We show that under some conditions, two constructions of nearby cycles over general bases coincide. More specifically, we show that under the assumption of $Psi$-factorizability, the constructions of unipotent nearby cycles over an affine space can be described using the theory of nearby cycles over general bases via the vanishing topos. In particular, this applies to nearby cycles of Satake sheaves on Beilinson-Drinfeld Grassmannians with parahoric ramification.
{"title":"Unipotent nearby cycles and nearby cycles over general bases","authors":"Andrew Salmon","doi":"10.5427/jsing.2024.27b","DOIUrl":"https://doi.org/10.5427/jsing.2024.27b","url":null,"abstract":"We show that under some conditions, two constructions of nearby cycles over general bases coincide. More specifically, we show that under the assumption of $Psi$-factorizability, the constructions of unipotent nearby cycles over an affine space can be described using the theory of nearby cycles over general bases via the vanishing topos. In particular, this applies to nearby cycles of Satake sheaves on Beilinson-Drinfeld Grassmannians with parahoric ramification.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140484686","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}
{"title":"Round fold maps of n-dimensional manifolds into (n-1)-dimensional Euclidean space","authors":"Naoki Kitazawa, O. Saeki","doi":"10.5427/jsing.2023.26a","DOIUrl":"https://doi.org/10.5427/jsing.2023.26a","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73283053","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}
Our aim is to precisely present a tame topology counterpart to canonical stratification of a Lie groupoid. We consider a definable Lie groupoid in semialgebraic, subanalytic, o-minimal over R, or more generally, Shiota's X-category. We show that there exists a canonical Whitney stratification of the Lie groupoid into definable strata which are invariant under the groupoid action. This is a generalization and refinement of results on real algebraic group action which J.N. Mather and V.A. Vassiliev independently stated with sketchy proofs. A crucial change to their proofs is to use Shiota's isotopy lemma and approximation theorem in the context of tame topology.
{"title":"Canonical stratification of definable Lie groupoids","authors":"Masato Tanabe","doi":"10.5427/jsing.2023.26d","DOIUrl":"https://doi.org/10.5427/jsing.2023.26d","url":null,"abstract":"Our aim is to precisely present a tame topology counterpart to canonical stratification of a Lie groupoid. We consider a definable Lie groupoid in semialgebraic, subanalytic, o-minimal over R, or more generally, Shiota's X-category. We show that there exists a canonical Whitney stratification of the Lie groupoid into definable strata which are invariant under the groupoid action. This is a generalization and refinement of results on real algebraic group action which J.N. Mather and V.A. Vassiliev independently stated with sketchy proofs. A crucial change to their proofs is to use Shiota's isotopy lemma and approximation theorem in the context of tame topology.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135650631","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 investigate Zariski multiples of plane curves Z 1 ,...,Z N such that each Z i is a union of a smooth quartic curve, some of its bitangents, and some of its 4-tangent conics. We show that, for plane curves of this type, the deformation types are equal to the homeomorphism types, and that the number of deformation types grows as O ( d 62 ) when the degree d of the plane curves tends to infinity.
{"title":"Zariski multiples associated with quartic curves","authors":"I. Shimada","doi":"10.5427/jsing.2022.24g","DOIUrl":"https://doi.org/10.5427/jsing.2022.24g","url":null,"abstract":". We investigate Zariski multiples of plane curves Z 1 ,...,Z N such that each Z i is a union of a smooth quartic curve, some of its bitangents, and some of its 4-tangent conics. We show that, for plane curves of this type, the deformation types are equal to the homeomorphism types, and that the number of deformation types grows as O ( d 62 ) when the degree d of the plane curves tends to infinity.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76778602","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 classify singularities at infinity of polynomials of degree 3 in 3 variables, f(x0, x1, x2) = f1(x0, x1, x2) + f2(x0, x1, x2) + f3(x0, x1, x2), fi homogeneous polynomial of degree i, i = 1, 2, 3. Based on this classification, we calculate the jump in the Milnor number of an isolated singularity at infinity, when we pass from the special fiber to a generic fiber. As an application of the results, we investigate the existence of global fibrations of degree 3 polynomials in C and search for information about the topology of the fibers in each equivalence class.
{"title":"Classification at infinity of polynomials of degree 3 in 3 variables","authors":"N. Ribeiro","doi":"10.5427/jsing.2022.25r","DOIUrl":"https://doi.org/10.5427/jsing.2022.25r","url":null,"abstract":"We classify singularities at infinity of polynomials of degree 3 in 3 variables, f(x0, x1, x2) = f1(x0, x1, x2) + f2(x0, x1, x2) + f3(x0, x1, x2), fi homogeneous polynomial of degree i, i = 1, 2, 3. Based on this classification, we calculate the jump in the Milnor number of an isolated singularity at infinity, when we pass from the special fiber to a generic fiber. As an application of the results, we investigate the existence of global fibrations of degree 3 polynomials in C and search for information about the topology of the fibers in each equivalence class.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84499867","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}
{"title":"Right Network-Preserving Diffeomorphisms","authors":"F. Antoneli, Ian M. Stewart","doi":"10.5427/jsing.2022.25a","DOIUrl":"https://doi.org/10.5427/jsing.2022.25a","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81929627","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}
{"title":"Critical principal singularities of hypersurfaces in Euclidean 4-spaces","authors":"R. Garcia, D. Lopes, J. Sotomayor","doi":"10.5427/jsing.2022.25i","DOIUrl":"https://doi.org/10.5427/jsing.2022.25i","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82163640","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}
{"title":"The Delta Invariant and Fiberwise Normalization for Families of isolated Non-Normal Singularities","authors":"G. Greuel, G. Pfister","doi":"10.5427/jsing.2022.25j","DOIUrl":"https://doi.org/10.5427/jsing.2022.25j","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78892060","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}
{"title":"Fundamental group of rational homology disk smoothings of surface singularities","authors":"E. Artal Bartolo, J. Wahl","doi":"10.5427/jsing.2022.24e","DOIUrl":"https://doi.org/10.5427/jsing.2022.24e","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72370978","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}
{"title":"Homologically trivial integrable deformations of germs of holomorphic functions","authors":"V. León, B. Scárdua","doi":"10.5427/jsing.2022.24d","DOIUrl":"https://doi.org/10.5427/jsing.2022.24d","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84124479","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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