Pub Date : 2022-10-01DOI: 10.1515/advgeom-2022-0015
A. Ivanov
Abstract The paper contributes to Majorana theory. Among the eight non-trivial Norton–Sakuma algebras, four algebras are closed on the set of Majorana generators. These algebras are 2A, 2B, 3C and 4B. The classification of Majorana representations restricted to the closed shapes was anticipated for a long time. In the present article the classification is achieved for shapes restricted to 2A, 3C and 4B and for the set of generating involutions in the target group forming a single conjugacy class. Timmesfeld’s classification of {3, 4}+-transposition groups reduces to consideration of just three groups: L3(2), G2(2)' and 3D4(2). Each of these groups possesses a unique Majorana representation of the required shape. Only the representation of L3(2), known before, is based on an embedding into the Monster.
{"title":"Closed Majorana representations of {3, 4}+-transposition groups","authors":"A. Ivanov","doi":"10.1515/advgeom-2022-0015","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0015","url":null,"abstract":"Abstract The paper contributes to Majorana theory. Among the eight non-trivial Norton–Sakuma algebras, four algebras are closed on the set of Majorana generators. These algebras are 2A, 2B, 3C and 4B. The classification of Majorana representations restricted to the closed shapes was anticipated for a long time. In the present article the classification is achieved for shapes restricted to 2A, 3C and 4B and for the set of generating involutions in the target group forming a single conjugacy class. Timmesfeld’s classification of {3, 4}+-transposition groups reduces to consideration of just three groups: L3(2), G2(2)' and 3D4(2). Each of these groups possesses a unique Majorana representation of the required shape. Only the representation of L3(2), known before, is based on an embedding into the Monster.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44516580","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 : 2022-10-01DOI: 10.1515/advgeom-2022-0022
H. Löwe
Abstract The present paper investigates 16-dimensional compact translation planes with automorphism groups of dimension d between 35 and 37; planes with groups of higher dimensions have been classified by Hähl. We obtain a complete classification for d = 37 (up to isomorphisms). It turns out that these planes have Lenz type V and are already described in a recent paper of Hähl and Meyer [10]. Moreover, we give a partial classification for d = 35 and d = 36. The latter case will be completely finished in a forthcoming paper [16] of the author, while the case where d = 35 is completed except for groups whose maximal compact subgroups are 9-dimensional.
{"title":"Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35","authors":"H. Löwe","doi":"10.1515/advgeom-2022-0022","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0022","url":null,"abstract":"Abstract The present paper investigates 16-dimensional compact translation planes with automorphism groups of dimension d between 35 and 37; planes with groups of higher dimensions have been classified by Hähl. We obtain a complete classification for d = 37 (up to isomorphisms). It turns out that these planes have Lenz type V and are already described in a recent paper of Hähl and Meyer [10]. Moreover, we give a partial classification for d = 35 and d = 36. The latter case will be completely finished in a forthcoming paper [16] of the author, while the case where d = 35 is completed except for groups whose maximal compact subgroups are 9-dimensional.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42092182","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 : 2022-10-01DOI: 10.1515/advgeom-2022-0020
G. Steinke
Abstract Kleinewillinghöfer classified in 1979 automorphism groups of Laguerre planes with respect to linearly transitive subgroups of central automorphisms and obtained a multitude of types. All feasible Kleinewillinghöfer types of 2-dimensional Laguerre planes were completely determined in 2021. In this paper we investigate the Kleinewillinghöfer types of 4-dimensional Laguerre planes with respect to the automorphism groups of these planes and show that of the 49 types Kleinewillinghöfer described, only twelve are feasible in 4-dimensional Laguerre planes. Examples of four of these type are provided.
{"title":"A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes","authors":"G. Steinke","doi":"10.1515/advgeom-2022-0020","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0020","url":null,"abstract":"Abstract Kleinewillinghöfer classified in 1979 automorphism groups of Laguerre planes with respect to linearly transitive subgroups of central automorphisms and obtained a multitude of types. All feasible Kleinewillinghöfer types of 2-dimensional Laguerre planes were completely determined in 2021. In this paper we investigate the Kleinewillinghöfer types of 4-dimensional Laguerre planes with respect to the automorphism groups of these planes and show that of the 49 types Kleinewillinghöfer described, only twelve are feasible in 4-dimensional Laguerre planes. Examples of four of these type are provided.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42041336","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 : 2022-10-01DOI: 10.1515/advgeom-2022-0023
R. Löwen
Abstract We describe the development of the mathematics of Helmut R. Salzmann (3. 11. 1930 – 8. 3. 2022) and the main difficulties he was facing, documenting his lifelong productivity and his far reaching influence. We include a comprehensive bibliography of his work.
{"title":"Helmut Salzmann and his legacy","authors":"R. Löwen","doi":"10.1515/advgeom-2022-0023","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0023","url":null,"abstract":"Abstract We describe the development of the mathematics of Helmut R. Salzmann (3. 11. 1930 – 8. 3. 2022) and the main difficulties he was facing, documenting his lifelong productivity and his far reaching influence. We include a comprehensive bibliography of his work.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48960833","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 : 2022-07-01DOI: 10.1515/advgeom-2022-0008
Daniel Naie
Abstract It is shown that a smooth surface lying on a nodal quartic hypersurface in ℙ4 is either regular or an elliptic conic bundle of degree 8. Furthermore, the latter configuration is shown to exist.
{"title":"Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities","authors":"Daniel Naie","doi":"10.1515/advgeom-2022-0008","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0008","url":null,"abstract":"Abstract It is shown that a smooth surface lying on a nodal quartic hypersurface in ℙ4 is either regular or an elliptic conic bundle of degree 8. Furthermore, the latter configuration is shown to exist.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43065642","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 : 2022-07-01DOI: 10.1515/advgeom-2022-0016
M. Brandt, David E. Speyer
Abstract We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. We give counterexamples to a few plausible statements about matroid subdivisions.
{"title":"Computation of Dressians by dimensional reduction","authors":"M. Brandt, David E. Speyer","doi":"10.1515/advgeom-2022-0016","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0016","url":null,"abstract":"Abstract We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. We give counterexamples to a few plausible statements about matroid subdivisions.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49460756","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 : 2022-07-01DOI: 10.1515/advgeom-2022-0014
H. Charalambous, K. Karagiannis, Sotiris Karanikolopoulos, A. Kontogeorgis
Abstract We present a necessary and sufficient condition for a maximal curve, defined over the algebraic closure of a finite field, to be realised as an HKG-cover. We use an approach via pole numbers in a rational point of the curve. For this class of curves, we compute their Weierstrass semigroup as well as the jumps of their higher ramification filtrations at this point, the unique ramification point of the cover.
{"title":"Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers","authors":"H. Charalambous, K. Karagiannis, Sotiris Karanikolopoulos, A. Kontogeorgis","doi":"10.1515/advgeom-2022-0014","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0014","url":null,"abstract":"Abstract We present a necessary and sufficient condition for a maximal curve, defined over the algebraic closure of a finite field, to be realised as an HKG-cover. We use an approach via pole numbers in a rational point of the curve. For this class of curves, we compute their Weierstrass semigroup as well as the jumps of their higher ramification filtrations at this point, the unique ramification point of the cover.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67145622","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 : 2022-07-01DOI: 10.1515/advgeom-2022-0010
D. Blázquez-Sanz, G. Casale, Juan Sebastián Díaz Arboleda
Abstract The Malgrange–Galois groupoid of Painlevé IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange–Galois groupoid of the Painlevé VI family including all parameters as new dependent variables. We conclude that it is the pseudogroup of transformations preserving the parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of a Painlevé VI equation depending analytically on the parameters does not satisfy any new partial differential equation (including derivatives with respect to parameters) which is not derived from Painlevé VI.
{"title":"The Malgrange–Galois groupoid of the Painlevé VI equation with parameters","authors":"D. Blázquez-Sanz, G. Casale, Juan Sebastián Díaz Arboleda","doi":"10.1515/advgeom-2022-0010","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0010","url":null,"abstract":"Abstract The Malgrange–Galois groupoid of Painlevé IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange–Galois groupoid of the Painlevé VI family including all parameters as new dependent variables. We conclude that it is the pseudogroup of transformations preserving the parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of a Painlevé VI equation depending analytically on the parameters does not satisfy any new partial differential equation (including derivatives with respect to parameters) which is not derived from Painlevé VI.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41379389","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 : 2022-05-28DOI: 10.1515/advgeom-2023-0006
I. Cardinali, H. Cuypers, L. Giuzzi, A. Pasini
Abstract A polar space 𝒮 is called symplectic if it admits a projective embedding ε : 𝒮 → PG(V) such that the image ε(𝒮) of 𝒮 by ε is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when 𝒮 admits different (non-isomorphic) embeddings, as it is the case when 𝒮 is defined over a field of characteristic 2.
{"title":"Characterizations of symplectic polar spaces","authors":"I. Cardinali, H. Cuypers, L. Giuzzi, A. Pasini","doi":"10.1515/advgeom-2023-0006","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0006","url":null,"abstract":"Abstract A polar space 𝒮 is called symplectic if it admits a projective embedding ε : 𝒮 → PG(V) such that the image ε(𝒮) of 𝒮 by ε is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when 𝒮 admits different (non-isomorphic) embeddings, as it is the case when 𝒮 is defined over a field of characteristic 2.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43120773","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 : 2022-04-18DOI: 10.1515/advgeom-2021-0039
Yaning Wang
Abstract In this paper, Hopf hypersurfaces in a complex projective plane ℂP2(c) or a complex hyperbolic plane ℂH2(c) with constant scalar curvature are classified. For a non-Hopf hypersurface in ℂP2(c) with constant scalar curvature r, it is proved that if the structure vector field is an eigenvector of the Ricci operator, then either r = 7c/2 or r = 3c/2. Moreover, these two cases are determined completely under an additional condition.
{"title":"Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature","authors":"Yaning Wang","doi":"10.1515/advgeom-2021-0039","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0039","url":null,"abstract":"Abstract In this paper, Hopf hypersurfaces in a complex projective plane ℂP2(c) or a complex hyperbolic plane ℂH2(c) with constant scalar curvature are classified. For a non-Hopf hypersurface in ℂP2(c) with constant scalar curvature r, it is proved that if the structure vector field is an eigenvector of the Ricci operator, then either r = 7c/2 or r = 3c/2. Moreover, these two cases are determined completely under an additional condition.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47084446","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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