splittable ideals and the resolutions of monomial ideals abstract we provide a new combinatorial approach to study the minimal free resolutions of edge ideals that is quadratic square free monomial ideals with this method we can recover most of the known results on resolutions of edge ideals with fuller generality and at the same time obtain new results past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or koszul homology groups our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph we also show that our technique extends successfully to the study of graded betti numbers of arbitrary square free monomial ideals viewed as facet ideals of simplicial complexes

deformations of quasi homogeneous surface singularities let a a be a finitely generated normal graded domain over e so that i 0 v spec a has an isolated singularity we say v is a quasi homogeneous singularity or admits a good ie action or is defined by weighted homogeneous polynomials we are concerned with two questions about the deformations of v there are two natural notions of equisingular deformation of v first one has deformations of weight 0 that is one perturbs the defining equations of vby terms of weight at least as big to make a good notion out of this dock sang rim observed one is really deforming a and the weight filtration ij where ij ai

last syzygies of 1 generic spaces abstract we consider determinantal varieties x of expected codimension defined by the maximal minors of a matrix m of linear forms representing a linear map eisenbud and popescu have conjectured that 1 generic linear maps have the property that the syzygy ideals i s of all last syzygies s of x coincide with i x we prove a geometric version of this conjecture for 1 generic linear maps the syzygy varieties syz s v i s of all last syzygies have the same support as x

moishezon s theorem and degeneration abstract let l be a tensor product of two very ample line bundles on the smooth projective complex threefold x under the hypothesis that h 0 x k x l 0 we show that the restriction map r pic x pic y is an isomorphism for very general y in the linear system l for such l this result recovers the noether lefschetz theorem of moishezon who extended the original topological and hodge theoretic arguments of lefschetz we give a degeneration argument which is almost completely algebraic in nature

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

the canonical syzygy conjecture for ribbons green s canonical syzygy conjecture asserts a simple relationship between the clifford index of a smooth projective curve and the shape of the minimal free resolution of its homogeneous ideal in the canonical embedding we prove the analogue of this conjecture formulated by bayer and eisenbud for a class of non reduced curves called ribbons our proof uses the results of voisin and hirschowitz ramanan on green s conjecture for general smooth curves

a de interlacing algorithm using markov random field model in this paper a motion compensated de interlacing algorithm using the markov random field mrf model is proposed the de interlacing problem is formulated as a maximum a posteriori map mrf problem the map solution is the one that minimizes an energy function which imposes discontinuity adaptive smoothness das spatial constraint on the de interlaced frame the edge direction information which is used to formulate the das constraint is implicitly indicated by weight vectors weights for 16 digitized directions generally large weights are assigned to along edge directions and relatively small weights are assigned to across edge directions as a local statistical based method the proposed weighting method should be more robust than traditional edge directed interpolation methods in deciding local edge directions the proposed algorithm is implemented by an iterative optimization process which guarantees convergence however a global optimal solution is not guaranteed due to computational complexity concern simulation results compare the proposed algorithm to other motion compensated de interlacing algorithms significant improvements of de interlaced edges are observed

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

fibred rational surfaces with extremal mordell weil lattices slope inequalities are given for fibred rational surfaces according as the clifford index of a general fibre for fibred rational surfaces of clifford index two the mordell weil lattices of maximal ranks are completely determined

the generic green lazarsfeld secant conjecture using lattice theory on special k3 surfaces calculations on moduli stacks of pointed curves and voisin s proof of green s conjecture on syzygies of canonical curves we prove the prym green conjecture on the naturality of the resolution of a general prym canonical curve of odd genus as well as many cases of the green lazarsfeld secant conjecture on syzygies of non special line bundles on general curves

on the projective normality of double coverings over a rational surface abstract we study the projective normality of a minimal surface x which is a ramified double covering over a rational surface s with dim k s 1 in particular horikawa surfaces the minimal surfaces of general type with k x 2 2 p g x 4 are of this type up to resolution of singularities let be the covering map from x to s we show that the z 2 invariant adjoint divisors k x r a are normally generated where the integer r 3 and a is an ample divisor on s

de interlacing using nonlocal costs and markov chain based estimation of interpolation methods a new method of de interlacing is proposed de interlacing is revisited as the problem of assigning a sequence of interpolation methods interpolators to a sequence of missing pixels of an interlaced frame field with this assumption our de interlacing algorithm de interlacer undergoes transitions from one interpolation method to another as it moves from one missing pixel position to the horizontally adjacent missing pixel position in a missing row of a field we assume a discrete countable state markov chain model on the sequence of interpolators markov chain states which are selected from a user defined set of candidate interpolators an estimation of the optimum sequence of interpolators with the aforementioned markov chain model requires the definition of an efficient cost function as well as a global optimization technique our algorithm introduces for the first time using a nonlocal cost nlc scheme the proposed algorithm uses the nlc to not only measure the fitness of an interpolator at a missing pixel position but also to derive an approximation for transition matrix tm of the markov chain of interpolators the tm in our algorithm is a frame variate matrix i e the algorithm updates the tm for each frame automatically the algorithm finally uses a viterbi algorithm to find the global optimum sequence of interpolators given the cost function defined and neighboring original pixels in hand next we introduce a new map based formulation for the estimation of the sequence of interpolators this time not by estimating the best sequence of interpolators but by successive estimations of the best interpolator at each missing pixel using forward backward algorithm simulation results prove that while competitive with each other on different test sequences the proposed methods one using viterbi and the other forward backward algorithm are superior to state of the art de interlacing algorithms proposed recently finally we propose motion compensated versions of our algorithm based on optical flow computation methods and discuss how it can improve the proposed algorithm

koszul homology and syzygies of veronese subalgebras a graded k algebra r has property np if it is generated in degree 1 has relations in degree 2 and the syzygies of order p on the relations are linear the green lazarsfeld index of r is the largest p such that it satisfies the property np our main results assert that under a mild assumption on the base field the cth veronese subring of a polynomial ring has green lazarsfeld index c 1 the same conclusion also holds for an arbitrary standard graded algebra provided c gg 0

a geometric effective nullstellensatz the purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of brownawell kollar and others on the effective nullstellensatz specifically we work on an arbitrary smooth complex projective variety x with the previous classical results corresponding to the case when x is projective space in this setting we prove a local effective nullstellensatz for ideal sheaves and a corresponding global division theorem for adjoint type bundles we also make explicit the connection with the intersection theory of fulton and macpherson finally constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties much of the previous activity in this area has been algebraic in nature and seems perhaps not well known in detail among geometers therefore we have felt it worthwhile to include here a rather extended introduction we start with an overview of the questions and earlier work on them then we present the set up and statement of our main theorem we conclude with a series of examples which can be read before the general result of what it yields in special cases

secant varieties and birational geometry we show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations these were first constructed as flips in the case of curves by m thaddeus via geometric invariant theory and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper we expose the finer structure of a second flip again for varieties of arbitrary dimension we also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties

singularities and syzygies of secant varieties of nonsingular projective curves in recent years the equations defining secant varieties and their syzygies have attracted considerable attention the purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies the main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus g is greater than 2g 2k p for nonnegative integers k and p then the k th secant variety of the curve has normal du bois singularities is arithmetically cohen macaulay and satisfies the property n k 2 p in addition the singularities of the secant varieties are further classified according to the genus of the curve and the castelnuovo mumford regularities are also obtained as well as one of the main technical ingredients we establish a vanishing theorem on the cartesian products of the curve which may have independent interests and may find applications elsewhere

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

on syzygies of non complete embedding of projective varieties let x be a non degenerate not necessarily linearly normal projective variety in mathbb p r recently the generalization of property np to non linearly normal projective varieties have been considered and its algebraic and geometric properties are studied extensively one of the generalizations is the property nd p for the saturated ideal ix eisenbud et al in compos math 141 1460 1478 2005 and the other is the property n p s for the graded module of the twisted global sections of mathcal o x 1 kwak and park in j reine angew math 582 87 105 2005 in this paper we are interested in the algebraic and geometric meaning of properties n p s for every p 0 and the syzygetic behaviors of isomorphic projections and hyperplane sections of a given variety with property n p s

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

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 discontinuity adaptive smoothness priors in computer vision a variety of analytic and probabilistic models in connection with markov random fields mrfs have been proposed in the last decade for solving low level vision problems involving discontinuities this paper presents a systematic study of these models and defines a general discontinuity adaptive da mrf model by analyzing the euler equation associated with the energy minimization it shows that the fundamental difference between different models lies in the behavior of interaction between neighboring points which is determined by the a priori smoothness constraint encoded into the energy function an important necessary condition is derived for the interaction to be adaptive to discontinuities to avoid oversmoothing this forms the basis on which a class of adaptive interaction functions aifs is defined the da model is defined in terms of the euler equation constrained by this class of aifs its solution is c sup 1 continuous and allows arbitrarily large but bounded slopes in dealing with discontinuities because of the continuous nature it is stable to changes in parameters and data a good property for regularizing ill posed problems experimental results are shown

computing graded betti tables of toric surfaces we present various facts on the graded betti table of a projectively embedded toric surface expressed in terms of the combinatorics of its defining lattice polygon these facts include explicit formulas for a number of entries as well as a lower bound on the length of the quadratic strand that we conjecture to be sharp and prove to be so in several special cases we also present an algorithm for determining the graded betti table of a given toric surface by explicitly computing its koszul cohomology and report on an implementation in sagemath it works well for ambient projective spaces of dimension up to roughly 25 depending on the concrete combinatorics although the current implementation runs in finite characteristic only as a main application we obtain the graded betti table of the veronese surface nu 6 p 2 subset of p 27 in characteristic 40 009 this allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary veronese surface nu d p 2

on the vanishing of higher syzygies of curves ii we verify green s conjecture for a generic k gonal curve of genus g for g geq k k 1 2

hilbert schemes and maximal betti numbers over veronese rings macaulay s theorem macaulay in proc lond math soc 26 531 555 1927 characterizes the hilbert functions of graded ideals in a polynomial ring over a field we characterize the hilbert functions of graded ideals in a veronese ring r the coordinate ring of a veronese embedding of pr 1 we also prove that the hilbert scheme which parametrizes all graded ideals in r with a fixed hilbert function is connected this is an analogue of hartshorne s theorem hartshorne in math ihes 29 5 48 1966 that hilbert schemes over a polynomial ring are connected furthermore we prove that each lex ideal in r has the greatest betti numbers among all graded ideals with the same hilbert function

on syzygies of calabi yau varieties and varieties of general type abstract we establish projective normality property n p and effective very ampleness for line bundles on a calabi yau variety as well as for pluricanonical bundles of a variety of general type we also prove a result on the degrees of generators of canonical rings under the condition that the image of the canonical map has minimal degree

an mrf based deinterlacing algorithm with exemplar based refinement in this paper we propose an mrf based deinterlacing algorithm that combines the benefits of rule based algorithms such as motion adaptation edge directed interpolation and motion compensation with those of an mrf formulation mrf based interpolation and enhancement algorithms are typically formulated as an optimization over pixel intensities or colors which can make them relatively slow in comparison our mrf based deinterlacing algorithm uses interpolation functions as labels we use seven interpolants three spatial three temporal and one for motion compensation the core dynamic programming algorithm is therefore sped up greatly over the direct use of intensity as labels we also show how an exemplar based learning algorithm can be used to refine the output of our mrf based algorithm the training set can be augmented with exemplars from static regions of the same video as a form of ldquoself learningrdquo

on the projective normality of line bundles of extremal degree we propose a new method using deformation theory to study the maximal rank conjecture for line bundles of extremal degree which can be viewed as the first case to test the conjecture we prove that maximal rank conjecture holds by our new method

curves and symmetric spaces ii let sym 3 c p k sym 3 k sym 3 k k p 13 a 1 a a det a be the veronese embedding of the space of symmetric matrices of degree 3 where a is the cofactor matrix of a the closure spg 3 6 of this image is a 6 dimensional homogeneous variety of the symplectic group sp 3 a canonical curve c 16 c p 8 of genus 9 over a perfect field k is isomorphic to a complete linear section of this projective variety spg 3 6 c p 13 unless c k k k being the algebraic closure is a covering of degree at most 5 of the projective line we prove this by means of linear systems of higher rank

a markov random field model based approach to image interpretation an image is segmented into a collection of disjoint regions that form the nodes of an adjacency graph and image interpretation is achieved through assigning object labels or interpretations to the segmented regions or nodes using domain knowledge extracted feature measurements and spatial relationships between the various regions the interpretation labels are modeled as a markov random field mrf on the corresponding adjacency graph and the image interpretation problem is then formulated as a maximum a posteriori map estimation rule given domain knowledge and region based measurements simulated annealing is used to find this best realization or optimal map interpretation this approach also provides a systematic method for organizing and representing domain knowledge through appropriate design of the clique functions describing the gibbs distribution representing the pdf of the underlying mrf a general methodology is provided for the design of the clique functions results of image interpretation experiments on synthetic and real world images are described

hilbert functions of cox rings of del pezzo surfaces to study syzygies of the cox rings of del pezzo surfaces we calculate important syzygetic invariants such as the hilbert functions the green lazarsfeld indices the projective dimensions and the castelnuovo mumford regularities using these computations as well as the natural multigrading structures by the picard groups of del pezzo surfaces and weyl group actions on picard lattices we determine the betti diagrams of the cox rings of del pezzo surfaces of degree at most four

analysis on some infinite modules inner projection and applications a projective scheme x is called quadratic if x is scheme theoretically cut out by homogeneous equations of degree 2 furthermore we say that x satisfies property n2 p if it is quadratic and the quadratic ideal has only linear syzygies up to the first p th steps in the present paper we compare the linear syzygies of the inner projections with those of x and obtain a theorem on embedded linear syzygies as one of our main results this is the natural projection analogue of restricting linear syzygies in the linear section case as an immediate corollary we show that the inner projections of x satisfy property n2 p 1 for any reduced scheme x with property n2 p moreover we also obtain the neccessary lower bound codimx p p p 1 2 which is sharp on the number of quadrics vanishing on x in order to satisfy n2 p and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme x these results admit an interesting syzygetic rigidity theorem on property n2 p which leads the classifications of extremal and next to extremal cases for these results we develop the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by m green this new method allows us to treat a wider class of projective schemes which cannot be covered by the koszul cohomology techniques because these are not projectively normal in general

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

syzygies of torsion bundles and the geometry of the level l modular variety over overline mathcal m _ g we formulate and in some cases prove three statements concerning the purity or more generally the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of l in its jacobian these statements can be viewed an analogues of green s conjecture and we verify them computationally for bounded genus we then compute the cohomology class of the corresponding non vanishing locus in the moduli space mathcal r g ell of twisted level l curves of genus g and use this to derive results about the birational geometry of mathcal r g ell for instance we prove that mathcal r g 3 is a variety of general type when g 11 and the kodaira dimension of mathcal r 11 3 is greater than or equal to 19 in the last section we explain probabilistically the unexpected failure of the prym green conjecture in genus 8 and level 2

smooth varieties of almost minimal degree abstract in this article we study non linearly normal smooth projective varieties x p r of deg x codim x p r 2 we first give geometric characterizations for x theorem 1 1 indeed x is the image of an isomorphic projection of smooth varieties x p r 1 of minimal degree also if x is not the veronese surface then there exists a smooth rational normal scroll y p r which contains x as a divisor linearly equivalent to h 2 f where h is the hyperplane section of y and f is a fiber of the projection morphism y p 1 by using these characterizations 1 we determine all the possible types of y from the type of x theorem 1 2 and 2 we investigate the relation between the betti diagram of x and the type of y theorem 1 3 in particular we clarify the relation between the number of generators of the homogeneous ideal of x and the type of y as an application we construct non linearly normal examples where the converse to theorem 1 1 in d eisenbud m green k hulek s popescu restriction linear syzygies algebra and geometry compos math 141 2005 1460 1478 fails to hold remark 2

asymptotic dirichlet problems for harmonic functions on riemannian manifolds we define the asymptotic dirichlet problem and give a sufficient condition for solving it this proves an existence of nontrivial bounded harmonic functions on certain classes of noncompact complete riemannian manifolds 0 introduction in this paper we will prove the existence of nonconstant bounded harmonic functions on certain classes of noncompact riemannian manifolds by defining and solving an asymptotically defined dirichlet problem for harmonic functions the motivation comes from the classical uniformization theorem of riemann surfaces which says that a simply connected riemann surface is biholo morphic to the riemann sphere s2 the complex plane c or the unit disk d this is a geometric theorem but its original proof due to koebe relies heavily on function theory the function theoretic interpretation of this theorem is the following among the noncompact simply connected surfaces c is characterized by the fact that it admits no nonconstant bounded harmonic functions and d by that it admits nonconstant bounded harmonic functions the geometric aspect of the uniformization theorem could be roughly stated as follows let m be a simply connected riemann surface equipped with a complete riemannian metric with gaussian curvature km if km c 0 then m is biholo morphic to s2 if km c 0 then m is biholomorphic to d if km is close to zero then m is biholomorphic to c the following precise version is due to greene and wu gw1 p 120