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

shape optimization for the steklov problem in higher dimensions abstract we show that the ball does not maximize the first nonzero steklov eigenvalue among all contractible domains of fixed boundary volume in r n when n 3 this is in contrast to the situation when n 2 where a result of weinstock from 1954 shows that the disk uniquely maximizes the first steklov eigenvalue among all simply connected domains in the plane having the same boundary length when n 3 we show that increasing the number of boundary components does not increase the normalized by boundary volume first steklov eigenvalue this is in contrast to recent results which have been obtained for surfaces and for convex domains

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

isoperimetric control of the steklov spectrum abstract we prove that the normalized steklov eigenvalues of a bounded domain in a complete riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain consequently the normalized steklov eigenvalues of a bounded domain in euclidean space hyperbolic space or a standard hemisphere are uniformly bounded above on a compact surface with boundary we obtain uniform bounds for the normalized steklov eigenvalues in terms of the genus we also establish a relationship between the steklov eigenvalues of a domain and the eigenvalues of the laplace beltrami operator on its boundary hypersurface

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

on asymptotic stability of linear time varying infinite dimensional systems abstract it is shown that every asymptotically equistable linear time varying infinite dimensional discrete time system x k 1 a k x k is uniformly asymptotically equistable if a k is a collectively compact sequence of bounded linear operators next this result is used to prove that for a broad class of linear retarded functional differential equations the notions of asymptotic equistability and uniform asymptotic equistability coincide

the geometry of the first non zero stekloff eigenvalue let mn g be a compact riemannian manifold with boundary and dimensionn 2 in this paper we discuss the first non zero eigenvalue problem begin align delta varphi 0 qquad on quad m frac partial varphi partial eta u 1 varphi qquad on quad partial m end align eqno 1 problem 1 is known as the stekloff problem because it was introduced by him in 1902 for bounded domains of the plane we discuss estimates of the eigenvalue 1in terms of the geometry of the manifold mn g in the two dimensional case we generalize payne s theorem p for bounded domains in the plane to non negative curvature manifolds in this case we show that 1 k0 wherekg k0andkgrepresents the geodesic curvature of the boundary in higher dimensionsn 3 for non negative ricci curvature manifolds we show that 1 k0 2 wherek0is a lower bound for any eigenvalue of the second fundamental form of the boundary we introduce an isoperimetric constant and prove a cheeger s type inequality for the stekloff eigenvalue

a comparison theorem for the first non zero steklov eigenvalue abstract let m n g be a complete riemannian manifold let p m and b r p be a geodesic ball of radius r with center at p assuming that r is smaller than the injectivity radius at p and an upper bound in the radial curvature we show a comparison principle for the first non zero steklov eigenvalue of b r p we give an example that shows that the hypothesis in the injectivity radius is necessary

which functions may be the upper bohl function of the diagonal discrete linear time varying systems abstract in this paper we give the necessary and sufficient conditions for a function f to be an upper bohl function of the diagonal discrete linear time varying systems

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

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

trace and inverse trace of steklov eigenvalues ii abstract in this paper we obtain some new estimates for the trace and inverse trace of steklov eigenvalues the estimates generalize some previous results of hersch payne schiffer 13 brock 2 raulot savo 21 and dittmar 5

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

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

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

koszul modules and green s conjecture we prove a strong vanishing result for finite length koszul modules and use it to derive green s conjecture for every g cuspidal rational curve over an algebraically closed field mathbf k with text char mathbf k 0 or text char mathbf k ge frac g 2 2 as a consequence we deduce that the general canonical curve of genus g satisfies green s conjecture in this range our results are new in positive characteristic whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by voisin our strategy involves establishing two key results of independent interest 1 we describe an explicit characteristic independent version of hermite reciprocity for mathfrak s mathfrak l 2 representations 2 we completely characterize in arbitrary characteristics the non vanishing behavior of the syzygies of the tangential variety to a rational normal curve

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

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

steklov eigenvalues and quasiconformal maps of simply connected planar domains we investigate isoperimetric upper bounds for sums of consecutive steklov eigenvalues of planar domains the normalization involves the perimeter and scale invariant geometric factors which measure deviation of the domain from roundness we prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace the proofs rely on a special class of quasiconformal mappings

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 boundary control of the poisson equation with the third boundary condition abstract this paper studies controllability of the poisson equation on the unit disk in c subject to the third boundary condition when the control is imposed on the boundary we use complex analytic methods to prove existence and uniqueness of the control when the parameter is a nonzero complex number but not a negative integer not an eigenvalue otherwise due to multiplicity of solutions to the underlying problem when is a negative integer controllability could only be obtained if proper additional conditions on the boundary are imposed

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

estimates for higher steklov eigenvalues in this paper motivated by the work of raulot and savo we generalize raulot savo s estimate for the first steklov eigenvalue of euclidean domains to higher steklov eigenvalues

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

on stability of linear time varying infinite dimensional discrete time systems the asymptotic behaviour of linear time varying infinite dimensional discrete lime systems is considered the introduced notions are weak power equistability power equistability uniform power equistability l p equistability uniform l p equistability and l p x equistability it is shown that they are identical a generalization of the concept of spectral radius of a single operator is also proposed it is proven that any time varying system is uniformly power equistable if and only if the generalized spectral radius of the sequence of the operators which define the system considered is less than one

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

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

the steklov and laplacian spectra of riemannian manifolds with boundary abstract given two compact riemannian manifolds m 1 and m 2 such that their respective boundaries 1 and 2 admit neighbourhoods 1 and 2 which are isometric we prove the existence of a constant c such that k m 1 k m 2 c for each k n the constant c depends only on the geometry of 1 2 this follows from a quantitative relationship between the steklov eigenvalues k of a compact riemannian manifold m and the eigenvalues k of the laplacian on its boundary our main result states that the difference k k is bounded above by a constant which depends on the geometry of m only in a neighbourhood of its boundary the proofs are based on a pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces in several situations the constant c is given explicitly in terms of bounds on the geometry of 1 2

on the number of upper bohl exponents for diagonal discrete time varying linear system abstract the bohl exponents similarly as the lyapunov exponents are one of the most important numerical characteristics of dynamical systems used in control theory by contrast with properties of the lyapunov characteristics properties of the bohl exponents are much less investigated in this paper we present some new properties of the upper bohl exponents of the diagonal discrete time varying linear system

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

remarks on the stability of linear infinite dimensional discrete time systems abstract various results are presented which complement the existing theory of stability for linear time varying infinite dimensional discrete time systems the concept of generalized spectral radius of a uniformly bounded sequence of elements of a normed algebra is studied in detail and several new formulas for computing this quantity are offered next the problem of stability of perturbed systems is considered finally a quantitative analysis of the so called perron condition is given

a sufficient condition on the exponential stability of two dimensional 2 d shift variant systems in this paper we consider a class of two dimensional 2 d shift variant systems based on the bound on the deviation of parameters a sufficient condition is derived which ensures exponential stability with different decaying rates along the two directions an example is presented to illustrate the result

the first steklov eigenvalue conformal geometry and minimal surfaces we consider the relationship of the geometry of compact riemannian manifolds with boundary to the first nonzero eigenvalue 1 of the dirichlet to neumann map steklov eigenvalue for surfaces with genus and k boundary components we obtain the upper bound 1l 2 k for 0 and k 1 this result was obtained by weinstock in 1954 and is sharp we attempt to find the best constant in this inequality for annular surfaces 0 and k 2 for rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball for a general class of not necessarily rotationally symmetric metrics on the annulus which we call supercritical we prove that 1 l is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary motivated by the annulus case we show that a proper submanifold of the ball is immersed by steklov eigenfunctions if and only if it is a free boundary solution we then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities we show that these bounds are only achieved when the manifold is minimally immersed by first steklov eigenfunctions we also use these ideas to show that any free boundary solution in two dimensions has area at least and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two dimensional case

variational aspects of laplace eigenvalues on riemannian surfaces abstract we study the existence and properties of metrics maximising the first laplace eigenvalue among conformal metrics of unit volume on riemannian surfaces we describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations the principal results include the general regularity properties of k extremal metrics and the existence of a partially regular 1 maximiser

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

conformal upper bounds for the eigenvalues of the laplacian and steklov problem in this paper we find upper bounds for the eigenvalues of the laplacian in the conformal class of a compact riemannian manifold m g these upper bounds depend only on the dimension and a conformal invariant that we call min conformal volume asymptotically these bounds are consistent with the weyl law and improve previous results by korevaar and yang and yau the proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions this method is interesting for itself and powerful as a further application of the method we obtain an upper bound for the eigenvalues of the steklov problem in a domain with c1 boundary in a complete riemannian manifold in terms of the isoperimetric ratio of the domain and the conformal invariant that we introduce

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

an isoperimetric inequality and the first steklov eigenvalue let mn g be a compact riemannian manifold with boundary in this paper we give upper and lower estimates for the first nonzero steklov eigenvalue 0inm 1 on m where 1 is a positive real number the estimate from below is for a star shaped domain on a manifold whose ricci curvature is bounded from below the upper estimate is for a convex manifold with nonnegative ricci curvature and is given in terms of the first nonzero eigenvalue for the laplacian on the boundary we prove a comparison theorem for simply connected domains in a simply connected manifold we exhibit annuli domains for which the comparison theorem fails to be true in j f escobar j funct anal 60 1997 544 556 we introduced the isoperimetric constant i m defined asi m inf mvol min vol 1 vol 2 where 1 m is a nonempty domain with boundary in the manifold m 2 m 1 and int m where int m is the interior of m we proved a cheeger s type inequality for 1 using the constant i m in this paper we give upper and lower estimates for the constant i in terms of isoperimetric constants of the boundary of m

sharp bounds for the first non zero stekloff eigenvalues abstract let m be an n 2 dimensional compact riemannian manifold with boundary and non negative ricci curvature consider the following two stekloff eigenvalue problems u 0 in m u p u on m 2 u 0 in m u u q u 0 on m where is the laplacian operator on m and denotes the outward unit normal on m the first non zero eigenvalues of the above problems will be denoted by p 1 and q 1 respectively in the present paper we prove that if the principle curvatures of the second fundamental form of m are bounded below by a positive constant c then p 1 1 1 1 n 1 c 2 n 1 c with equality holding if and only if is isometric to an n dimensional euclidean ball of radius 1 c here 1 denotes the first non zero eigenvalue of the laplacian of m we also show that if the mean curvature of m is bounded below by a positive constant c then q 1 n c with equality holding if and only if m is isometric to an n dimensional euclidean ball of radius 1 c finally we show that q 1 a v and that if the equality holds and if there is a point x 0 m such that the mean curvature of m at x 0 is no less than a n v then m is isometric to an n dimensional euclidean ball being a and v the area of m and the volume of m respectively

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

modular properties of nodal curves on k3 surfaces in this article we are going to address the following issues 1 the first is a rigidity property for pairs s c consisting of a general projective k3 surface s and a curve c obtained as the normalization of a nodal hyperplane section hat c hookrightarrow s we prove that a non trivial deformation of a pair s c induces a non trivial deformation of c 2 the second question concerns the wahl map of curves c obtained as above we prove that the wahl map of the normalization of a nodal curve contained in a general projective k3 surface is non surjective in both cases we impose upper bounds on the number of nodes of hat c

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

comparison of steklov eigenvalues on a domain and laplacian eigenvalues on its boundary in riemannian manifolds abstract we prove that in riemannian manifolds the k th steklov eigenvalue on a domain and the square root of the k th laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions as an application we derive a weyl type upper bound for steklov eigenvalues a pohozaev type identity for harmonic functions on the domain and the min max variational characterization of both eigenvalues are important ingredients

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

siegel metric and curvature of the moduli space of curves we study the curvature of the moduli space mg of curves of genus g with the siegel metric induced by the period map j mg ag we give an explicit formula for the holomorphic sectional curvature of mg along a schiffer variationp for p a point on the curve x in terms of the holomorphic sectional curvature of ag and the second gaussian map finally we extend the kahler form of the siegel metric as a closed current on m g and we determine its cohomology class as a multiple of