Pub Date : 2021-04-01DOI: 10.1515/advgeom-2020-0017
Takayuki Morifuji, Anh T. Tran
Abstract In this paper, we explicitly calculate the highest degree term of the hyperbolic torsion polynomial of an infinite family of pretzel knots. This gives supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. The verification of the genus part of the conjecture for this family of knots also follows from the work of Agol and Dunfield [1] or Porti [19].
{"title":"Hyperbolic torsion polynomials of pretzel knots","authors":"Takayuki Morifuji, Anh T. Tran","doi":"10.1515/advgeom-2020-0017","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0017","url":null,"abstract":"Abstract In this paper, we explicitly calculate the highest degree term of the hyperbolic torsion polynomial of an infinite family of pretzel knots. This gives supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. The verification of the genus part of the conjecture for this family of knots also follows from the work of Agol and Dunfield [1] or Porti [19].","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"21 1","pages":"265 - 272"},"PeriodicalIF":0.5,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43297894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-01DOI: 10.1515/advgeom-2020-0034
Taro Hayashi
Abstract General K3 surfaces obtained as double covers of the n-th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces.
{"title":"Double cover K3 surfaces of Hirzebruch surfaces","authors":"Taro Hayashi","doi":"10.1515/advgeom-2020-0034","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0034","url":null,"abstract":"Abstract General K3 surfaces obtained as double covers of the n-th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"21 1","pages":"221 - 225"},"PeriodicalIF":0.5,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44133106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-01DOI: 10.1515/advgeom-2020-0035
N. Hungerbühler, Gideon Villiger
Abstract In the Euclidean plane, two circles that intersect or are tangent clearly do not carry a finite Steiner chain of circles. We show that such exotic Steiner chains exist in finite Miquelian Möbius planes of odd order. We obtain explicit conditions in terms of the order of the plane and the capacitance of the two carrier circles for the existence, length, and number of Steiner chains.
{"title":"Exotic Steiner chains in Miquelian Möbius planes of odd order","authors":"N. Hungerbühler, Gideon Villiger","doi":"10.1515/advgeom-2020-0035","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0035","url":null,"abstract":"Abstract In the Euclidean plane, two circles that intersect or are tangent clearly do not carry a finite Steiner chain of circles. We show that such exotic Steiner chains exist in finite Miquelian Möbius planes of odd order. We obtain explicit conditions in terms of the order of the plane and the capacitance of the two carrier circles for the existence, length, and number of Steiner chains.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"21 1","pages":"207 - 220"},"PeriodicalIF":0.5,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0035","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47631427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-30DOI: 10.1515/advgeom-2023-0002
S. A. Secci
Abstract We prove a classification result for smooth complex Fano fourfolds of Picard number 3 having a prime divisor of Picard number 1, after a characterisation result in arbitrary dimension by Casagrande and Druel [5]. These varieties are obtained by blowing-up a ℙ1-bundle over a smooth Fano variety of Picard number 1 along a codimension 2 subvariety. We study in detail the case of dimension 4, and show that they form 28 families. We compute the main numerical invariants, determine the base locus of the anticanonical system, and study their deformations to give an upper bound to the dimension of the base of the Kuranishi family of a general member.
{"title":"Fano fourfolds having a prime divisor of Picard number 1","authors":"S. A. Secci","doi":"10.1515/advgeom-2023-0002","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0002","url":null,"abstract":"Abstract We prove a classification result for smooth complex Fano fourfolds of Picard number 3 having a prime divisor of Picard number 1, after a characterisation result in arbitrary dimension by Casagrande and Druel [5]. These varieties are obtained by blowing-up a ℙ1-bundle over a smooth Fano variety of Picard number 1 along a codimension 2 subvariety. We study in detail the case of dimension 4, and show that they form 28 families. We compute the main numerical invariants, determine the base locus of the anticanonical system, and study their deformations to give an upper bound to the dimension of the base of the Kuranishi family of a general member.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"23 1","pages":"267 - 280"},"PeriodicalIF":0.5,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46339592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-11DOI: 10.1515/advgeom-2020-0020
S. Cynk
Abstract We give a formula for the Hodge numbers of a three-dimensional hypersurface in a weighted projective space with only ordinary triple points as singularities.
摘要给出了仅以普通三点为奇异点的加权投影空间中三维超曲面的Hodge数的一个公式。
{"title":"Hodge numbers of hypersurfaces in ℙ4 with ordinary triple points","authors":"S. Cynk","doi":"10.1515/advgeom-2020-0020","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0020","url":null,"abstract":"Abstract We give a formula for the Hodge numbers of a three-dimensional hypersurface in a weighted projective space with only ordinary triple points as singularities.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"21 1","pages":"293 - 298"},"PeriodicalIF":0.5,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47410888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-05DOI: 10.1515/advgeom-2020-0033
Tomoya Miura, S. Maeta
Abstract We show that any triharmonic Riemannian submersion from a 3-dimensional space form into a surface is harmonic. This is an affirmative partial answer to the submersion version of the generalized Chen conjecture. Moreover, a non-existence theorem for f -biharmonic Riemannian submersions is also presented.
{"title":"Triharmonic Riemannian submersions from 3-dimensional space forms","authors":"Tomoya Miura, S. Maeta","doi":"10.1515/advgeom-2020-0033","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0033","url":null,"abstract":"Abstract We show that any triharmonic Riemannian submersion from a 3-dimensional space form into a surface is harmonic. This is an affirmative partial answer to the submersion version of the generalized Chen conjecture. Moreover, a non-existence theorem for f -biharmonic Riemannian submersions is also presented.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":" 19","pages":"163 - 168"},"PeriodicalIF":0.5,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0033","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41311219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-14DOI: 10.1515/advgeom-2023-0005
Lin Zhou
Abstract We compute the Chow groups of smooth Gushel–Mukai varieties of dimension 5.
摘要计算了5维光滑Gushel-Mukai变元的Chow群。
{"title":"Chow groups of Gushel–Mukai fivefolds","authors":"Lin Zhou","doi":"10.1515/advgeom-2023-0005","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0005","url":null,"abstract":"Abstract We compute the Chow groups of smooth Gushel–Mukai varieties of dimension 5.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"23 1","pages":"295 - 303"},"PeriodicalIF":0.5,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46560440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-14DOI: 10.1515/advgeom-2022-0013
Auguste Hébert
Abstract Masures are generalizations of Bruhat–Tits buildings. They were introduced by Gaussent and Rousseau in order to study Kac–Moody groups over valued fields. We prove that the intersection of two apartments of a masure is convex. Using this, we simplify the axiomatic definition of masures given by Rousseau.
{"title":"A new axiomatics for masures II","authors":"Auguste Hébert","doi":"10.1515/advgeom-2022-0013","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0013","url":null,"abstract":"Abstract Masures are generalizations of Bruhat–Tits buildings. They were introduced by Gaussent and Rousseau in order to study Kac–Moody groups over valued fields. We prove that the intersection of two apartments of a masure is convex. Using this, we simplify the axiomatic definition of masures given by Rousseau.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"22 1","pages":"513 - 522"},"PeriodicalIF":0.5,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47461537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1515/advgeom-2020-0026
Drew Johnson, A. Polishchuk
Abstract We study birational projective models of 𝓜2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M2,2(𝓩).
{"title":"Birational models of 𝓜2,2 arising as moduli of curves with nonspecial divisors","authors":"Drew Johnson, A. Polishchuk","doi":"10.1515/advgeom-2020-0026","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0026","url":null,"abstract":"Abstract We study birational projective models of 𝓜2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M2,2(𝓩).","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"21 1","pages":"23 - 43"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0026","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41378364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1515/advgeom-2021-frontmatter1
{"title":"Frontmatter","authors":"","doi":"10.1515/advgeom-2021-frontmatter1","DOIUrl":"https://doi.org/10.1515/advgeom-2021-frontmatter1","url":null,"abstract":"","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2021-frontmatter1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46825967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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