In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous $k$-algebras $A$ in terms of the dimension $d$, the codimension $c$, the initial degree $q$, and the length of the local cohomology modules of $A$. Next, we introduce the notion of Buchsbaum $k$-algebras with minimal multiplicity of degree $q$, and give several characterizations for those rings. In particular, we will show that those algebras have linear free resolutions. Further, we will give many examples of those algebras.
In this paper, we study non-Cohen--Macaulay Buchsbaum Stanley--Reisner rings with linear free resolution. In particular, for given integers $c$, $d$, $q$ with $c \ge 1$, $2 \le q \le d$, we give an upper bound $h_{c,d,q}$ on the dimension of the unique non-vanishing homology $\widetilde{H}_{q-2}(Δ;k)$ of a $d$-dimensional Buchsbaum ring $k[Δ]$ with $q$-linear resolution and codimension $c$. Also, we discuss about existence for such Buchsbaum rings with $\dim_k \widetilde{H}_{q-2}(Δ;k) = h$ for any $h$ with $0 \le h \le h_{c,d,q}$, and prove an existence theorem in the case of $q=d=3$ using the…
We generalize an improved Lech bound, due to Huneke, Smirnov, and Validashti, for the Buchsbaum-Rim multiplicity and mixed multiplicity. We reduce the problem to the graded case and then to the polynomial ring case. There we use complete reductions, studied by Rees, to prove sharper bounds for the mixed multiplicity in low dimensions before proving the general case.
We study the Buchsbaum-Rim multiplicity, ab inicio, through a Koszul-Čech spectral sequence. We show that the Buchsbaum-Rim multiplicity is the arithmetic genus (Euler characteristic) of Koszul homology sheaves on a projective space over the base scheme.
Let A be a Buchsbaum local ring with the maximal ideal m and let e(A) denote the multiplicity of A. Let Q be a parameter ideal in A and put I=Q:m. Then the equality I^2=QI holds true, if e(A)=2 and depthA>0. The assertion is no longer true, unless e(A)=2. Counterexamples are given.
In this article we prove that the Buchsbaum-Rim multiplicity $e(F/N)$ of a parameter module $N$ in a free module $F=A^r$ is bounded above by the colength $\ell_A(F/N)$. Moreover, we prove that once the equality $\ell_A(F/N)=e(F/N)$ holds true for some parameter module $N$ in $F$, then the base ring $A$ is Cohen-Macaulay.
In this article, we compute the Buchsbaum-Rim function of two variables associated to a direct sum of cyclic modules and give a formula for the last positive associated Buchsbaum-Rim multiplicity in terms of the ordinary Hilbert-Samuel multiplicity of an ideal. This is a generalization of a formula for the last positive Buchsbaum-Rim multiplicity given by Kirby and Rees.
The associated Buchsbaum-Rim multiplicities of a module are a descending sequence of non-negative integers. These invariants of a module are a generalization of the classical Hilbert-Samuel multiplicity of an ideal. In this article, we compute the associated Buchsbaum-Rim multiplicity of a direct sum of cyclic modules and give a formula for the second to last positive Buchsbaum-Rim multiplicity in terms of the ordinary Buchsbaum-Rim and Hilbert-Samuel multiplicities. This is a natural generalization of a formula given by Kirby and Rees.
Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and its Hilbert-Samuel multiplicity using linkage theory. As applications, we establish several multiplicity formula that express the Buchsbaum-Rim multiplicity of M in terms of the Hilbert-Samuel multiplicities of ideals related to a certain minimal reduction U of M. Moreover, there exists m-primary Bourbaki ideals I and J of the modules F and M respectively such…
In the paper we prove Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems on $\mathbb P^2$ for $m=7$, 8, 9, 10, i.e. systems of curves of given degree passing through points in general position with multiplicities at least $m,...,m,m_0$, where $m=7$, 8, 9, 10, $m_0$ is arbitrary.
Let $R$ be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of $R$ by means of local $j$-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion $A\subseteq B$ of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of $A$ in terms of that of $B$ and of local $j$-multiplicities of hyperplane sections along ${\rm Proj}\,(B)$. Our formulas can be used to find the multiplicity of special fiber ring…
Our aim in this paper is to obtain a formula for the multiplicity of the maximal homogeneous ideal of the multi-graded extended Rees algebra. This formula generalizes the one obtained in [8] by Katz and Verma for the ordinary extended Rees algebra.
In this note we compare the a-invariant of a homogeneous algebra B to the a-invariant of a subalgebra A. In particular we show that if $A \subset B$ is a finite homogeneous inclusion of standard graded domains over an algebraically closed field with A normal and B of minimal multiplicity then A has minimal multiplicity. In some sense these results are algebraic generalizations of Hurwitz type theorems.
Let $R$ be a polynomial ring over a field of characteristic zero and let $I \subset R$ be a graded ideal of height $N$ which is minimally generated by $N+1$ homogeneous polynomials. If $I=(f_1,...,f_{N+1})$ where $f_i$ has degree $d_i$ and $(f_1,...,f_N)$ has height $N$, then the multiplicity of $R/I$ is bounded above by $\prod_{i=1}^N d_i - \max\{1, \sum_{i=1}^N (d_i-1) - (d_{N+1}-1) \}$.
Herzog, Huneke, and Srinivasan have conjectured that for any homogeneous $k$-algebra, the multiplicity is bounded above by a function of the maximal degrees of the syzygies and below by a function of the minimal degrees of the syzygies. The goal of this paper is to establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field $k$ for $k$-algebras $k[x_1, ..., x_n]/I$ being $I$ a determinantal ideal of arbitrary codimension.
We show that the Hilbert-Kunz multiplicity is a rational number for an R_+-primary homogeneous ideal I=(f_1, ..., f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. Moreover we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f_1, . . ., f_n) on the projective curve Y = Proj R.
Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two invariants of $R/J$ and the degree of $F$. We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.
This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ a $m \times n$-matrix of variables, we utilize Gröbner bases to give a closed form the length $λ( k[X] / (I_2(X) + \mathfrak{m}^{ \lceil sq \rceil} + \mathfrak{m}^{[q]} ))$ where $s \in \mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $\mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {\it polynomial} function of $q$ for all $s$.
Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.
Let Z be a fat point scheme in P^2 supported on general points. Here we prove that if the multiplicities are at most 3 and the length of Z is sufficiently high then the number of generators of the homogeneous ideal I_Z in each degree is as small as numerically possible. Since it is known that Z has maximal Hilbert function, this implies that Z has the expected minimal free resolution.
Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.
The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect the containment in the integral closure or the tight closure of a parameter ideal using F-thresholds. We formulate a conjecture bounding $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals, and $J$ is generated by a system of parameters. We prove the conjecture when $J$ is a monomial ideal in a polynomial …
The $F$-threshold $c^J(\a)$ of an ideal $\a$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We study a conjecture formulated in an earlier paper \cite{HMTW} by the same authors together with M. Mustaţă, which bounds $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\a$ and $J$ are generated by homogeneous systems of parameters in a Noetherian graded $k$-algebra. We a…
Consider a rational projective plane curve C parameterized by three homogeneous forms h1,h2,h3 of the same degree d in the polynomial ring R=k[x,y] over the field k. Extracting a common factor, we may harmlessly assume that the ideal I=(h1,h2,h3)R has height two. Let phi be a homogeneous minimal Hilbert-Burch matrix for the row vector [h1,h2,h3]. So, phi is a 3 by 2 matrix of homogeneous forms from R; the entries in column m have degree dm, with d1 \le d2 and d1+d2=d. The Rees algebra $cal R$ of I is the subring k[h1t,h2t,h3t] of the polynomial ring k[t]. The bi-projective spectrum of $cal R$ …
Resultant multiplicity via projective degrees and applications to tensor eigenvalues
Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）；Inria Paris and Sorbonne University（法国国家数字与信息技术研究所巴黎分部和索邦大学）；Department of Mathematics, National and Kapodistrian University of Athens（雅典国立卡波季斯特里安大学数学系）
Patterns on numerical semigroups are multivariate linear polynomials, and they are said to be admissible if there exists a numerical semigroup such that evaluated at any nonincreasing sequence of elements of the semigroup gives integers belonging to the semigroup. In a first approach, only homogeneous patterns where analized. In this contribution we study conditions for an eventually non-homogeneous pattern to be admissible, and particularize this study to the case the independent term of the pattern is a multiple of the multiplicity of the semigroup. Moreover, for the so called strongly admis…
Consider a rational projective curve C of degree d over an algebraically closed field k. There are n homogeneous forms g_1,...,g_n of degree d in B=k[x,y] which parameterize C in a birational, base point free, manner. We study the singularities of C by studying a Hilbert-Burch matrix phi for the row vector [g_1,...,g_n]. In the "General Lemma" we use the generalized row ideals of phi to identify the singular points on C, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch. Let p be a singular point on the parameterized planar curve C which c…
