Let $ρ_C$ be the regularity of the Hilbert function of a projective curve $C$ in $\mathbb P^n_K$ over an algebraically closed field $K$ and $α_1,...,α_{n-1}$ be minimal degrees for which there exists a complete intersection of type $(α_1,...,α_{n-1})$ containing the curve $C$. Then the Castelnuovo-Mumford regularity of $C$ is upper bounded by $\max\{ρ_C+1,α_1+...+α_{n-1}-(n-2)\}$. We study and, for space curves, refine the above bound providing several examples.
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 to 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.
The notion of regularity has been used by S. Kleiman in the construction of bounded families of ideals or sheaves with given Hilbert polynomial, a crucial point in the construction of Hilbert or Picard scheme. In a related direction, Kleiman proved that if I is an equidimensional reduced ideal in a polynomial ring S over an algebraically closed field, then the coefficients of the Hilbert polynomial of R = S/I can be bounded by the dimension and the multiplicity of R. Srinivas and Trivedi proved that the corresponding result does not hold for a local domain. However, they proved that there exis…
Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial $p(z)$. Experimental evidences led us to consider the idea that $m_{p(z)}$ could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $m_p(z)^{\varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ of …
In this paper, we consider $\mathbb{Z}^{r}-$graded modules on the $\mathrm{Cl}(X)$ $-$graded Cox ring $\mathbb{C}[x_{1},\dotsc,x_{r}]$ of a smooth complete toric variety $X$. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive $\mathbb{Z}^{s+r+2}$-graded modules over non-standard bigraded polynomial rings $\mathbb{C}[x_{0},\dotsc,x_{s},y_{0},\dotsc,y_{r}]$. In this case, we give sharp bounds for the multigraded regularity index of their multig…
$V$ is a complete intersection scheme in a multiprojective space if it can be defined by an ideal $I$ with as many generators as $\textrm{codim}(V)$. We investigate the multigraded regularity of complete intersections scheme in $\mathbb{P}^n\times \mathbb{P}^m$. We explicitly compute many values of the Hilbert functions of $0$-dimensional complete intersections. We show that these values only depend upon $n,m$, and the bidegrees of the generators of $I$. As a result, we provide a sharp upper bound for the multigraded regularity of $0$-dimensional complete intersections.
The study of the stratification associated to the number of generators of the ideals in the punctual Hilbert scheme of points on the affine plain goes back to the '70s. In this paper, we present an elegant formula for the E-polynomials of these strata.
In this paper we consider the Hilbert scheme $Hilb_{p(t)}^n$ parameterizing subschemes of $P^n$ with Hilbert polynomial $p(t)$, and we investigate its locus containing points corresponding to schemes with regularity lower than or equal to a fixed integer $r'$. This locus is an open subscheme of $Hilb_{p(t)}^n$ and, for every $s\geq r'$, we describe it as a locally closed subscheme of the Grasmannian $Gr_{p(s)}^{N(s)}$ given by a set of equations of degree $\leq \mathrm{deg}(p(t))+2$ and linear inequalities in the coordinates of the Plücker embedding.
We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann's regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert schemes.
Using the methods developed in [LQW], math.AG/0009132, we obtain a second set of generators for the cohomology ring of the Hilbert scheme of points on an arbitrary smooth projective surface over the field of complex numbers. These generators have clear and simple geometric as well as algebraic descriptions.
We prove the existence of ternary forms admitting apolar sets of points of cardinality equal to the Waring rank, but having different Hilbert function and different regularity. This is done exploiting liaison theory and Cayley-Bacharach properties for sets of points in the projective plane
We establish a form of the Gotzmann representation of the Hilbert polynomial based on rank and generating degrees of a module, which allow for a generalization of Gotzmann's Regularity Theorem. Under an additional assumption on the generating degrees, the Gotzmann regularity bound becomes sharp. An analoguous modification of the Macaulay representation is used along the way, which generalizes the theorems of Macaulay and Green, and Gotzmann's Persistence Theorem.
In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal $I$ in terms of the degrees of its generators. We assume that $I$ defines a local complete intersection with log canonical singularities.
We establish bounds for the Castelnuovo-Mumford regularity of a finitely generated graded module and its symmetric powers in terms of the degrees of the generators of the module and the degrees of their relations. We extend to modules (and improve) the known bounds for homogenous ideal in a polynomial ring established by Galligo, Guisti, Caviglia and Sbarra.
Categorical characterizations of regularity for algebraic stacks
Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano（米兰大学恩里克斯数学系）；School of Mathematics & Statistics The University of Melbourne（墨尔本大学数学与统计学院）
Bounds for the Castelnuovo-Mumford regularity of Ext modules, over a polynomial ring over a field, are given in terms of the initial degrees, Castelnuovo-Mumford regularities and number of generators of the two graded modules involved. These general bounds are refined in the case the second module is the ring. Other estimates, for instance on the size of graded pieces of these modules, are given. We also derive a bound on the homological degree in terms of the Castelnuovo-Mumford regularity. This answers positively a question raised by Vasconcelos.
We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded a-invariant, and all modules with bounded Castelnuovo-Mumford regularity. The first of these cones is infinite-dimensional and simplicial, the second is finite-dimensional but neither simplicial nor polyhedral, and the third is finite-dimensional and simplicial.
We give a bound on the Castelnuovo-Mumford regularity of a homogeneous ideals I, in a polynomial ring A, in terms of number of variables and the degrees of generators, when the dimension of A/I is at most two. This bound improves the one obtained by Caviglia and Sbarra in this case. In the continuation of the examples constructed by Clare D'Cruz and the first author, we use families of monomial curves to construct homogeneous ideals showing that these bounds are quite sharp.
We study bounds for the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular our aim is to give a positive answer to a question posed by Bayer and Mumford, by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyze Giusti's proof, which provides the result in characteristic 0, giving some insight on the combinatorial properties needed in that context. For the general case we provide a new argument which employs Bayer and Stillma…
The NumericalHilbert package for Macaulay2 includes algorithms for computing local dual spaces of polynomial ideals, and related local combinatorial data about its scheme structure. These techniques are numerically stable, and can be used with floating point arithmetic over the complex numbers. They provide a viable alternative in this setting to purely symbolic methods such as standard bases. In particular, these methods can be used to compute initial ideals, local Hilbert functions and Hilbert regularity.
We derive new bounds for the Castelnuovo-Mumford regularity of the ideal sheaf of a complex projective manifold of any dimension. They depend linearly on the coefficients of the Hilbert polynomial, and are optimal for rational scrolls, but most likely not for other varieties. Our proof is based on an observation of Lazarsfeld in his approach for surfaces and does not require the (full) projection step. We obtain a bound for each partial linear projection of the given variety, as long as a certain vanishing condition on the fibers of a general projection holds.
Let $D$ be a smooth divisor on a non singular surface $S$. We compute Betti numbers of the relative Hilbert scheme of points of $S$ relative to $D$. In the case of $\PP^2$ and a line in it, we give an explicit set of generators and relations for the cohomology groups of this space.
Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers.
We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$and genus $g \geq 4$ in ${\Bbb P}^r$, in the case when $d \geq 2g+1 $ and $r \leq d-g$. We express the classes of the generators in terms of some ``natural'' divisor classes.
