Search

In this paper, to any subset A ⊂ Z n we explicitly associate a unique monomial projection Y n , d A of a Veronese variety, whose Hilbert function coincides with the cardinality of the t-fold sumsets t A . This link allows us to tackle the classical problem of determining the polynomial p A ∈ Q [ t ] such that | t A | = p A (t) for all t ≥ t 0 and the minimum integer n 0 (A) ≤ t 0 for which this condition is satisfied, i.e. the so-called phase transition of | t A | . We use the Castelnuovo–Mumford regularity and the geometry of Y n , d A to describe the polynomial p A (t) and to derive new bounds for n 0 (A) under some technical assumptions on the convex hull of A ; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties Y n , d A . [ABSTRACT FROM AUTHOR]

Let I be a homogeneous ideal in a polynomial ring S. In this paper, we extend the study of the asymptotic behavior of the minimum distance function δI of I and give bounds for its stabilization point, rI, when I is an F -pure or a square-free monomial ideal. These bounds are related with the dimension and the Castelnuovo–Mumford regularity of I.

The aim of this note is to exploit a new relationship between additive combinatorics and the geometry of monomial projective curves. We associate to a finite set of non-negative integers A = { a 1 , … , a n } a monomial projective curve C A ⊂ P k n - 1 such that the Hilbert function of C A and the cardinalities of s A : = { a i 1 + ⋯ + a i s ∣ 1 ≤ i 1 ≤ ⋯ ≤ i s ≤ n } agree. The singularities of C A determines the asymptotic behaviour of |sA|, equivalently the Hilbert polynomial of C A , and the asymptotic structure of sA. We show that some additive inverse problems can be translate to the rigidity of Hilbert polynomials and we improve an upper bound of the Castelnuovo-Mumford regularity of monomial projective curves by using results of additive combinatorics. [ABSTRACT FROM AUTHOR]

For a given d-dimensional polyhedral complex Δ and a given degree k, we consider the vector space of piecewise polynomial functions on Δ of degree at most k with a different smoothness condition on each pair of adjacent d-faces of Δ. This is a finite dimensional vector space. The fundamental problem in Approximation Theory is to compute the dimension of this vector space. It is known that the dimension is given by a polynomial for sufficiently large k via commutative algebra. By using the technique of McDonald and Schenck [3] and extending their result to a plane polyhedral complex Δ with varying smoothness conditions, we determine this polynomial. This gives a complete answer for the dimension. At the end we discuss some examples through this technique. [ABSTRACT FROM AUTHOR]

Let ( R , m ) be a Noetherian local ring of dimension d > 0 and depth R ≥ d − 1 . Let Q be a parameter ideal of R . In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient e i ( Q ) under certain assumptions on the depth of associated graded ring G ( Q ) . For 2 ≤ i ≤ d , we show that (1) e i ( Q ) ≤ 0 provided depth G ( Q ) ≥ d − 2 and (2) e i ( Q ) ≥ − λ R ( H m d − 1 ( R ) ) provided depth G ( Q ) ≥ d − 1 . It is proved that e 3 ( Q ) ≤ 0 . Further, we obtain a necessary condition for the vanishing of the last coefficient e d ( Q ) . As a consequence, we characterize the vanishing of e 2 ( Q ) . Our results generalize [5, Theorem 3.2] and [11, Corollary 4.5] . [ABSTRACT FROM AUTHOR]

We study consequences, for a standard graded algebra, of extremal behavior in Green's Hyperplane Restriction Theorem. First, we extend his Theorem 4 from the case of a plane curve to the case of a hypersurface in a linear space. Second, assuming a certain Lefschetz condition, we give a connection to extremal behavior in Macaulay's theorem. We apply these results to show that ( 1 , 19 , 17 , 19 , 1 ) is not a Gorenstein sequence, and as a result we classify the sequences of the form ( 1 , a , a − 2 , a , 1 ) that are Gorenstein sequences. [ABSTRACT FROM AUTHOR]

Let ( A , m ) be a local complete intersection ring of dimension d and let I be an m -primary ideal. Let M be a maximal Cohen–Macaulay A -module. For i = 0 , 1 , ⋯ , d , let e i I ( M ) denote the i th Hilbert-coefficient of M with respect to I . We prove that for i = 0 , 1 , 2 , the function j ↦ e i I ( Syz j A ( M ) ) is of quasi-polynomial type with period 2. Let G I ( M ) be the associated graded module of M with respect to I . If G I ( A ) is Cohen–Macaulay and dim ⁡ A ≤ 2 we also prove that the functions j ↦ depth G I ( Syz 2 j + i A ( M ) ) are eventually constant for i = 0 , 1 . Let ξ I ( M ) = lim l → ∞ ⁡ depth G I l ( M ) . Finally we prove that if dim ⁡ A = 2 and G I ( A ) is Cohen–Macaulay then the functions j ↦ ξ I ( Syz 2 j + i A ( M ) ) are eventually constant for i = 0 , 1 . [ABSTRACT FROM AUTHOR]

Let X = m P 1 + ⋯ + m P n + k be a fat point subscheme of P n , where S u p p ( X ) consists of n + k distinct points which generate P n . We study the regularity index τ ( X ) of X , which is the least degree in which the Hilbert function of X equals its Hilbert polynomial. We prove that the generalized Segre's bound for τ ( X ) holds if n ≥ 4 and there are k + 3 points of S u p p ( X ) on a linear subspace Λ ≃ P 3 . We assume S u p p ( X ) is not in general position and call d the least integer for which there exists a linear subspace Λ of dimension d containing at least d + 2 points of S u p p ( X ) . We prove that the generalized Segre's bound holds for simple points when either 3 ≤ k ≤ n + 1 and d > k − 3 or k = 4 with no restriction on d . For m ≥ 2 we prove the generalized Segre's bound when S u p p ( X ) consists of n + 4 points and either there are at least 3 points on a line or at least 5 points on a plane or at least 6 points on a linear subspace Λ ≃ P 3 . Finally we prove that, in general, 2 m − 1 ≤ τ ( X ) ≤ 2 m when 3 ≤ k ≤ n − 1 and d > k − 1 , and we extend this result to the non-equimultiple case. We also provide cases in which the previous bound gives the generalized Segre's bound. [ABSTRACT FROM AUTHOR]

We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a Grassmannian are given by a single Fitting ideal. [ABSTRACT FROM AUTHOR]

We study Gorenstein h -vectors ( h 0 , h 1 , … , h e ) of socle degree e with h 1 ≥ h i for each i , and find a necessary and sufficient condition that there exists a nonunimodal Gorenstein sequence in terms of socle degree e and codimension h 1 . In particular, we prove that there exist nonunimodal Gorenstein h -vectors if and only if h 1 ≥ 4 e − 3 for e ≥ 4 . We also find infinitely many cases of non-Gorenstein h -vectors having the lower bound in [15] . This result generalizes the recent work [17] that the h -vector ( 1 , 12 , 11 , 12 , 1 ) is not a Gorenstein sequence. [ABSTRACT FROM AUTHOR]

Let R be a Cohen–Macaulay local ring of dimension d with infinite residue field. Let I be an R -ideal that has analytic spread ℓ ( I ) = d , satisfies the G d condition and the weak Artin–Nagata property A N d − 2 − . We provide a formula relating the length λ ( I n + 1 / J I n ) to the difference P I ( n ) − H I ( n ) , where J is a general minimal reduction of I , P I ( n ) and H I ( n ) are respectively the generalized Hilbert–Samuel polynomial and the generalized Hilbert–Samuel function. We then use it to establish formulas to compute the generalized Hilbert coefficients of I . As an application, we extend Northcott's inequality to non- m -primary ideals. Furthermore, when equality holds, we prove that the ideal I enjoys nice properties. Indeed, if this is the case, then the reduction number of I is at most one and the associated graded ring of I is Cohen–Macaulay. We also recover results of G. Colomé-Nin, C. Polini, B. Ulrich and Y. Xie on the positivity of the generalized first Hilbert coefficient j 1 ( I ) . Our work extends that of S. Huckaba, C. Huneke and A. Ooishi to ideals that are not necessarily m -primary. [ABSTRACT FROM AUTHOR]

For a scheme of fat points Z defined by the saturated ideal I Z , the regularity index computes the Castelnuovo–Mumford regularity of the Cohen–Macaulay ring R / I Z . For points in “general position” we improve the bound for the regularity index computed by Segre for P 2 and generalised by Catalisano, Trung and Valla for P n . Moreover, we prove that the generalised Segre's bound conjectured by Fatabbi and Lorenzini holds for n + 3 arbitrary points in P n . We propose a modification of Segre's conjecture for arbitrary points and we discuss some evidences. [ABSTRACT FROM AUTHOR]

Let M be a finitely generated module of dimension d and depth t over a Noetherian local ring ( A, $${\mathfrak{m}}$$ ) and I an $${\mathfrak{m}}$$ -primary ideal. In the main result it is shown that the last t Hilbert coefficients $${e_{d-t+1}(I,M),\ldots, e_{d}(I,M)}$$ are bounded below and above in terms of the first d − t + 1 Hilbert coefficients $${e_{0}(I,M),\ldots,e_{d-t}(I,M)}$$ and d. [ABSTRACT FROM AUTHOR]

LetRbe a Cohen–Macaulay local ring with a canonical module ωR. LetIbe an 𝔪-primary ideal ofRandM, a maximal Cohen–MacaulayR-module. We call the functionn⟼ℓ(HomR(M, ωR/In+1ωR)) thedual Hilbert–Samuel function of M with respect to I. By a result of Theodorescu, this function is of polynomial type. We study its first two normalized coefficients. In particular, we analyze the case whenRis Gorenstein. [ABSTRACT FROM PUBLISHER]

Let A = S / J be a standard artinian graded algebra over the polynomial ring S . A theorem of Macaulay dictates the possible growth of the Hilbert function of A from any degree to the next, and if this growth is the maximal possible then strong consequences have been given by Gotzmann. It can be phrased in terms of the base locus of the linear system defined by the relevant component(s) of J . If J is the artinian reduction of the ideal of a finite set of points in projective space then this maximal growth for A was shown by Bigatti, Geramita and the second author to imply strong geometric consequences for the points. We now suppose that the growth of the Hilbert function is one less than maximal. This again has (not as) strong consequences for the base locus defined by the relevant component. And when J is the artinian reduction of the ideal of a finite set of points in projective space, we prove that almost maximal growth again forces geometric consequences. [ABSTRACT FROM AUTHOR]

Artinian ideals $${I\subset R: =k[x_1,..., x_n]}$$ generated by m general forms of given degree have attracted a great deal of attention recently, and one of the main challenging problems is to determine its Hilbert function. The Hilbert function of R/ I was conjectured by Fröberg, and it is well known that the Weak Lefschetz property imposes severe constraints on the possible Hilbert functions. In this short note, we will focus our attention on Artinian ideals $${I \subset R}$$ generated by m general forms all of the same degree d and we analyze whether the Weak Lefschetz property is satisfied. More precisely, for m = n or n ≤ 4 the Weak Lefschetz property holds, and our goal will be to prove that for any integers n and d, the Weak Lefschetz property also holds provided m falls into the interval $${[\frac{1}{d+1} \alpha_{n,d}, \alpha_{n,d}]}$$ where $${\alpha_{n,d}={n+d-1\choose d}}$$ . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the minimal free resolution of certain level algebras which have minimal Hilbert Function among Artinian graded algebras of given socle degree and type. As a consequence of this results, we discover a new class of Hilbert Functions which do not support a unique smallest set of graded Betti numbers and another class which do not support all the possible resolutions between its maximum and minimum elements. [ABSTRACT FROM AUTHOR]

Hilbert–Kunz multiplicity is known to be a very mysterious invariant of a ring or an ideal. We will show a very beautiful formula on Hilbert–Kunz multiplicity for integrally closed ideals in two-dimensional Gorenstein rational singularities. In the proof, “McKay correspondence” and “Riemann–Roch formula” play essential roles. Also this formula gives a new significance to “good ideals”. [ABSTRACT FROM AUTHOR]