1,207 research outputs found
On a class of power ideals
In this paper we study the class of power ideals generated by the forms
where is a fixed
primitive -root of unity and for all . For
, by using a -grading on ,
we compute the Hilbert series of the associated quotient rings via a simple
numerical algorithm. We also conjecture the extension for . Via Macaulay
duality, those power ideals are related to schemes of fat points with support
on the points in . We
compute Hilbert series, Betti numbers and Gr\"obner basis for such
-dimensional schemes. This explicitly determines the Hilbert series of the
power ideal for all : that this agrees with our conjecture for is
supported by several computer experiments
On some invariants in numerical semigroups and estimations of the order bound
We study suitable parameters and relations in a numerical semigroup S. When S
is the Weierstrass semigroup at a rational point P of a projective curve C, we
evaluate the Feng-Rao order bound of the associated family of Goppa codes.
Further we conjecture that the order bound is always greater than a fixed value
easily deduced from the parameters of the semigroup: we also prove this
inequality in several cases
An application of type sequences to the blowing-up
Let I be an m-primary ideal of a one-dimensional, analytically irreducible
and residually rational local Noetherian domain R. Given the blowing-up of R
along I we establish connections between the type-sequence of R and classical
invariants like multiplicity, genus and reduction exponent of I.Comment: 19 page
On semigroup rings with decreasing Hilbert function
In this paper we study the Hilbert function HR of one-dimensional semigroup
rings R = k[[S]]. For some classes of semigroups, by means of the notion of
support of the elements in S, we give conditions on the generators of S in
order to have decreasing HR. When the embedding dimension v and the
multiplicity e verify v + 3 ? e ? v + 4, the decrease of HR gives explicit
description of the Apery set of S. In particular for e = v+3, we classify the
semigroups with e = 13 and HR decreasing, further we show that HR is
non-decreasing if e < 12. Finally we deduce that HR is non-decreasing for every
Gorenstein semigroup ring with e ? v + 4
On the Hilbert function of general fat points in
We study the bi-graded Hilbert function of ideals of general fat points with
same multiplicity in . Our first tool is the
multiprojective-affine-projective method introduced by the second author in
previous works with A.V. Geramita and A. Gimigliano where they solved the case
of double points. In this way, we compute the Hilbert function when the
smallest entry of the bi-degree is at most the multiplicity of the points. Our
second tool is the differential Horace method introduced by J. Alexander and A.
Hirschowitz to study the Hilbert function of sets of fat points in standard
projective spaces. In this way, we compute the entire bi-graded Hilbert
function in the case of triple points.Comment: 25 pages; minor changes (Remark 1.7 added and Example 3.13 improved
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