Let A and B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra that contains A. Denote by A' the integral closure of A in B. We show that A' is determined by finitely many unique discrete valuation rings. Our result generalizes Rees' classical valuation theorem for ideals. We also obtain a variant of Zariski's main theorem.
{"title":"A Valuation Theorem for Noetherian Rings","authors":"Antoni Rangachev","doi":"10.1307/mmj/20206022","DOIUrl":"https://doi.org/10.1307/mmj/20206022","url":null,"abstract":"Let A and B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra that contains A. Denote by A' the integral closure of A in B. We show that A' is determined by finitely many unique discrete valuation rings. Our result generalizes Rees' classical valuation theorem for ideals. We also obtain a variant of Zariski's main theorem.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"100 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76208951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Generalizing the involution length of the complex hyperbolic plane, we obtain that the α -length of PU(2 , 1) is 4, that is, every element of PU(2 , 1) can be decomposed as the product of at most 4 special elliptic isometries with parameter α . We also describe the isometries that can be written as the product of 2 or 3 such special elliptic isometries.
{"title":"The Length of PU(2,1) Relative to Special Elliptic Isometries with Fixed Parameter","authors":"Felipe de Aguilar Franco","doi":"10.1307/mmj/20206013","DOIUrl":"https://doi.org/10.1307/mmj/20206013","url":null,"abstract":"Generalizing the involution length of the complex hyperbolic plane, we obtain that the α -length of PU(2 , 1) is 4, that is, every element of PU(2 , 1) can be decomposed as the product of at most 4 special elliptic isometries with parameter α . We also describe the isometries that can be written as the product of 2 or 3 such special elliptic isometries.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"12 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74598415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Felix Klein in course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of $60$ points in ${mathbb P}^3$. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the $60$ reflection planes in the group $G_{31}$ in the Shephard-Todd list. In the present note we show that the $60$ points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree $6$. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of $60$ points is a cone with a single singularity of multiplicity $6$ and the other has three singular points of multiplicities $4,2$ and $2$. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in ${mathbb P}^3$ with the surprising property that their general projection to ${mathbb P}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of $24$ points in ${mathbb P}^3$ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property.
菲利克斯·克莱因在研究正二十面体及其对称性的过程中遇到了一个高度对称的构型$60$点在${mathbb P}^3$中。这种配置以各种形式出现，也许最引人注目的是在Shephard-Todd列表中$G_{31}$组中$60$反射面对偶点的配置。在本报告中，我们表明，从最近开始的两条研究路径的角度来看，$60$点显示出有趣的特性。首先，它们产生了两个完全不同的6次意想不到的曲面。Cook II, Harbourne, Migliore, Nagel在2018年引入了意想不到的超表面。与$60$点的配置相关的一个意想不到的曲面是一个具有单个多重奇点$6$的圆锥，另一个具有三个多重奇点$4,2$和$2$。其次，Chiantini和Migliore在2020年观察到${mathbb P}^3$中存在非平凡的点集，它们到${mathbb P}^2$的一般投影是一个完全相交。他们发现了一组这样的集合，他们称之为网格。他们论文的附录描述了${mathbb P}^3$中$24$点的奇异构型，它不是网格，但具有其一般投影是完全相交的显著性质。我们证明Klein构型也不是一个网格，它投射到一个完整的交叉点。我们还确定了它的固有子集，它们具有相同的性质。＼
{"title":"Unexpected Properties of the Klein Configuration of 60 Points in P3","authors":"Piotr Pokora, T. Szemberg, J. Szpond","doi":"10.1307/mmj/20216141","DOIUrl":"https://doi.org/10.1307/mmj/20216141","url":null,"abstract":"Felix Klein in course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of $60$ points in ${mathbb P}^3$. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the $60$ reflection planes in the group $G_{31}$ in the Shephard-Todd list. In the present note we show that the $60$ points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree $6$. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of $60$ points is a cone with a single singularity of multiplicity $6$ and the other has three singular points of multiplicities $4,2$ and $2$. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in ${mathbb P}^3$ with the surprising property that their general projection to ${mathbb P}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of $24$ points in ${mathbb P}^3$ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property. ","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"36 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85250072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A conjecture, known as the Shokurov-Koll'ar connectedness principle, predicts the following. Let $(X,B)$ be a pair, and let $f colon X rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$; then, for any point $s in S$, the intersection $f^{-1} (s) cap mathrm{Nklt}(X,B)$ has at most two connected components, where $mathrm{Nklt}(X,B)$ denotes the non-klt locus of $(X,B)$. This conjecture has been extensively studied in characteristic zero, and it has been recently settled in that context. In this work, we consider this conjecture in the setup of positive characteristic algebraic geometry. We prove this conjecture holds for threefolds in characteristic $p>5$, and, under the same assumptions, we characterize the cases in which $mathrm{Nklt}(X,B)$ fails to be connected.
一个被称为Shokurov-Kollár连通性原理的猜想预测了以下情况。设$(X,B)$为一对，设$f colon X rightarrow S$为$-(K_X + B)$ nef / $S$的缩略语;然后，对于任意点$s in S$，相交$f^{-1} (s) cap mathrm{Nklt}(X,B)$最多有两个连通分量，其中$mathrm{Nklt}(X,B)$表示$(X,B)$的非klt轨迹。这一猜想在特征零中得到了广泛的研究，最近在这一背景下得到了解决。在本工作中，我们在正特征代数几何的建立中考虑这个猜想。我们证明了这个猜想在特征$p>5$中三倍成立，并且，在相同的假设下，我们描述了$mathrm{Nklt}(X,B)$不连接的情况。
{"title":"Connectedness Principle for 3-Folds in Characteristic p>5","authors":"Stefano Filipazzi, J. Waldron","doi":"10.1307/mmj/20216143","DOIUrl":"https://doi.org/10.1307/mmj/20216143","url":null,"abstract":"A conjecture, known as the Shokurov-Koll'ar connectedness principle, predicts the following. Let $(X,B)$ be a pair, and let $f colon X rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$; then, for any point $s in S$, the intersection $f^{-1} (s) cap mathrm{Nklt}(X,B)$ has at most two connected components, where $mathrm{Nklt}(X,B)$ denotes the non-klt locus of $(X,B)$. This conjecture has been extensively studied in characteristic zero, and it has been recently settled in that context. In this work, we consider this conjecture in the setup of positive characteristic algebraic geometry. We prove this conjecture holds for threefolds in characteristic $p>5$, and, under the same assumptions, we characterize the cases in which $mathrm{Nklt}(X,B)$ fails to be connected.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"35 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72805598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study trisections of smooth, compact non-orientable 4-manifolds, and introduce trisections of non-orientable 4-manifolds with boundary. In particular, we prove a non-orientable analogue of a classical theorem of Laudenbach-Poenaru. As a consequence, trisection diagrams and Kirby diagrams of closed non-orientable 4-manifolds exist. We discuss how the theory of trisections may be adapted to the setting of non-orientable 4-manifolds with many examples.
{"title":"Trisections of Nonorientable 4-Manifolds","authors":"Maggie Miller, Patrick Naylor","doi":"10.1307/mmj/20216127","DOIUrl":"https://doi.org/10.1307/mmj/20216127","url":null,"abstract":"We study trisections of smooth, compact non-orientable 4-manifolds, and introduce trisections of non-orientable 4-manifolds with boundary. In particular, we prove a non-orientable analogue of a classical theorem of Laudenbach-Poenaru. As a consequence, trisection diagrams and Kirby diagrams of closed non-orientable 4-manifolds exist. We discuss how the theory of trisections may be adapted to the setting of non-orientable 4-manifolds with many examples.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"57 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76901734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose that every non-minimal bridge position of a knot $K$ is perturbed. We show that if $L$ is a $(2, 2q)$-cable link of $K$, then every non-minimal bridge position of $L$ is also perturbed.
{"title":"Nonminimal Bridge Position of 2-Cable Links","authors":"Jung Hoon Lee","doi":"10.1307/mmj/20216060","DOIUrl":"https://doi.org/10.1307/mmj/20216060","url":null,"abstract":"Suppose that every non-minimal bridge position of a knot $K$ is perturbed. We show that if $L$ is a $(2, 2q)$-cable link of $K$, then every non-minimal bridge position of $L$ is also perturbed.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"6 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90352223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Croke-Kleiner admissible groups firstly introduced by Croke-Kleiner belong to a particular class of graph of groups which generalize fundamental groups of $3$--dimensional graph manifolds. In this paper, we show that if $G$ is a Croke-Kleiner admissible group, acting geometrically on a CAT(0) space $X$, then a finitely generated subgroup of $G$ has finite height if and only if it is strongly quasi-convex. We also show that if $G curvearrowright X$ is a flip CKA action then $G$ is quasi-isometric embedded into a finite product of quasi-trees. With further assumption on the vertex groups of the flip CKA action $G curvearrowright X$, we show that $G$ satisfies property (QT) that is introduced by Bestvina-Bromberg-Fujiwara.
{"title":"Croke–Kleiner Admissible Groups: Property (QT) and Quasiconvexity","authors":"H. Nguyen, Wen-yuan Yang","doi":"10.1307/mmj/20216045","DOIUrl":"https://doi.org/10.1307/mmj/20216045","url":null,"abstract":"Croke-Kleiner admissible groups firstly introduced by Croke-Kleiner belong to a particular class of graph of groups which generalize fundamental groups of $3$--dimensional graph manifolds. In this paper, we show that if $G$ is a Croke-Kleiner admissible group, acting geometrically on a CAT(0) space $X$, then a finitely generated subgroup of $G$ has finite height if and only if it is strongly quasi-convex. We also show that if $G curvearrowright X$ is a flip CKA action then $G$ is quasi-isometric embedded into a finite product of quasi-trees. With further assumption on the vertex groups of the flip CKA action $G curvearrowright X$, we show that $G$ satisfies property (QT) that is introduced by Bestvina-Bromberg-Fujiwara.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"21 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73981490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cayley Trees do Not Determine the Maximal Zero-Free Locus of the Independence Polynomial","authors":"Pjotr Buys","doi":"10.1307/mmj/1599206419","DOIUrl":"https://doi.org/10.1307/mmj/1599206419","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"36 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83058798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we will introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we will develop a Littlewood-Paley decomposition and explore its connection to the existence of the spectral gap for random walks. Moreover, under the dimension condition alone, we will prove a multi-scale entropy gain result `a la Bourgain-Gamburd and Tao.
{"title":"Locally Random Groups","authors":"Keivan Mallahi-Karai, A. Mohammadi, A. Golsefidy","doi":"10.1307/mmj/20217213","DOIUrl":"https://doi.org/10.1307/mmj/20217213","url":null,"abstract":"In this work, we will introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we will develop a Littlewood-Paley decomposition and explore its connection to the existence of the spectral gap for random walks. Moreover, under the dimension condition alone, we will prove a multi-scale entropy gain result `a la Bourgain-Gamburd and Tao.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"69 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75977104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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