We prove that for each positive integer $n$ there exists a positive number $epsilon_n$ so that $n$-dimensional toric quotient singularities satisfy the ACC for mld's on the interval $(0,epsilon_n)$. In the course of the proof, we will show a geometric Jordan property for finite automorphism groups of affine toric varieties.
{"title":"Small Quotient Minimal Log Discrepancies","authors":"Joaqu'in Moraga","doi":"10.1307/mmj/20205985","DOIUrl":"https://doi.org/10.1307/mmj/20205985","url":null,"abstract":"We prove that for each positive integer $n$ there exists a positive number $epsilon_n$ so that $n$-dimensional toric quotient singularities satisfy the ACC for mld's on the interval $(0,epsilon_n)$. In the course of the proof, we will show a geometric Jordan property for finite automorphism groups of affine toric varieties.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"82 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90474397","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}
For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standard Bernstein-Sato ideals for reduced, generic arrangements: we compute the Bernstein-Sato ideal for a factorization into linear forms and we compute its zero locus for other factorizations.
{"title":"A Note on Bernstein–Sato Varieties for Tame Divisors and Arrangements","authors":"Daniel Bath","doi":"10.1307/mmj/20206011","DOIUrl":"https://doi.org/10.1307/mmj/20206011","url":null,"abstract":"For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standard Bernstein-Sato ideals for reduced, generic arrangements: we compute the Bernstein-Sato ideal for a factorization into linear forms and we compute its zero locus for other factorizations.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"33 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89264898","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 perform a variation of geometric invariant theory stability analysis for 2nd Hilbert points of bi-log-canonically embedded pointed curves of genus 2 . As a result, we give a GIT construction of the log canonical models M ¯ 2 , 1 ( α ) for α = 2 / 3 ± ϵ and obtain a VGIT presentation of the second flip in the Hassett–Keel program for the moduli space of pointed genus 2 curves.
{"title":"VGIT Presentation of the Second Flip of M ¯ 2 , 1","authors":"M. Fedorchuk, M. Grimes","doi":"10.1307/MMJ/1596700815","DOIUrl":"https://doi.org/10.1307/MMJ/1596700815","url":null,"abstract":"We perform a variation of geometric invariant theory stability analysis for 2nd Hilbert points of bi-log-canonically embedded pointed curves of genus 2 . As a result, we give a GIT construction of the log canonical models M ¯ 2 , 1 ( α ) for α = 2 / 3 ± ϵ and obtain a VGIT presentation of the second flip in the Hassett–Keel program for the moduli space of pointed genus 2 curves.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"5 10 1","pages":"487-514"},"PeriodicalIF":0.9,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86987543","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}
For blow-ups of threefolds along ( − 1 , − 1 ) -curves, we use the degeneration formula and the absolute/relative correspondence to obtain some closed blow-up formulae for Gromov–Witten invariants and generalized BPS numbers.
{"title":"Gromov–Witten Invariants Under Blow-Ups Along ( − 1 , − 1 ) -Curves","authors":"Hua-Zhong Ke","doi":"10.1307/MMJ/1596700816","DOIUrl":"https://doi.org/10.1307/MMJ/1596700816","url":null,"abstract":"For blow-ups of threefolds along ( − 1 , − 1 ) -curves, we use the degeneration formula and the absolute/relative correspondence to obtain some closed blow-up formulae for Gromov–Witten invariants and generalized BPS numbers.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"23 1","pages":"515-531"},"PeriodicalIF":0.9,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82287350","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}
Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this article, we show that there are densely many $overline{mathbb Q}$-rational points with maximal arithmetic degree (i.e. whose arithmetic degree is equal to the first dynamical degree) for self-morphisms on projective varieties. For unirational varieties and abelian varieties, we show that there are densely many rational points with maximal arithmetic degree over a sufficiently large number field. We also give a generalization of a result of Kawaguchi and Silverman in the appendix.
{"title":"Zariski Density of Points with Maximal Arithmetic Degree","authors":"K. Sano, Takahiro Shibata","doi":"10.1307/mmj/20205960","DOIUrl":"https://doi.org/10.1307/mmj/20205960","url":null,"abstract":"Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this article, we show that there are densely many $overline{mathbb Q}$-rational points with maximal arithmetic degree (i.e. whose arithmetic degree is equal to the first dynamical degree) for self-morphisms on projective varieties. For unirational varieties and abelian varieties, we show that there are densely many rational points with maximal arithmetic degree over a sufficiently large number field. We also give a generalization of a result of Kawaguchi and Silverman in the appendix.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"15 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75878723","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 paper we revisit Ribenboim's notion of higher derivations of modules and relate it to the recent work of De Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the Kahler differentials of the Hasse-Schmidt algebra as a consequence of the fact that the Hasse-Schmidt algebra functors commute.
{"title":"Higher Derivations of Modules and the Hasse–Schmidt Module","authors":"C. Chiu, L. N. Macarro","doi":"10.1307/mmj/20205958","DOIUrl":"https://doi.org/10.1307/mmj/20205958","url":null,"abstract":"In this paper we revisit Ribenboim's notion of higher derivations of modules and relate it to the recent work of De Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the Kahler differentials of the Hasse-Schmidt algebra as a consequence of the fact that the Hasse-Schmidt algebra functors commute.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"46 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90326675","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}
Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic $p$, we show that a similar result holds in equicharacteristic $0$ under the additional hypothesis that the symbolic Rees algebra of $I$ is noetherian.
{"title":"Expected Resurgence of Ideals Defining Gorenstein Rings","authors":"Eloísa Grifo, C. Huneke, Vivek Mukundan","doi":"10.1307/mmj/20206004","DOIUrl":"https://doi.org/10.1307/mmj/20206004","url":null,"abstract":"Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic $p$, we show that a similar result holds in equicharacteristic $0$ under the additional hypothesis that the symbolic Rees algebra of $I$ is noetherian.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"31 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89926884","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}
The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that shows that they are not, in general, equivalent. We also present the relation of generating sequences and key polynomials.
{"title":"Generating Sequences and Key Polynomials","authors":"M. S. Barnab'e, J. Novacoski","doi":"10.1307/mmj/20205953","DOIUrl":"https://doi.org/10.1307/mmj/20205953","url":null,"abstract":"The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that shows that they are not, in general, equivalent. We also present the relation of generating sequences and key polynomials.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"2014 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83385475","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}
For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family are the Gaussian polytopes defined as convex hulls of i.i.d. samples from a non-degenerate Gaussian distribution in $mathbb R^d$. The second family are convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are non-trivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a special case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known but it seems that no detailed proofs can be found in the existing literature.
{"title":"Angle Sums of Random Polytopes","authors":"Thomas Godland, Z. Kabluchko, D. Zaporozhets","doi":"10.1307/mmj/20206021","DOIUrl":"https://doi.org/10.1307/mmj/20206021","url":null,"abstract":"For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family are the Gaussian polytopes defined as convex hulls of i.i.d. samples from a non-degenerate Gaussian distribution in $mathbb R^d$. The second family are convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are non-trivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a special case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known but it seems that no detailed proofs can be found in the existing literature.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"30 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75279802","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":"The Behavior at Infinity of p-Harmonic Measure in an Infinite Slab","authors":"Dante DeBlassie, R. Smits","doi":"10.1307/mmj/1592877615","DOIUrl":"https://doi.org/10.1307/mmj/1592877615","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"14 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87273575","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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