A positive structure on the varieties of critical points of master functions for KZ equations is introduced. It comes as a combination of the ideas from classical works by G.Lusztig and a previous work by E.Mukhin and the second named author.
{"title":"Positive Populations","authors":"V. Schechtman, A. Varchenko","doi":"10.5427/jsing.2020.20p","DOIUrl":"https://doi.org/10.5427/jsing.2020.20p","url":null,"abstract":"A positive structure on the varieties of critical points of master functions for KZ equations is introduced. It comes as a combination of the ideas from classical works by G.Lusztig and a previous work by E.Mukhin and the second named author.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83599395","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 Image-Computing Spectral Sequence computes the homology of the image of a finite map from the alternating homology of the multiple point spaces of the map. A related spectral sequence was obtained by Gabrielov, Vorobjob and Zell which computes the homology of the image of a closed map from the homology of $k$-fold fibred products of the map. We give new proofs of these results, in case the map can be triangulated. Thanks to work of Hardt, this holds for a very wide range of maps, and in particular for most of the finite maps of interest in singularity theory. The proof seems conceptually simpler and more canonical than earlier proofs.
{"title":"Multiple points of a simplicial map and image-computing spectral sequences","authors":"J. Cisneros-Molina, D. Mond","doi":"10.5427/jsing.2022.24h","DOIUrl":"https://doi.org/10.5427/jsing.2022.24h","url":null,"abstract":"The Image-Computing Spectral Sequence computes the homology of the image of a finite map from the alternating homology of the multiple point spaces of the map. A related spectral sequence was obtained by Gabrielov, Vorobjob and Zell which computes the homology of the image of a closed map from the homology of $k$-fold fibred products of the map. We give new proofs of these results, in case the map can be triangulated. Thanks to work of Hardt, this holds for a very wide range of maps, and in particular for most of the finite maps of interest in singularity theory. The proof seems conceptually simpler and more canonical than earlier proofs.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75382275","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 the generic bifurcations of the Minkowski symmetry set for 1-parameter families of plane curves are classified and the necessary and sufficient geometric criteria for each type are given. The Minkowski symmetry set is an analogue of the standard Euclidean symmetry set, and is defined to be the locus of centres of all its bitangent pseudo-circles. It is shown that the list of possible bifurcation types are different to those that occur in the list of possible types for the Euclidean symmetry set.
{"title":"Minkowski symmetry sets for 1-parameter families of plane curves","authors":"Graham M. Reeve","doi":"10.5427/jsing.2022.25q","DOIUrl":"https://doi.org/10.5427/jsing.2022.25q","url":null,"abstract":"In this paper the generic bifurcations of the Minkowski symmetry set for 1-parameter families of plane curves are classified and the necessary and sufficient geometric criteria for each type are given. The Minkowski symmetry set is an analogue of the standard Euclidean symmetry set, and is defined to be the locus of centres of all its bitangent pseudo-circles. It is shown that the list of possible bifurcation types are different to those that occur in the list of possible types for the Euclidean symmetry set.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81872833","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 show that a Cohen-Macaulay analytic singularity can be arbitrarily closely approximated by germs of Nash sets which are also Cohen-Macaulay and share the same Hilbert-Samuel function. We also prove that every analytic singularity is topologically equivalent to a Nash singularity with the same Hilbert-Samuel function.
{"title":"Equisingular algebraic approximation of real and complex analytic germs","authors":"J. Adamus, Aftab Patel","doi":"10.5427/jsing.2020.20n","DOIUrl":"https://doi.org/10.5427/jsing.2020.20n","url":null,"abstract":"We show that a Cohen-Macaulay analytic singularity can be arbitrarily closely approximated by germs of Nash sets which are also Cohen-Macaulay and share the same Hilbert-Samuel function. We also prove that every analytic singularity is topologically equivalent to a Nash singularity with the same Hilbert-Samuel function.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80669213","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 Brasselet number of a function $f$ with nonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, we consider two function-germs $f,g:(X,0)rightarrow(mathbb{C},0)$ such that $f$ has isolated singularity at the origin and $g$ has a stratified one-dimensional critical set. We use the Brasselet number to study the local topology a deformation $tilde{g}$ of $g$ defined by $tilde{g}=g+f^N,$ where $Ngg1$ and $Ninmathbb{N}$. As an application of this study, we present a new proof of the Le-Iomdin formula for the Brasselet number.
{"title":"Some notes on the local topology of a deformation of a function-germ with a one-dimensional critical set","authors":"H. Santana","doi":"10.5427/jsing.2022.25t","DOIUrl":"https://doi.org/10.5427/jsing.2022.25t","url":null,"abstract":"The Brasselet number of a function $f$ with nonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, we consider two function-germs $f,g:(X,0)rightarrow(mathbb{C},0)$ such that $f$ has isolated singularity at the origin and $g$ has a stratified one-dimensional critical set. We use the Brasselet number to study the local topology a deformation $tilde{g}$ of $g$ defined by $tilde{g}=g+f^N,$ where $Ngg1$ and $Ninmathbb{N}$. As an application of this study, we present a new proof of the Le-Iomdin formula for the Brasselet number.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76237201","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}
Javier Carvajal-Rojas, Linquan Ma, Thomas Polstra, Karl Schwede, Kevin Tucker
We further the classification of rational surface singularities. Suppose $(S, mathfrak{n}, mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $mathfrak{n}$. The classification of such functions are used to show that if $(R, mathfrak{m}, mathcal{k})$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0, p > 5)$, then there exists a split finite cover of $mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.
{"title":"Covers of rational double points in mixed characteristic","authors":"Javier Carvajal-Rojas, Linquan Ma, Thomas Polstra, Karl Schwede, Kevin Tucker","doi":"10.5427/jsing.2021.23h","DOIUrl":"https://doi.org/10.5427/jsing.2021.23h","url":null,"abstract":"We further the classification of rational surface singularities. Suppose $(S, mathfrak{n}, mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $mathfrak{n}$. The classification of such functions are used to show that if $(R, mathfrak{m}, mathcal{k})$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0, p > 5)$, then there exists a split finite cover of $mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74251705","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 main result in this paper is to supply a recursive formula, on the number of minimal primes, for the colength of a fractional ideal in terms of the maximal points of the value set of the ideal itself. The fractional ideals are taken in the class of complete admissible rings, a more general class of rings than those of algebroid curves. For such rings with two or three minimal primes, a closed formula for that colength is provided, so improving results by Barucci, D'Anna and Fr"oberg.
{"title":"On the Colength of Fractional Ideals","authors":"Edison Marcavillaca Nino de Guzm'an, A. Hefez","doi":"10.5427/jsing.2020.21g","DOIUrl":"https://doi.org/10.5427/jsing.2020.21g","url":null,"abstract":"The main result in this paper is to supply a recursive formula, on the number of minimal primes, for the colength of a fractional ideal in terms of the maximal points of the value set of the ideal itself. The fractional ideals are taken in the class of complete admissible rings, a more general class of rings than those of algebroid curves. For such rings with two or three minimal primes, a closed formula for that colength is provided, so improving results by Barucci, D'Anna and Fr\"oberg.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86552657","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 characteristic $p=2$, we compute the singularities of Kummer varieties arising from products of elliptic curves. This result is generalized to Kummer varieties associated to ordinary abelian varieties.
{"title":"Wild Singularities of Kummer Varieties","authors":"Benedikt Schilson","doi":"10.5427/jsing.2020.20m","DOIUrl":"https://doi.org/10.5427/jsing.2020.20m","url":null,"abstract":"In characteristic $p=2$, we compute the singularities of Kummer varieties arising from products of elliptic curves. This result is generalized to Kummer varieties associated to ordinary abelian varieties.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88851083","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}
Generically, the singular complex analytic vector fields $X$ on the Riemann sphere $widehat{mathbb{C}}_{z}$ belonging to the family $$ mathscr{E}(r,d)=Big{ X(z)=frac{1}{P(z)} text{e}^{E(z)}frac{partial}{partial z} bigvert P, Einmathbb{C}[z], deg(P)=r, deg(E)=d Big}, $$ have an essential singularity of finite 1-order at infinity and a finite number of poles on the complex plane. We describe $X$, particularly the singularity at $inftyinwidehat{mathbb{C}}_{z}$. In order to do so, we use the natural $correspondence$ between $Xinmathscr{E}(r,d)$, a global singular analytic distinguished parameter $Psi_X=int omega_X$, and the Riemann surface $mathcal{R}_X$ of the distinguished parameter. We introduce $(r,d)$-$configuration trees$ $Lambda_X$: combinatorial objects that completely encode the Riemann surfaces $mathcal{R}_X$ and singular flat metrics associated to $Xinmathscr{E}(r,d)$. This provides an alternate `dynamical' coordinate system and an analytic classification of $mathscr{E}(r,d)$. Furthermore, the phase portrait of $mathscr{Re}(X)$ on $mathbb{C}$ is decomposed into $mathscr{Re}(X)$-invariant regions: half planes and finite height strip flows. The germ of $X$ at $infty in widehat{mathbb{C}}$ is described as an admissible word (equivalent to certain canonical angular sectors). The structural stability of the phase portrait of $mathscr{Re}(X)$ is characterized by using $Lambda_X$ and the number of topologically equivalent phase portraits of $mathscr{Re}(X)$ is bounded.
{"title":"Dynamics of singular complex analytic vector fields with essential singularities II","authors":"Alvaro Alvarez-Parrilla, Jesús Muciño-Raymundo","doi":"10.5427/jsing.2022.24a","DOIUrl":"https://doi.org/10.5427/jsing.2022.24a","url":null,"abstract":"Generically, the singular complex analytic vector fields $X$ on the Riemann sphere $widehat{mathbb{C}}_{z}$ belonging to the family $$ mathscr{E}(r,d)=Big{ X(z)=frac{1}{P(z)} text{e}^{E(z)}frac{partial}{partial z} bigvert P, Einmathbb{C}[z], deg(P)=r, deg(E)=d Big}, $$ have an essential singularity of finite 1-order at infinity and a finite number of poles on the complex plane. We describe $X$, particularly the singularity at $inftyinwidehat{mathbb{C}}_{z}$. \u0000In order to do so, we use the natural $correspondence$ between $Xinmathscr{E}(r,d)$, a global singular analytic distinguished parameter $Psi_X=int omega_X$, and the Riemann surface $mathcal{R}_X$ of the distinguished parameter. \u0000We introduce $(r,d)$-$configuration trees$ $Lambda_X$: combinatorial objects that completely encode the Riemann surfaces $mathcal{R}_X$ and singular flat metrics associated to $Xinmathscr{E}(r,d)$. This provides an alternate `dynamical' coordinate system and an analytic classification of $mathscr{E}(r,d)$. Furthermore, the phase portrait of $mathscr{Re}(X)$ on $mathbb{C}$ is decomposed into $mathscr{Re}(X)$-invariant regions: half planes and finite height strip flows. The germ of $X$ at $infty in widehat{mathbb{C}}$ is described as an admissible word (equivalent to certain canonical angular sectors). The structural stability of the phase portrait of $mathscr{Re}(X)$ is characterized by using $Lambda_X$ and the number of topologically equivalent phase portraits of $mathscr{Re}(X)$ is bounded.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81332410","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}
Atsufumi Honda, K. Naokawa, K. Saji, M. Umehara, Kotaro Yamada
In the second, fourth and fifth authors' previous work, a duality on generic real analytic cuspidal edges in the Euclidean 3-space $boldsymbol R^3$ preserving their singular set images and first fundamental forms, was given. Here, we call this an `isometric duality'. When the singular set image has no symmetries and does not lie in a plane, the dual cuspidal edge is not congruent to the original one. In this paper, we show that this duality extends to generalized cuspidal edges in $boldsymbol R^3$, including cuspidal cross caps, and $5/2$-cuspidal edges. Moreover, we give several new geometric insights on this duality.
{"title":"Duality on generalized cuspidal edges preserving singular set images and first fundamental forms","authors":"Atsufumi Honda, K. Naokawa, K. Saji, M. Umehara, Kotaro Yamada","doi":"10.5427/jsing.2020.22e","DOIUrl":"https://doi.org/10.5427/jsing.2020.22e","url":null,"abstract":"In the second, fourth and fifth authors' previous work, a duality on generic real analytic cuspidal edges in the Euclidean 3-space $boldsymbol R^3$ preserving their singular set images and first fundamental forms, was given. Here, we call this an `isometric duality'. When the singular set image has no symmetries and does not lie in a plane, the dual cuspidal edge is not congruent to the original one. In this paper, we show that this duality extends to generalized cuspidal edges in $boldsymbol R^3$, including cuspidal cross caps, and $5/2$-cuspidal edges. Moreover, we give several new geometric insights on this duality.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81388223","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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