We give a recursive formula, expressed in terms of the characteristic tuples, for the Betti numbers of the boundary of the Milnor fiber of an irreducible quasi-ordinary surface. The singular locus of the surface consists of two components, and for each component we introduce a sequence of increasingly simpler surfaces. Our recursion depends on a detailed comparison of these two sequences. In the final section, we indicate how we expect pieces of these associated surfaces to glue together to reconstruct the Milnor fiber and its boundary.
{"title":"On the Milnor Fiber Boundary of a Quasi-Ordinary Surface","authors":"G. Kennedy, Lee J. McEwan","doi":"10.5427/JSING.2019.19C","DOIUrl":"https://doi.org/10.5427/JSING.2019.19C","url":null,"abstract":"We give a recursive formula, expressed in terms of the characteristic tuples, for the Betti numbers of the boundary of the Milnor fiber of an irreducible quasi-ordinary surface. The singular locus of the surface consists of two components, and for each component we introduce a sequence of increasingly simpler surfaces. Our recursion depends on a detailed comparison of these two sequences. In the final section, we indicate how we expect pieces of these associated surfaces to glue together to reconstruct the Milnor fiber and its boundary.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"7 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76833587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using a relation due to Katz linking up additive and multiplicative convolutions, we make explicit the behaviour of some Hodge invariants by middle multiplicative convolution, following [DS13] and [Mar18a] in the additive case. Moreover, the main theorem gives a new proof of a result of Fedorov computing the Hodge invariants of hypergeometric equations.
{"title":"Middle multiplicative convolution and hypergeometric equations","authors":"Nicolas Martin","doi":"10.5427/jsing.2021.23k","DOIUrl":"https://doi.org/10.5427/jsing.2021.23k","url":null,"abstract":"Using a relation due to Katz linking up additive and multiplicative convolutions, we make explicit the behaviour of some Hodge invariants by middle multiplicative convolution, following [DS13] and [Mar18a] in the additive case. Moreover, the main theorem gives a new proof of a result of Fedorov computing the Hodge invariants of hypergeometric equations.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"2 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85301093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is a continuation of authors' research interrupted in the year 2010. Derived are recursive relations describing for the first time all infinitesimal symmetries of special 2-flags (sometimes also misleadingly called `Goursat 2-flags'). When algorithmized to the software level, they will give an answer filling in the gap in knowledge as of 2010: on one side the local finite classification of special 2-flags known in lengths not exceeding four, on the other side the existence of a continuous numerical modulus of that classification in length seven.
{"title":"Symmetries of special 2-flags","authors":"P. Mormul, F. Pelletier","doi":"10.5427/jsing.2020.21k","DOIUrl":"https://doi.org/10.5427/jsing.2020.21k","url":null,"abstract":"This work is a continuation of authors' research interrupted in the year 2010. Derived are recursive relations describing for the first time all infinitesimal symmetries of special 2-flags (sometimes also misleadingly called `Goursat 2-flags'). When algorithmized to the software level, they will give an answer filling in the gap in knowledge as of 2010: on one side the local finite classification of special 2-flags known in lengths not exceeding four, on the other side the existence of a continuous numerical modulus of that classification in length seven.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"6 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74049438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is a survey article on recognition problem of frontal singularities. We specify geometrically several frontal singularities and then we solve the recognition problem of such singularities, giving explicit normal forms. We combine the recognition results by K. Saji and several arguments on openings, which was performed for the classification of singularities of tangent surfaces (tangent developables) by the author. As an application of our solutions of recognition problem of frontal singularities, we announce the classification of singularities appearing in tangent surfaces of generic null curves which are ruled by null geodesics in general Lorentz three-manifolds, mentioning related recognition results and open problems.
{"title":"Recognition Problem of Frontal Singularities","authors":"G. Ishikawa","doi":"10.5427/jsing.2020.21i","DOIUrl":"https://doi.org/10.5427/jsing.2020.21i","url":null,"abstract":"This is a survey article on recognition problem of frontal singularities. We specify geometrically several frontal singularities and then we solve the recognition problem of such singularities, giving explicit normal forms. We combine the recognition results by K. Saji and several arguments on openings, which was performed for the classification of singularities of tangent surfaces (tangent developables) by the author. As an application of our solutions of recognition problem of frontal singularities, we announce the classification of singularities appearing in tangent surfaces of generic null curves which are ruled by null geodesics in general Lorentz three-manifolds, mentioning related recognition results and open problems.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"31 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87259705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. S. Carvalho, B. Oréfice-Okamoto, J. N. Tomazella
We study deformations of holomorphic function germs $f:(X,0)tomathbb C$ where $(X,0)$ is an ICIS. We present conditions for these deformations to have constant Milnor number, Euler obstruction and Bruce-Roberts number.
{"title":"$mu$-constant deformations of functions on an ICIS","authors":"R. S. Carvalho, B. Oréfice-Okamoto, J. N. Tomazella","doi":"10.5427/jsing.2019.19i","DOIUrl":"https://doi.org/10.5427/jsing.2019.19i","url":null,"abstract":"We study deformations of holomorphic function germs $f:(X,0)tomathbb C$ where $(X,0)$ is an ICIS. We present conditions for these deformations to have constant Milnor number, Euler obstruction and Bruce-Roberts number.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"144 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75996591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Singular quadratic mappings creating Kato’s chaos are given.
给出了产生加藤混沌的奇异二次映射。
{"title":"Kato's chaos created by quadratic mappings associated with spherical orthotomic curves","authors":"T. Nishimura","doi":"10.5427/jsing.2020.21l","DOIUrl":"https://doi.org/10.5427/jsing.2020.21l","url":null,"abstract":"Singular quadratic mappings creating Kato’s chaos are given.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"7 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89357171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The results of Kolmogorov, Arnold, and Moser on the stability of quasi-periodic motions spanning lagrangian tori in Hamiltonian systems are of fundamental importance and led to the development of KAM theory. Over the years, many variations of these results on quasi-periodic motions have been considered. In this paper, we present a more conceptual way of attacking such problems by considering the particular case of quasi-periodic motions on symplectic tori.
{"title":"Quasi-periodic motions on symplectic tori","authors":"M. Garay, A. Kessi, D. Straten, N. Yousfi","doi":"10.5427/jsing.2023.26c","DOIUrl":"https://doi.org/10.5427/jsing.2023.26c","url":null,"abstract":"The results of Kolmogorov, Arnold, and Moser on the stability of quasi-periodic motions spanning lagrangian tori in Hamiltonian systems are of fundamental importance and led to the development of KAM theory. Over the years, many variations of these results on quasi-periodic motions have been considered. In this paper, we present a more conceptual way of attacking such problems by considering the particular case of quasi-periodic motions on symplectic tori.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"5 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89059540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the last twenty years a number of papers appeared aiming to construct locally free replacements of the sheaf of principal parts for families of Gorenstein curves. The main goal of this survey is to present to the widest possible audience of mathematical readers a catalogue of such constructions, discussing the related literature and reporting on a few applications to classical problems in Enumerative Algebraic Geometry.
{"title":"Jet Bundles on Gorenstein Curves and Applications","authors":"Letterio Gatto, Andrea T. Ricolfi","doi":"10.5427/jsing.2020.21d","DOIUrl":"https://doi.org/10.5427/jsing.2020.21d","url":null,"abstract":"In the last twenty years a number of papers appeared aiming to construct locally free replacements of the sheaf of principal parts for families of Gorenstein curves. The main goal of this survey is to present to the widest possible audience of mathematical readers a catalogue of such constructions, discussing the related literature and reporting on a few applications to classical problems in Enumerative Algebraic Geometry.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"30 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78082766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a bouquet decomposition for the determinantal Milnor fiber of an essentially isolated determinantal singularity of arbitrary type $(m,n,t)$. The building blocks in the decomposition are (suspensions of) hyperplane sections of the associated generic determinantal variety $M_{m,n}^t$ in general position off the origin.
{"title":"Bouquet decomposition for Determinantal Milnor fibers","authors":"M. Zach","doi":"10.5427/jsing.2020.22m","DOIUrl":"https://doi.org/10.5427/jsing.2020.22m","url":null,"abstract":"We provide a bouquet decomposition for the determinantal Milnor fiber of an essentially isolated determinantal singularity of arbitrary type $(m,n,t)$. The building blocks in the decomposition are (suspensions of) hyperplane sections of the associated generic determinantal variety $M_{m,n}^t$ in general position off the origin.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"201 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83693134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M setminus L$ which can be realized as the singular point set of a generic map $M to mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $mathbb{R}^2$.
{"title":"Linking between singular locus and regular fibers","authors":"O. Saeki","doi":"10.5427/jsing.2020.21n","DOIUrl":"https://doi.org/10.5427/jsing.2020.21n","url":null,"abstract":"Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M setminus L$ which can be realized as the singular point set of a generic map $M to mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $mathbb{R}^2$.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"14 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78563538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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