For a given topological type of a normal surface singularity, there are various types of complex structures which realize it. We are interested in the following problem: Find the maximum of the geometric genus and a condition for that the maximal ideal cycle coincides with the undamental cycle on the minimal good resolution. In this paper, we study weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersection singularities from the perspective of the problem.
{"title":"Weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersections","authors":"Tomohiro Okuma","doi":"10.5427/jsing.2021.23j","DOIUrl":"https://doi.org/10.5427/jsing.2021.23j","url":null,"abstract":"For a given topological type of a normal surface singularity, there are various types of complex structures which realize it. We are interested in the following problem: Find the maximum of the geometric genus and a condition for that the maximal ideal cycle coincides with the undamental cycle on the minimal good resolution. In this paper, we study weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersection singularities from the perspective of the problem.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88093169","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 middle homology of the Milnor fiber of a quasihomogeneous polynomial with an isolated singularity is a ${mathbb Z}$-lattice and comes equipped with an automorphism of finite order, the integral monodromy. Orlik (1972) made a precise conjecture, which would determine this monodromy in terms of the weights of the polynomial. Here we prove this conjecture for the cycle type singularities. A paper of Cooper (1982) with the same aim contained two mistakes. Still it is very useful. We build on it and correct the mistakes. We give additional algebraic and combinatorial results.
{"title":"The integral monodromy of the cycle type singularities","authors":"C. Hertling, Makiko Mase","doi":"10.5427/jsing.2022.25l","DOIUrl":"https://doi.org/10.5427/jsing.2022.25l","url":null,"abstract":"The middle homology of the Milnor fiber of a quasihomogeneous polynomial with an isolated singularity is a ${mathbb Z}$-lattice and comes equipped with an automorphism of finite order, the integral monodromy. Orlik (1972) made a precise conjecture, which would determine this monodromy in terms of the weights of the polynomial. Here we prove this conjecture for the cycle type singularities. A paper of Cooper (1982) with the same aim contained two mistakes. Still it is very useful. We build on it and correct the mistakes. We give additional algebraic and combinatorial results.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76673136","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 this paper we discuss the behavior of the curvature lines of a transversal eq¨uiaffine vector field along a surface in 3-space at isolated umbilical points. Mathematics Subject Classification (2010). 53A15, 53A05.
{"title":"Curvature lines of a transversal equiaffine vector field along a surface in 3-space","authors":"M. Craizer, Ronaldo Garcia","doi":"10.5427/jsing.2022.25g","DOIUrl":"https://doi.org/10.5427/jsing.2022.25g","url":null,"abstract":". In this paper we discuss the behavior of the curvature lines of a transversal eq¨uiaffine vector field along a surface in 3-space at isolated umbilical points. Mathematics Subject Classification (2010). 53A15, 53A05.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74590748","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}
Bucsra Karadeniz, H. Mourtada, Camille Pl'enat, M. Tosun
We consider the question whether one can construct an embedded resolution of singularities of a singular variety $Xsubset textbf{A}^n$ from the data of the irreducible components of the spaces of jets (of $X$) centered at the singular locus of $X.$ We show that the answer is no in general and that it is yes for some birational models of rational triple surface singularities.
{"title":"The embedded Nash problem of birational models of rational triple singularities","authors":"Bucsra Karadeniz, H. Mourtada, Camille Pl'enat, M. Tosun","doi":"10.5427/jsing.2020.22u","DOIUrl":"https://doi.org/10.5427/jsing.2020.22u","url":null,"abstract":"We consider the question whether one can construct an embedded resolution of singularities of a singular variety $Xsubset textbf{A}^n$ from the data of the irreducible components of the spaces of jets (of $X$) centered at the singular locus of $X.$ We show that the answer is no in general and that it is yes for some birational models of rational triple surface singularities.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84271901","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 purely inseparable morphisms of degree $p$ between rational double points (RDPs) in characteristic $p$. Using such morphisms, we show that any RDP admit a finite smooth covering.
{"title":"Purely inseparable coverings of rational double points in positive characteristic","authors":"Y. Matsumoto","doi":"10.5427/jsing.2022.24b","DOIUrl":"https://doi.org/10.5427/jsing.2022.24b","url":null,"abstract":"We classify purely inseparable morphisms of degree $p$ between rational double points (RDPs) in characteristic $p$. Using such morphisms, we show that any RDP admit a finite smooth covering.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73189377","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 this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of itself. The parabolic set itself is detected in this way, and each arc is given a sign, which changes at the special points, where the family has an additional degree of symmetry. Other points of $M$ which are detected by the family of reflexions include inflexion points of real and imaginary type, and the first of these is also associated with sign changes on the parabolic set. We show how to compute the special points globally for the case where $M$ is given in Monge form and give some examples illustrating the birth of special parbolic points in a 1-parameter family of surfaces. The tool we use from singularity theory is the contact classification of certain symmetric maps from the plane to the plane and we give the beginning of this classification, including versal unfoldings which we relate to the geometry of $M$.
{"title":"Reflexion maps and geometry of surfaces in R^4","authors":"P. Giblin, S. Janeczko, M. Ruas","doi":"10.5427/jsing.2020.21e","DOIUrl":"https://doi.org/10.5427/jsing.2020.21e","url":null,"abstract":"In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of itself. The parabolic set itself is detected in this way, and each arc is given a sign, which changes at the special points, where the family has an additional degree of symmetry. Other points of $M$ which are detected by the family of reflexions include inflexion points of real and imaginary type, and the first of these is also associated with sign changes on the parabolic set. We show how to compute the special points globally for the case where $M$ is given in Monge form and give some examples illustrating the birth of special parbolic points in a 1-parameter family of surfaces. The tool we use from singularity theory is the contact classification of certain symmetric maps from the plane to the plane and we give the beginning of this classification, including versal unfoldings which we relate to the geometry of $M$.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89186963","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}
For a smooth surface in $mathbb{R}^3$ this article contains local study of certain affine equidistants, that is loci of points at a fixed ratio between points of contact of parallel tangent planes (but excluding ratios 0 and 1 where the equidistant contains one or other point of contact). The situation studied occurs generically in a 1-parameter family, where two parabolic points of the surface have parallel tangent planes at which the unique asymptotic directions are also parallel. The singularities are classified by regarding the equidistants as critical values of a 2-parameter unfolding of maps from $mathbb{R}^4$ to $mathbb{R}^3$. In particular, the singularities that occur near the so-called `supercaustic chord', joining the two special parabolic points, are classified. For a given ratio along this chord either one or three special points are identified at which singularities of the equidistant become more special. Many of the resulting singularities have occurred before in the literature in abstract classifications, so the article also provides a natural geometric setting for these singularities, relating back to the geometry of the surfaces from which they are derived.
{"title":"Equidistants for families of surfaces","authors":"P. Giblin, Graham M. Reeve","doi":"10.5427/jsing.2020.21f","DOIUrl":"https://doi.org/10.5427/jsing.2020.21f","url":null,"abstract":"For a smooth surface in $mathbb{R}^3$ this article contains local study of certain affine equidistants, that is loci of points at a fixed ratio between \u0000points of contact of parallel tangent planes (but excluding ratios 0 and 1 where the equidistant contains one or other point of contact). The situation studied occurs generically in a 1-parameter family, where two parabolic points of the surface have parallel tangent planes at which the unique asymptotic directions are also parallel. The singularities are classified by regarding the equidistants as critical values of a 2-parameter unfolding of maps from $mathbb{R}^4$ to $mathbb{R}^3$. In particular, the singularities that occur near the so-called `supercaustic chord', joining the two special parabolic points, are classified. For a given ratio along this chord either one or three special points are identified at which singularities of the equidistant become more special. Many of the resulting singularities have occurred before in the literature in abstract classifications, so the article also provides a natural geometric setting for these singularities, relating back to the geometry of the surfaces from which they are derived.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84747285","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 give a counterexample to a conjecture of Eyral on the existence of deformation retracts to intersections of Whitney stratifications embedded in a smooth manifold. We then prove that the conjecture holds if the stratifications are definable in some o-minimal structure without assuming any regularity conditions. Moreover, we also show that the conjecture holds for Whitney stratifications if they intersect transversally.
{"title":"Deformation retracts to intersections of Whitney stratifications","authors":"S. Trivedi, D. Trotman","doi":"10.5427/jsing.2020.22s","DOIUrl":"https://doi.org/10.5427/jsing.2020.22s","url":null,"abstract":". We give a counterexample to a conjecture of Eyral on the existence of deformation retracts to intersections of Whitney stratifications embedded in a smooth manifold. We then prove that the conjecture holds if the stratifications are definable in some o-minimal structure without assuming any regularity conditions. Moreover, we also show that the conjecture holds for Whitney stratifications if they intersect transversally.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84022646","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}
. Let M be a connected compact surface with boundary. A C ∞ map M → R 2 is admissible if it is non-singular on a neighborhood of the boundary. For a C ∞ stable map f : M → R 2 , denote by c ( f ) and n ( f ), i ( f ) the number of cusps and nodes, connected components of the set of singular points respectively. In this paper, we introduce the notion of admissibly homotopic among C ∞ maps M → R 2 , and we will determine the minimal number c + n for each admissibly homotopy class.
． 设M是一个有边界的连通紧曲面。如果一个C∞映射M→r2在边界的邻域上是非奇异的，则该映射是允许的。对于一个C∞稳定映射f: M→r2，分别用C (f)和n (f)， i (f)表示奇异点集合的顶点数和节点数。本文引入了C∞映射M→r2中的可容许同伦的概念，并确定了每个可容许同伦类的最小值C + n。
{"title":"Apparent contours of stable maps of surfaces with boundary into the plane","authors":"Takahiro Yamamoto","doi":"10.5427/jsing.2020.22h","DOIUrl":"https://doi.org/10.5427/jsing.2020.22h","url":null,"abstract":". Let M be a connected compact surface with boundary. A C ∞ map M → R 2 is admissible if it is non-singular on a neighborhood of the boundary. For a C ∞ stable map f : M → R 2 , denote by c ( f ) and n ( f ), i ( f ) the number of cusps and nodes, connected components of the set of singular points respectively. In this paper, we introduce the notion of admissibly homotopic among C ∞ maps M → R 2 , and we will determine the minimal number c + n for each admissibly homotopy class.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80950655","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":"On the topology of a resolution of isolated singularities, II","authors":"V. Di Gennaro, D. Franco","doi":"10.5427/jsing.2020.20e","DOIUrl":"https://doi.org/10.5427/jsing.2020.20e","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91084969","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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