cayley bacharach schemes and their canonical modules a set of s points in p d is called a cayley bacharach scheme cb scheme if every subset of s 1 points has the same hilbert function we investigate the consequences of this weak uniformity the main result characterizes cb schemes in terms of the structure of the canonical module of their projective coordinate ring from this we get that the hilbert function of a cb scheme x has to satisfy growth conditions which are only slightly weaker than the ones given by harris and eisenbud for points with the uniform position property we also characterize cb schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points

on bounding the number of generators for fat point ideals on the projective plane abstract let x be the surface obtained by blowing up general points p1 pn of the projective plane over an algebraically closed ground field k and let l be the pullback to x of a line on the plane if c is a rational curve on x with c l d then for every t there is a natural map c o c t x o x l c o c t d given by multiplication on simple tensors the ranks of such maps are determined as a function of t d and m where m is the largest multiplicity of c at any of the points pi if i is the ideal defining the fat point subscheme z m1p1 mnpn p2 and is the least degree in which i has generators then the ranks of the maps c o c t x o x l c o c t d can be used for bounding the number of generators of i in degrees t 1

on the cohen macaulay type of s lines in an 1 abstract let a be the coordinate ring of s lines through the origin in a n 1 k we discuss what it means for these lines to be in sufficiently general position and then with this restriction on the lines we attempt to verify our belief that the cohen macaulay type of a depends only on s and n we characterize those s and n for which a is a gorenstein ring i e of c m type 1 and explicitly calculate the cohen macaulay type in several other cases we then extend some of our results to the case of an arbitrary reduced curve whose tangent cone consists of lines in sufficiently general position finally we calculate the hubert samuel polynomials of the curves we have been considering

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

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

multiple solutions of h systems and rellich s conjecture soit x y r 2 x 2 y 2 1 on considere une fonction u r 3 satisfaisant le systeme h u 2hu x u y sur avec l une des conditions suivantes soit dirichlet u sur soit plateau u x 2 u y 2 u x u y 0 sur u et u non decroissante sur

cayley bacharach theorems and conjectures a theorem of pappus of alexandria proved in the fourth century a d began a long development in algebraic geometry in its changing expressions one can see reflected the changing concerns of the field from synthetic geometry to projective plane curves to riemann surfaces to the modern development of schemes and duality we survey this development historically and use it to motivate a brief treatment of a part of duality theory we then explain one of the modern developments arising from it a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them we give a proof of the conjectures in a new

on the first infinitesimal neighborhood of a linear configuration of points in p2 abstract we consider the following open questions fix a hilbert function h that occurs for a reduced zero dimensional subscheme of p 2 among all subschemes x with hilbert function h what are the possible hilbert functions and graded betti numbers for the first infinitesimal neighborhood z of x i e the double point scheme supported on x is there a minimum h min and maximum h max such function the numerical information encoded in h translates to a type vector which allows us to find unions of points on lines called linear configurations with hilbert function h we give necessary and sufficient conditions for the hilbert function and graded betti numbers of the first infinitesimal neighborhoods of all such linear configurations to be the same even for those h for which the hilbert functions or graded betti numbers of the resulting double point schemes are not uniquely determined we give one depending only on h that does occur we prove the existence of h max in general and discuss h min our methods include liaison techniques

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

objective fluxes in a multi scale continuum description of sparse medium dynamics abstract we discuss reasons justifying a multi scale continuum description of sparse media which do not admit a choice of a representative volume element remaining permanent in time with constant mass we suggest the choice of objective derivatives for time varying terms in the balance equations pertinent to the scheme that we analyze to avoid some problems connected with s o 3 based changes in observers which emerge within the setting of the standard theory of gases when we start from it to get continuum models the scheme discussed here can be reduced to versions of the averaged regularizations of the navier stokes equations in our approach we have primarily in mind the continuum description of bodies like macro molecular fluids or granular ones even if what we propose can be significant in a broader setting

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

mechanics of affine bodies towards affine dynamical symmetry abstract in this paper we discuss certain dynamical models of affine bodies including problems of partial separability and integrability there are some reasons to expect that the suggested models are dynamically viable and that on the fundamental level of physical phenomena the large affine symmetry of dynamical laws is more justified and desirable than the restricted invariance under isometries

multiple solutions of inhomogeneous h systems with zero dirichlet boundary conditions in this paper we study the existence of multiple solutions to the dirichlet problem of the inhomogeneous h system u 2hux uy f in u 0 where r2 is a bounded smooth domain h 0 a constant and f h 1 r3 is a given function by ekeland s variational principle and the mountain pass theorem we prove that for f 0 satisfying some assumptions the problem has at least two solutions in h01 r3 this is not the case if f 0 and is simply connected 16

covariance in plasticity covariance is imposed to the second law of thermodynamics and consequences are shown for elastic perfectly plastic bodies in this setting pointwise balances of standard and configurational action

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

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

on the resolution of a curve lying on a smooth cubic surface in ℙ 3 let c be any reduced and irreducible curve lying on a smooth cubic surface in p 3 in this paper we determine the graded betti numbers of the ideal sheaf j c

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

resolutions of ideals of quasiuniform fat point subschemes of the notion of a quasiuniform fat point subscheme z p 2 is introduced and conjectures for the hilbert function and minimal free resolution of the ideal i defining z are put forward in a large range of cases it is shown that the hilbert function conjecture implies the resolution conjecture in addition the main result gives the first determination of the resolution of the mth symbolic power i m n of an ideal defining n general points of p 2 when both m and n are large in particular for infinitely many m for each of infinitely many n and for infinitely many n for every m 2 resolutions in other cases such as fat points with tails are also given except where an explicit exception is made all results hold for an arbitrary algebraically closed field k as an incidental result a bound for the regularity of i m n is given which is often a significant improvement on previously known bounds

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

geometric consequences of extremal behavior in a theorem of macaulay f s macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer valued function which determine when such a function can be the hilbert function of a standard graded k algebra we investigate some algebraic and geometric consequences which arise from the extremal cases of macaulay s theorem our work also builds on the fundamental work of g gotzmann our principal applications are to the study of hilbert functions of zeroschemes with uniformity conditions as a consequence we have new strong limitations on the possible hilbert functions of the points which arise as a general hyperplane section of an irreducible curve