the volatility cluster by the waiting time distribution which can be used successfully to distinguish between different models

a dichotomy theorem for the approximate counting of complex weighted bounded degree boolean csps we determine the computational complexity of approximately counting the total weight of variable assignments for every complex weighted boolean constraint satisfaction problem or csp with any number of additional unary i e arity 1 constraints particularly when degrees of input instances are bounded from above by a fixed constant all degree 1 counting csps are obviously solvable in polynomial time when the instance s degree is more than two we present a dichotomy theorem that classifies all counting csps admitting free unary constraints into exactly two categories this classification theorem extends to complex weighted problems an earlier result on the approximation complexity of unweighted counting boolean csps of bounded degree the framework of the proof of our theorem is based on a theory of signature developed from valiant s holographic algorithms that can efficiently solve seemingly intractable counting csps despite the use of arbitrary complex weight our proof of the classification theorem is rather elementary and intuitive due to an extensive use of a novel notion of limited t constructibility for the remaining degree 2 problems in contrast they are as hard to approximate as holant problems which are a generalization of counting csps

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

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

long time fluctuations in a dynamical model of stock market indices financial time series typically exhibit strong fluctuations that cannot be described by a gaussian distribution recent empirical studies of stock market indices examined whether the distribution p r of returns r tau after some time tau can be described by a truncated levy stable distribution l alpha r with some index 0 2 namely beyond the range of levy stable distributions our results are in agreement with both empirical studies and reconcile the apparent disagreement between their results

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

expressiveness of matchgates matchgates have been used to simulate classically in polynomial time assemblies of quantum gates in this paper it is shown that every matchgate for 2 input 2 output functions has to obey a certain set of five polynomial identities it is also shown that no matchgate can realize a nontrivial control gate for any number of inputs on the other hand it is proved that classical boolean formulae can be expressed as matchcircuits of polynomial size

the complexity of weighted boolean csp this paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted boolean constraint satisfaction problem the problem is parameterized by a finite set mathcal f of nonnegative functions that may be used to assign weights to the configurations feasible solutions of a problem instance classical constraint satisfaction problems correspond to the special case of 0 1 valued functions we show that computing the partition function i e the sum of the weights of all configurations is text sf fp text sf p complete unless either 1 every function in mathcal f is of product type or 2 every function in mathcal f is pure affine in the remaining cases computing the partition function is in p

three themes of syzygies we present three exciting themes of syzygies where major progress was made recently boij soderberg theory stillman s question and syzygies over complete intersections

families of nodal curves on projective threefolds and their regularity via postulation of nodes the main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds precisely given x a smooth projective threefold e a rank two vector bundle on x l a very ample line bundle on x and k 0 0 integers and denoted by v e l k the subscheme of p h 0 e l k parametrizing global sections of e l k whose zero loci are irreducible and nodal curves on x we present a new cohomological description of the tangent space t s v e l k at a point s 2 v e l k this description enable us to determine effective and uniform upper bounds for which are linear polynomials in k such that the family v e lk is smooth and of the expected dimension regular for short the almost sharpness of our bounds is shown by some interesting examples furthermore when x is assumed to be a fano or a calaby yau threefold we study in detail the regularity property of a point s 2 v e lk related to the postulation of the nodes of its zero locus c v s x roughly speaking when the nodes of c are assumed to be in general position either on x or on an irreducible divisor of x having at worst log terminal singularities or to lie on a l c i and subcanonical curve in x we find upper bounds onwhich are respectively cubic quadratic and linear polynomials in k ensuring the regularity of v e l k at s finally when x p 3 we also discuss some interesting geometric properties of the curves given by sections parametrized by v e ox k

approximation complexity of complex weighted degree two counting constraint satisfaction problems constraint satisfaction problems have been studied in numerous fields with practical and theoretical interests in recent years major breakthroughs have been made in a study of counting constraint satisfaction problems or csps in particular a computational complexity classification of bounded degree csps has been discovered for all degrees except for two where the degree of an input instance is the maximal number of times that each input variable appears in a given set of constraints despite the efforts of recent studies however a complexity classification of degree 2 csps has eluded from our understandings this paper challenges this open problem and gives its partial solution by applying two novel proof techniques t 2 constructibility and parametrized symmetrization which are specifically designed to handle arbitrary constraints under randomized approximation preserving reductions we partition entire constraints into four sets and we classify the approximation complexity of all degree 2 csps whose constraints are drawn from two of the four sets into two categories problems computable in polynomial time or problems that are at least as hard as sat our proof exploits a close relationship between complex weighted degree 2 csps and holant problems which are a natural generalization of complex weighted csps