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Let A be a finite subset of an abelian group (G, +). Let h $ge$ 2 be an integer. If |A| $ge$ 2 and the cardinality |hA| of the h-fold iterated sumset hA = A + $times$ $times$ $times$ + A is known, what can one say about |(h -- 1)A| and |(h + 1)A|? It is known that |(h -- 1)A| $ge$ |hA| (h--1)/h , a consequence of Pl{u}nneckes inequality. Here we improve this bound with a new approach. Namely, we model the sequence |hA| h$ge$0 with the Hilbert function of a standard graded algebra. We then apply Macaulays 1927 theorem on the growth of Hilbert functions, and more specifically a recent condensed version of it. Our bound implies |(h -- 1)A| $ge$ $theta$(x, h) |hA| (h--1)/h for some factor $theta$(x, h) > 1, where x is a real number closely linked to |hA|. Moreover, we show that $theta$(x, h) asymptotically tends to e $approx$ 2.718 as |A| grows and h lies in a suitable range varying with |A|.
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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 had shown earlier that for a standard graded ring $R$ and a graded ideal $I$ in characteristic $p>0$, with $ell(R/I) <infty$, there exists a compactly supported continuous function $f_{R, I}$ whose Riemann integral is the HK multiplicity $e_{HK}(R, I)$. We explore further some other invariants, namely the shape of the graph of $f_{R, {bf m}}$ (where ${bf m}$ is the graded maximal ideal of $R$) and the maximum support (denoted as $alpha(R,I)$) of $f_{R, I}$. In case $R$ is a domain of dimension $dgeq 2$, we prove that $(R, {bf m})$ is a regular ring if and only if $f_{R, {bf m}}$ has a symmetry $f_{R, {bf m}}(x) = f_{R, {bf m}}(d-x)$, for all $x$. If $R$ is strongly $F$-regular on the punctured spectrum then we prove that the $F$-threshold $c^I({bf m})$ coincides with $alpha(R,I)$. As a consequence, if $R$ is a two dimensional domain and $I$ is generated by homogeneous elements of the same degree, thene have (1) a formula for the $F$-threshold $c^I({bf m})$ in terms of the minimum strong Harder-Narasimahan slope of the syzygy bundle and (2) a well defined notion of the $F$-threshold $c^I({bf m})$ in characteristic $0$. This characterisation readily computes $c^{I(n)}({bf m})$, for the set of all irreducible plane trinomials $k[x,y,z]/(h)$, where ${bf m} = (x,y,z)$ and $I(n) = (x^n, y^n, z^n)$.
In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area of dynamic mathematical activity. Motivated by the ever increasing interest in this field, our goal is to gather together many new developments of this theory into one place, and to present them using a unifying approach which gives self-contained and easier proofs. In this text we shall discuss many results by different authors, following essentially the direction typified by the pioneering work of J. Sally. Our personal view of the subject is most visibly expressed by the presentation of Chapters 1 and 2 in which we discuss the use of the superficial elements and related devices. Basic techniques will be stressed with the aim of reproving recent results by using a more elementary approach. Over the past few years several papers have appeared which extend classical results on the theory of Hilbert functions to the case of filtered modules. The extension of the theory to the case of general filtrations on a module has one more important motivation. Namely, we have interesting applications to the study of graded algebras which are not associated to a filtration, in particular the Fiber cone and the Sally-module. We show here that each of these algebras fits into certain short exact sequences, together with algebras associated to filtrations. Hence one can study the Hilbert function and the depth of these algebras with the aid of the know-how we got in the case of a filtration.
It remains an open problem to classify the Hilbert functions of double points in $mathbb{P}^2$. Given a valid Hilbert function $H$ of a zero-dimensional scheme in $mathbb{P}^2$, we show how to construct a set of fat points $Z subseteq mathbb{P}^2$ of double and reduced points such that $H_Z$, the Hilbert function of $Z$, is the same as $H$. In other words, we show that any valid Hilbert function $H$ of a zero-dimensional scheme is the Hilbert function of a set of a positive number of double points and some reduced points. For some families of valid Hilbert functions, we are also able to show that $H$ is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines.
Here we compute Hilbert-Kunz functions of any nontrivial ruled surface over ${bf P}^1_k$, with respect to all ample line bundles on it.
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