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Asymptotic growth of symbolic powers, mixed multiplicities, and convex bodies

符号幂、混合多重性和凸体的渐近增长

基本信息

批准号：
2303605
负责人：
Jonathan Montano
金额：
$ 13.98万
依托单位：
Arizona State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-10-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2303605&HistoricalAwards=false
关键词：
Asymptotic growth symbolic powers mixed

项目摘要

This project is in commutative algebra which is the area of mathematics that studies properties of polynomial equations in several variables. Various phenomena across science and engineering can be modeled by systems of polynomial equations, which leads to natural interactions with other fields such as applied mathematics, biology, computer science, and mathematical physics. For systems of polynomial equations, one studies its sets of solutions, called varieties, and the corresponding shapes they describe in high-dimensional spaces. As the number of equations and variables grow larger, one useful approach to understand these shapes is to study the algebraic functions defined on them. Using this approach, many properties such as smoothness and degrees of varieties can be translated into algebraic terms. One such strategy is to study properties of varieties by analyzing the asymptotic behavior of sequences of algebraic objects called ideals. This project contributes to this line of research with problems originating from two directions that involve the study of these properties of ideals. The PI will also be involved in mentoring students and in organizing academic events, focusing on benefiting graduate students and underrepresented groups in mathematics.The symbolic powers of an ideal encode important algebraic and geometric information of the ideal and the variety it defines. The PI will investigate the asymptotic behavior of homological invariants of symbolic powers such as, the number of generators, the Castelnuovo-Mumford regularity, and the projective dimension. For the sequence of numbers of generators, the PI intends to investigate if it always has polynomial complexity, building of previous results by the PI and his collaborators. A general result in this direction would have important consequences on the arithmetic rank, Frobenius complexity, and Kodaira dimension of divisors. For the other two sequences, regularities and projective dimensions, the PI will focus on regular rings of positive characteristic. In such rings, a new class of ideals is defined for which it has been previously shown the limit of these sequences exist; this class includes several types of determinantal ideals, as well as the square-free monomial ideals. The PI will investigate the existence of these limits for more general classes of ideals. The relation between multiplicities and convex bodies is an important research topic lying in the interaction of Commutative Algebra, Algebraic Geometry, and Combinatorics. In recent years, this line of research has gained much activity due to the introduction of Newton-Okounkov bodies and mixed multiplicities of filtrations of ideals. The PI and his collaborators intend to find conditions for the non-vanishing of mixed multiplicities of multigraded algebras. The PI will also investigate a notion of mixed multiplicities for filtrations of not necessarily zero dimensional ideals and the relation of these new multiplicities with the mixed volumes of certain Newton-Okounkov bodies.This project is jointly funded by the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目是在交换代数这是数学领域的研究性质的多项式方程在几个变量。科学和工程中的各种现象都可以通过多项式方程系统来建模，这导致了与其他领域的自然相互作用，如应用数学，生物学，计算机科学和数学物理。对于多项式方程组，人们研究它的解集，称为簇，以及它们在高维空间中描述的相应形状。随着方程和变量的数量越来越多，理解这些形状的一个有用的方法是研究定义在它们上面的代数函数。使用这种方法，许多性质，如光滑度和度的品种可以转化为代数术语。其中一个策略是通过分析称为理想的代数对象序列的渐近行为来研究簇的性质。这个项目有助于这一行的研究问题源于两个方向，涉及研究这些性质的理想。PI还将参与指导学生和组织学术活动，重点是使研究生和数学中代表性不足的群体受益。理想的符号力量编码了理想及其定义的多样性的重要代数和几何信息。PI将研究符号幂的同调不变量的渐近行为，例如生成器的数量，Castelnuovo-Mumford正则性和投影维数。对于生成器的数量序列，PI打算研究它是否总是具有多项式复杂度，建立PI及其合作者以前的结果。在这个方向上的一般结果将有重要后果的算术秩，弗罗贝纽斯复杂性，和科代拉维数的除数。对于其他两个序列，双维和投影维数，PI将专注于正特征的正则环。在这样的环中，定义了一类新的理想，之前已经证明了这些序列的极限存在;这类理想包括几种类型的行列式理想，以及无平方根的单项理想。PI将研究这些极限对于更一般的理想类的存在性。重数与凸体之间的关系是交换代数、代数几何和组合数学相互作用的一个重要研究课题。近年来，由于Newton-Okounkov体和理想滤子的混合多重性的引入，这条研究路线获得了很多活动。PI和他的合作者打算找到多重代数的混合重数不为零的条件。该项目由数学科学部和刺激竞争研究既定计划（EPSCoR）共同资助，主要研究不一定为零维理想的过滤的混合多重性概念以及这些新的多重性与某些Newton-Okounkov体的混合体积的关系。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。