Duong Quoc Viet
This paper investigates the relationship between multiplicities and the degree sequence of ideals in graded algebras, gives multiplicity equations of graded rings via the degree sequence of ideals, and characterizes mixed multiplicities and multiplicities of Rees rings in terms of the degree sequence of ideals. As an application, the paper answered an open question on the multiplicity of Rees rings.
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In this paper, we present an algorithm for computing the minimal reductions of $\mathfrak{m}$-primary ideals of Cohen--Macaulay local rings. Using this algorithm, we are able to compute the Hilbert--Samuel multiplicities and solve the membership problem for the integral closure of $\mathfrak{m}$-primary ideals.
In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities of almost complete intersections. We also introduce a new class of ideals that extends the family of monomial complete intersections, and give a formula for their multiplicity, as well as a visual interpretation of this invariant.
Buchsbaum-Rim Multiplicities as Hilbert-Samuel Multiplicities
研究了具有极大理想m的正则2维局部环上有限长有限生成模M在自由模F中的Buchsbaum-Rim重数.首先，我们利用联系理论研究了m-准素理想的色长及其Hilbert-Samuel重数。作为应用，我们建立了几个表示M的Buchsbaum-Rim重数的重数公式，这些重数是用与M的某个极小约化U有关的理想的Hilbert-Samuel重数表示的.此外，分别存在模F和M的m-准素Bourbaki理想I和J，使得F/M同构于I/J。后一部分是有关的图形解释的多重性的情况下，单项理想。
We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of $m$-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of $r$ filtrations on an analytically unramified local ring $R$ come from the coefficients of a suitable homogeneous polynomial in $r$ variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of $m$-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterizat…
A note on joint reductions and mixed multiplicities
设$(A, \frak m)$是具有极大理想$\frak{m}$和无限剩余域$k = A/\frak{m}.$的Noether局部环，$J$是$A$的$\frak m$ -准素理想，$I_1,...,I_s$理想，$M$是一个生成的$A$ -模.本文将$(I_1,..., I_s,J)$关于$M$的混合重数解释为它们的联合约化的重数。这推广了Rees关于混合重数的定理（定理2.4）。作为一个应用，我们表明，混合多重性也是里斯的表面序列的多重性。
Multiplicities Associated to Graded Families of Ideals
我们证明了与分次理想族相关的重数极限在非常一般的条件下存在。我们的大部分结果适用于解析非分歧等特征局部环，具有完美剩余域。我们给出了一些应用程序，包括一个“体积=多重性”公式，推广公式的Lazarsfeld和Mustata，并证明了Ulrich和Validashti的多重性存在作为一个限制理想在相当一般的环，包括分析的局部域。我们还证明了一个渐近的“加法公式”的限制的多重性，和公式限制增长的估值，这回答了一个问题所提出的作者，起亚Dalili和奥尔加Kashcheyeva。我们的证明的灵感来自奥肯科夫的哲学，计算限制的多重性的体积切片的适当的锥所产生的半群确定的适当的过滤上的一个家庭的代数对象。
Asymptotic Multiplicities
给出了分次理想族作为极限的重数存在性定理的几个新的应用，包括分次理想族的一个非常一般的Minkowski型不等式，局部体积作为极限的一个非常一般的存在性公式和模的重数作为极限的一个非常一般的存在性定理.
We show the existence (and define) the mixed multiplicities of arbitrary graded families of ideals under mild assumptions. In particular, our methods and results are valid for the case of arbitrary $\mathfrak{m}$-primary graded families. Furthermore, we provide a far-reaching "Volume = Multiplicity formula" for the mixed multiplicities of graded families of ideals.
Hilbert functions of multigraded algebras, mixed multiplicities of ideals and their applications
本文综述了关于多重分次代数的Hilbert函数和理想的混合重数的主要结果，包括它们在计算具有孤立奇点的复解析超曲面的Milnor数、爆破代数的重数和多面体的混合体积中的应用。
Mixed multiplicities of ideals versus mixed volumes of polytopes
本文的主要结果将格多面体的混合体积解释为理想的混合重数，将理想的混合重数解释为Samuel重数。特别是，我们可以给一个纯粹的代数证明伯恩斯坦定理断言，共同零点的数量系统的洛朗多项式方程在环面是有界的混合体积以上的牛顿多面体。
Minimal values of multiplicities of ideals have a strong relation with the depth of blowup algebras. In this paper, we introduce the notion of Goto-minimal $j$-multiplicity for ideals of maximal analytic spread. In a Cohen-Macaulay ring, inspired by the work of S. Goto, A. Jayanthan, T. Puthenpurakal, and J. Verma, we study the interplay among this new notion, the notion of minimal $j$-multiplicity introduced by C. Polini and Y. Xie, and the Cohen-Macaulayness of the fiber cone of ideals satisfying certain residual assumptions. We are also able to provide a bound on the reduction number of ide…
Multiplicities of semidualizing modules
交换诺特环R上的模C是半对偶的，如果Hom_R（C，C）\cong R且Ext^i_R（C，C）= 0，对所有i \geq 1.对于某些局部Cohen-Macaulay环（R，m），证明了对于所有半对偶R-模C和所有m-准素理想J，Hilbert-Samuel重数eR（J;C）= eR（J;R）是相等的.我们所研究的环类包括由定义射影空间中胖点方案的理想或单项式理想所决定的环类.
This paper defines the Euler-Poincaré characteristic of joint reductions of ideals which concerns the maximal terms in the Hilbert polynomial; characterizes the positivity of mixed multiplicities in terms of minimal joint reductions; proves the additivity and other elementary properties for mixed multiplicities. The results of the paper together with the results of [17] seem to show a natural and nice picture of mixed multiplicities of maximal degrees.
Let $\mathfrak{q}$ denote an $\mathfrak{m}$-primary ideal of a $d$-dimensional local ring $(A, \mathfrak{m}).$ Let $\underline{a} = a_1,\ldots,a_d \subset \mathfrak{q}$ be a system of parameters. Then there is the following inequality for the multiplicities $c \cdot e(\mathfrak{q};A) \leq e(\underline{a};A)$ where $c$ denotes the product of the initial degrees of $a_i$ in the form ring $G_A(\mathfrak{q}).$ The aim of the paper is a characterization of the equality as well as a description of the difference by various homological methods via Koszul homology. To this end we have to characterize …
The original mixed multiplicity theory considered the class of mixed multiplicities concerning the terms of highest total degree in the Hilbert polynomial. This paper defines a broader class of mixed multiplicities that concern the maximal terms in this polynomial, and gives many results, which are not only general but also more natural than many results in the original mixed multiplicity theory.
Let $S$ be a polynomial ring over an algebraic closed field $k$ and $ \mathfrak p =(x,y,z,w) $ a homogeneous height four prime ideal. We give a finite characterization of the degree two component of ideals primary to $\mathfrak p$, with multiplicity $e \leq 3$. We use this result to give a tight bound on the projective dimension of almost complete intersections generated by five quadrics with $e \leq 3$.
On multiplicities of graded sequences of ideals
推广了Ein-Lazarsfeld-Smith（math.AG/0202303）的一个结果，证明了对于任意零维理想序列，序列的重数与其体积相等.这是通过对单项式理想的变形来完成的。作为我们结果的结果，我们得到一个公式，该公式根据幂I^m的初始单项理想的重数来计算理想I的重数。我们用这个给一个新的证明之间的不平等的多重性和日志规范阈值由于德Fernex，艾因和作者。
The $j$-multiplicity plays an important role in the intersection theory of Stückrad-Vogel cycles, while recent developments confirm the connections between the $ε$-multiplicity and equisingularity theory. In this paper we establish, under some constraints, a relationship between the $j$-multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the $j$-multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the $j$- and $ε$-multiplicity of ideals defining determinantal varieties: The fo…
Suppose that $R$ is an excellent local domain with maximal ideal $m_R$. The theory of multiplicities and mixed multiplicities of $m_R$-primary ideals extends to (possibly non Noetherian) filtrations of $R$ by $m_R$-primary ideals, and many of the classical theorems for $m_R$-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for $m_R$-primary ideals do not hold for filtrations. In this article, we consider multiplicities and mixed multiplicities of $R$ by $m…
Convex bodies and multiplicities of ideals
我们将R^n中的凸区域与一大类局部代数（包括解析不可约局部区域）中的子空间的m-准素分次序列，特别是理想的m-准素分次序列联系起来。这些凸区域编码关于塞缪尔多重性的信息。这是在精神的理论格罗布纳基地和牛顿多面体一方面，和理论的牛顿奥肯科夫机构的线性系的另一方面。我们用这个给一个新的证明，以及推广的Brunn-Minkowski不等式的多重性由于Teissier和Rees-Sharp。
Note on bounds for multiplicities
令S=K[x_1，...，x_n]是一个多项式环，R=S/I是一个分次K-代数，其中I是S中的一个分次理想. Herzog，Huneke和Srinivasan证明了R的重数是由R在S上的最小分次自由分解的最大位移的函数上界的.证明了codim（R）=2时的猜想，推广了Herzog，Srinivasan和Gold的结果.我们也给出了一个证明的情况下，我是分量线性的界限。例如，稳定理想和无平方稳定理想都属于这类理想。
Mixed Multiplicities of Multi-Graded Algebras Over Noetherian Local Rings
设S是Noether局部局部环A上的标准分次代数.本文首先用Hilbert-Samuel重数来表示S的混合重数，将S的混合重数解释为S的商模的Hilbert-Samuel重数。作为应用，我们得到了理想的混合重数的公式，涵盖了Trung-Verma 2007 [TV]的主要结果。
Positivity of mixed multiplicities
本文研究了局部环（A，m）中任意标准双分次代数的混合重数和两个理想I，J的混合重数，其中I是m-准素理想，J是任意理想.主要结果是他们的积极性，可以用来有效地计算他们的标准。我们还证明了一个双分次代数的正混合重数的值域是刚性的，如果代数满足第一链条件，并且在余维1中连通，并且这个值域对于理想的混合重数总是刚性的。这些结果可用于研究解析超曲面的μ-不变量、射影空间爆破得到的有理簇的次数以及相交理论中Stuckrad-Vogel圈的次数.
The weak Lefschetz property and mixed multiplicities of monomial ideals
本文利用等生成单项式理想的分析扩张、混合重数与双有理单项式映射的联系，以关联理想的混合重数给出平方自由约化满足弱 Lefschetz 性质的充要条件，并推广至正特征情形。
Let R be the local ring of a point on a variety X over an algebraically closed field k. We make a connection between the notion of mixed (Samuel) multiplicity of m-primary ideals in R and intersection theory of subspaces of rational functions on X which deals with the number of solutions of systems of equations. From this we readily deduce several properties of mixed multiplicities. In particular, we prove a (reverse) Alexandrov-Fenchel inequality for mixed multiplicities due to Teissier and Rees-Sharp. As an application in convex geometry we obtain a proof of a (reverse) Alexandrov-Fenchel in…
Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $γ> 0$ such that for all $n \ge 0$, $$\ell_R(R/I_{n+1}) - \ell_R(R/I_n) < γn^{d-1}.$$
We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show the $j$-multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the $j$-multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute t…
This paper gives the recursion formula for mixed multiplicities of maximal degrees with respect to joint reductions of ideals, which is one of important results in the mixed multiplicity theory. Using this result, we give consequences on the relationship between mixed multiplicities and the Hilbert-Samuel multiplicity under most essential assumptions, that is a substantial progress in the problem of expressing mixed multiplicities into the Hilbert-Samuel multiplicity.
Multiplicities of weakly graded families of ideals
本文推广线性下有界弱分次理想族的重数概念，证明相关极限存在性、体积重数公式与Minkowski不等式，探究特定形式弱分次理想族的重数性质、不等式取等条件，推广Rees的结果并研究相关长度函数的渐近行为。
Algorithms for computing mixed multiplicities, mixed volumes and sectional Milnor numbers
本研究推出MixedMultiplicity程序包，通过推广Cox等人的成果获取理想多重Rees代数的定义方程，可计算诺特环理想混合重数，还能求解凸格点多面体混合体积与孤立奇点超曲面的截面Milnor数。
In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $R$ is less than the dimension of $R$, which holds for instance if $R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $m_R$-primary ideals hold fo…
We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra.
Multiplicities of graded families of linear series and ideals
本文给出了域k上的射影概形的渐近极限存在的简单的充要条件。当I_n是m_R-准素理想的分次族时，给出了d维局部环R上极限\lim_{i\rightarrow\infty} \frac{\ell_R(R/I_n)}{n^d}存在的充要条件.我们给出了一些应用。
Filter-regular sequences and mixed multiplicities
设$S$是Artinian局部环$A$上的一个标准生成的多重分次代数，$M$是一个标准生成的多重分次$S$ -模.本文回答了$M$的混合重数为正的问题，并用$A$ -模的长度刻画了它们.作为应用，我们得到了理想的混合重数的结果（命题4.4和定理4.5），并恢复了Risler和Teissier [Te]（见注4.8），Trung和Verma [TV]（见注4.6）的结果.
Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations
我们给出了一个广泛的一类n元组的理想$(I_1,..., I_n)$在$Ø_n$使用固定的牛顿过滤所提供的信息的Eschojasiewicz指数的表达式。为了得到这个表达式，我们考虑Rees混合多重性的Rejojasiewicz指数的重新表述。因此，我们得到了一类广义的半加权齐次函数$(\mathbb{C}^n,0)\to (\mathbb{C},0)$，对于这类函数，其梯度映射$\nabla f$的最大可能值为零.
Multiplicities and degree functions in local rings via intersection products
本文通过定义诺特局部环上真双有理概形的相交积，证明了里斯次数函数定理的新证明，并推广了拉马努詹公式以计算希尔伯特-塞缪尔重数，进而用相交理论表达混合重数及其多重线性性质。
A Survey of Some Results for Mixed Multiplicities
本文证明了[13，15，16，25]中的表表序列是[18]中的弱（FC）序列。由此可见，Viet在2000年的[18，定理3.4]覆盖了Risler和Teissier在1973年的结果; Trung在2001年的结果; Trung和Verma在2007年的结果。本文统一了文献[13，15，16，17，18，25]中关于理想的混合重数的结果。
Computing mixed multiplicities, mixed volumes, and sectional Milnor numbers
本文为综述性工作，介绍了基于推广Cox等人结果的多重Rees代数方程计算的Macaulay2新算法，可高效计算诺特环理想混合重数、凸格点多胞体混合体积与孤立奇点超曲面的截面Milnor数。
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.
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.
We find simple necessary and sufficient conditions on a local ring $R$ of dimension $d$ for the limit $$ \lim_{i\rightarrow\infty}\frac{\ell_R(R/I_n)}{n^d} $$ to exist whenever $\{I_n\}$ is a graded family of $m_R$-primary ideals, and give a number of applications. We also give simple necessary and sufficient conditions on projective schemes over a field $k$ for asymptotic limits of the growth of all graded linear series of a fixed Kodaira-Iitaka dimension to exist.
We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective degrees of rational maps determined by perfect ideals of height two or by Gorenstein ideals of height three.
Multiplicities and a dimension inequality for unmixed modules
我们证明了下面的结果，这是由最近的工作Kurano和Roberts对塞尔的积极性猜想的动机。设（R，m）是具有有限生成模M的局部环，使得R/ann（M）是拟非混合的且包含域，p和q是M支撑下的素理想，使得p是解析非分歧的，p+q是m-准素的，e（M_p）=e（M）.则dim（R/p）+dim（R/q）\leq dim（M）。在混合特征的特殊情况下，我们也证明了一个类似的定理。最后，我们提供了几个例子来解释我们的假设，以及一个例子，一个非悬链线，局部域R的素理想p，使e（R_p）>e（R）=1。
In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we provide simple algebraic and geometric proofs. Connections with mixed multiplicities of ideals are explained.
Let $S=\mathbb{C}[x_{ij}]$ be a polynomial ring of $m\times n$ variables over $\mathbb{C}$ and let $I$ be the determinantal ideal of maximal minors of $S$. Using the representation theoretic techniques introduced in arXiv:1305.1719, arXiv:1309.0617 and arXiv:1611.00415, we prove the existence of the generalized $j$-multiplicities $ε^j(I)$ defined by Dao and Montaño in arXiv:1705.05033. We will also give a closed formula of $ε^j(I)$, which generalized the results in arXiv:1308.0582 and arXiv:1912.02917 in the maximal minors case.
The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathc…
Multiplicities of Jumping Numbers
本文研究任意维光滑簇中乘子理想跳跃数的重数，证明重数函数为拟多项式，从而庞加莱级数有理，并探讨最高次数分量与里斯赋值及单项式理想情形。
Multiplicities of graded families of ideals on Noetherian local rings
本文研究了Noetherian局部环上分级理想族的乘数概念，推广了传统理想乘数，并证明了多个经典定理在该框架下的泛化，包括混合乘数、Rees定理和Minkowski不等式。
Suppose that R is an analytically irreducible or excellent local domain with maximal ideal m_R. We consider multiplicities and mixed multiplicities of R by filtrations of m_R-primary ideals. We show that the theorem of Teissier, Rees and Sharp, and Katz, characterizing equality in the Minkowski inequality for multiplicities of ideals, is true for divisorial filtrations, and for the larger category of bounded filtrations. This theorem is not true for arbitrary filtrations of m_R-primary ideals.
On some multiplicity and mixed multiplicity formulas (Forum Math. 26(2014), 413-442)
本文给出了多分次模$M$的混合重数和任意理想的混合重数的可加性和约化公式，并建立了$M.$的所有混合重数之和的递推公式。作为这些公式的应用，我们得到了多分次Rees模的重数的递推公式。
Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties
通过Kazhdan-Lusztig理想的Groebner退化，研究了Schubert簇在完备旗簇中的点的Hilbert-Samuel重数.在covexillary的情况下，我们给出了一个积极的组合规则的多重性，通过建立（与Groebner基）减少和等维的限制，其斯坦利-Reisner单纯复形是同胚的可壳球或球体。我们表明，多重计数的方面，这个复杂的数量。我们还得到了局部环的Hilbert级数的一个公式。特别地，我们的工作给出了Grassmannian Schubert簇的重数规则，为[Lakshmibai-Weyman '90]，[Rosenthal-Zelevinsky' 01]，[Krattenthaler '01]，[Kreiman-Lakshmibai' 04]和[Woo-Yong '09]的公式提供了替代陈述和证明.我们建议我们的方法扩展到一般情况下。
Resolutions of fat point ideals involving 8 general points of P2
主要结果提供了一个算法，用于确定最小的自由分解的理想的脂肪点子方案的${\bf P}^2$涉及多达8个一般点与任意的多重性，结果保持在代数闭域的任何特性。该算法的工作原理是在某些情况下给出一个公式，在其他情况下对这些情况进行简化，不涉及Gröbner基，因此非常快，即使对于非常大的多重性。在某些情况下，还可以使用$n>8$获得部分信息。
For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension étale in codimension one. Our result can be applied to bound the size of the local étale fundamental group of a singularity in terms of F-signature, recovering a recent result of Carvajal-Rojas, Schwede, and Tucker, and differential signature, providing the first characteristic-free effective bound.
On the Equality of Ordinary and Symbolic Powers of Ideals
我们考虑以下关于理想的普通和象征力量的平等的问题。在一个正则局部环中，如果一个一维素理想的普通幂和符号幂直到它的高度都是相同的，那么它们对所有的幂都是相同的吗？我们提供了支持的证据，一个积极的答案类的素理想定义单项曲线或环的低多重性。
Let $S=\mathbb{C}[x_{ij}]$ be a polynomial ring of $m\times n$ generic variables (resp. a polynomial ring of $(2n+1) \times (2n+1)$ skew-symmetric variables) over $\mathbb{C}$ and let $I$ (resp. Pf) be the determinantal ideal of maximal minors (resp. sub-maximal pfaffians) of $S$. Using the representation theoretic techniques introduced in the work of Raicu et al, we study the asymptotic behavior of the length of the local cohomology module of determinantal and pfaffian thickenings for suitable choices of cohomological degrees. This asymptotic behavior is also defined as a notion of multiplict…
Generalized Hilbert-Kunz Multiplicity for Families of Ideals
本文系统研究正特征Noetherian局部环中理想族的广义Hilbert-Kunz重数，引入Amao型重数这一新渐近不变量，证明p-理想族的广义Hilbert-Kunz重数是Amao型重数的极限。
Joint reductions and mixed Buchsbaum-Rim multiplicities of modules and a joint-reduction-number-zero theorem
本文提出了 Noetherian 局部环上模的联合约化与混合 Buchsbaum-Rim 重数的新定义，证明了二维正则局部环上整闭模的联合约化数零定理，并揭示了模的混合重数与 Koszul 复形特征及理想重数间的联系。
Counterexamples to Minkowski-type inequalities and the Rees converse for modules via mixed Buchsbaum-Rim Multiplicities
本文构造混合Buchsbaum-Rim重数，证明其系数乘积公式，给出Minkowski型不等式等失效的反例，明确一维情形下Rees逆命题不成立，仅在特定条件下成立，并提出高维逆命题问题。
We investigate the relationship among several numerical invariants associated to a (free) projective hypersurface $V$: the sequence of mixed multiplicities of its Jacobian ideal, the Hilbert polynomial of its Milnor algebra, and the sequence of exponents when $V$ is free. As a byproduct, we obtain explicit equations for some of the homaloidal surfaces in the projective 3-dimensional space constructed by C. Ciliberto, F. Russo and A. Simis.
The paper investigates the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities in families of ideals. It is shown that Hilbert-Samuel multiplicity is upper semicontinuous almost generally and that Hilbert-Kunz multiplicity is upper semicontinuous in families of finite type. Surprisingly, our machinery can be applied for families over Z and yields a partial solution to the question about characteristic zero Hilbert-Kunz multiplicity posed by Brenner, Li, and Miller. Another application is that for an affine ring the infimum in the definition of F-rational signature, an invariant defined …
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