Pub Date : 2024-05-16DOI: 10.1017/s001708952400017x
Vincent Grandjean, Roger Oliveira
Let $sigma _q ,:,{{mathbb{R}}^q} to{textbf{S}}^qsetminus N_q$ be the inverse of the stereographic projection with center the north pole $N_q$ . Let $W_i$ be a closed subset of ${mathbb{R}}^{q_i}$ , for $i=1,2$ . Let $Phi ,:,W_1 to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $sigma _{q_2}circ Phi circ sigma _{q_1}^{-1}$ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_1}$ with value $N_{q_2}$ whenever
{"title":"Stereographic compactification and affine bi-Lipschitz homeomorphisms","authors":"Vincent Grandjean, Roger Oliveira","doi":"10.1017/s001708952400017x","DOIUrl":"https://doi.org/10.1017/s001708952400017x","url":null,"abstract":"\u0000\t <jats:p>Let <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952400017X_inline1.png\"/>\u0000\t\t<jats:tex-math>\u0000$sigma _q ,:,{{mathbb{R}}^q} to{textbf{S}}^qsetminus N_q$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> be the inverse of the stereographic projection with center the north pole <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952400017X_inline2.png\"/>\u0000\t\t<jats:tex-math>\u0000$N_q$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. Let <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952400017X_inline3.png\"/>\u0000\t\t<jats:tex-math>\u0000$W_i$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> be a closed subset of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952400017X_inline4.png\"/>\u0000\t\t<jats:tex-math>\u0000${mathbb{R}}^{q_i}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, for <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952400017X_inline5.png\"/>\u0000\t\t<jats:tex-math>\u0000$i=1,2$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. Let <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952400017X_inline6.png\"/>\u0000\t\t<jats:tex-math>\u0000$Phi ,:,W_1 to W_2$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952400017X_inline7.png\"/>\u0000\t\t<jats:tex-math>\u0000$sigma _{q_2}circ Phi circ sigma _{q_1}^{-1}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952400017X_inline8.png\"/>\u0000\t\t<jats:tex-math>\u0000$N_{q_1}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> with value <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952400017X_inline9.png\"/>\u0000\t\t<jats:tex-math>\u0000$N_{q_2}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> whenever <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140971318","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 : 2024-05-09DOI: 10.1017/s0017089524000181
Azer Akhmedov
We prove the Girth Alternative for finitely generated subgroups of $PL_o(I)$ . We also prove that a finitely generated subgroup of Homeo $_{+}(I)$ which is sufficiently rich with hyperbolic-like elements has infinite girth.
{"title":"Girth Alternative for subgroups of","authors":"Azer Akhmedov","doi":"10.1017/s0017089524000181","DOIUrl":"https://doi.org/10.1017/s0017089524000181","url":null,"abstract":"We prove the <jats:italic>Girth Alternative</jats:italic> for finitely generated subgroups of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000181_inline2.png\"/> <jats:tex-math> $PL_o(I)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also prove that a finitely generated subgroup of <jats:italic>Homeo</jats:italic><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000181_inline3.png\"/> <jats:tex-math> $_{+}(I)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> which is sufficiently rich with hyperbolic-like elements has infinite girth.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936343","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 : 2024-05-02DOI: 10.1017/s0017089524000168
Sandip Singh, Shashank Vikram Singh
We show the thinness of $7$ of the $40$ hypergeometric groups having a maximally unipotent monodromy in $mathrm{Sp}(6)$ .
我们证明了$mathrm{Sp}(6)$中具有最大单势单色性的$40$超几何群中$7$的稀疏性。
{"title":"Thinness of some hypergeometric groups in","authors":"Sandip Singh, Shashank Vikram Singh","doi":"10.1017/s0017089524000168","DOIUrl":"https://doi.org/10.1017/s0017089524000168","url":null,"abstract":"We show the thinness of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000168_inline2.png\"/> <jats:tex-math> $7$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000168_inline3.png\"/> <jats:tex-math> $40$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> hypergeometric groups having a maximally unipotent monodromy in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000168_inline4.png\"/> <jats:tex-math> $mathrm{Sp}(6)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140834195","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 : 2024-04-22DOI: 10.1017/s0017089524000107
Giuseppe Bargagnati
We show that for $n neq 1,4$ , the simplicial volume of an inward tame triangulable open $n$ -manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $pi _1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.
{"title":"Simplicial volume of manifolds with amenable fundamental group at infinity","authors":"Giuseppe Bargagnati","doi":"10.1017/s0017089524000107","DOIUrl":"https://doi.org/10.1017/s0017089524000107","url":null,"abstract":"We show that for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline1.png\" /> <jats:tex-math> $n neq 1,4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the simplicial volume of an inward tame triangulable open <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline2.png\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-manifold <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline3.png\" /> <jats:tex-math> $M$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with amenable fundamental group at infinity at each end is finite; moreover, we show that if also <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline4.png\" /> <jats:tex-math> $pi _1(M)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is amenable, then the simplicial volume of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000107_inline5.png\" /> <jats:tex-math> $M$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140636483","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 : 2024-04-17DOI: 10.1017/s0017089524000120
Karthika Rajeev, Anitha Thillaisundaram
The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently, a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an $s$ -generated weakly branch, but not branch, group that acts on the $m$ -adic tree, for $s,mge 2$ . A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.
{"title":"Maximal subgroups of a family of iterated monodromy groups","authors":"Karthika Rajeev, Anitha Thillaisundaram","doi":"10.1017/s0017089524000120","DOIUrl":"https://doi.org/10.1017/s0017089524000120","url":null,"abstract":"The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently, a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000120_inline1.png\" /> <jats:tex-math> $s$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-generated weakly branch, but not branch, group that acts on the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000120_inline2.png\" /> <jats:tex-math> $m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-adic tree, for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000120_inline3.png\" /> <jats:tex-math> $s,mge 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140613816","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 : 2024-04-15DOI: 10.1017/s0017089524000077
Maciej Borodzik, Anthony Conway, Wojciech Politarczyk
This paper describes how to compute algorithmically certain twisted signature invariants of a knot $K$ using twisted Blanchfield forms. An illustration of the algorithm is implemented on $(2,q)$ -torus knots. Additionally, using satellite formulas for these invariants, we also show how to obstruct the sliceness of certain iterated torus knots.
{"title":"Twisted Blanchfield pairings and twisted signatures III: Applications","authors":"Maciej Borodzik, Anthony Conway, Wojciech Politarczyk","doi":"10.1017/s0017089524000077","DOIUrl":"https://doi.org/10.1017/s0017089524000077","url":null,"abstract":"This paper describes how to compute algorithmically certain twisted signature invariants of a knot <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000077_inline1.png\" /> <jats:tex-math> $K$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> using twisted Blanchfield forms. An illustration of the algorithm is implemented on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000077_inline2.png\" /> <jats:tex-math> $(2,q)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-torus knots. Additionally, using satellite formulas for these invariants, we also show how to obstruct the sliceness of certain iterated torus knots.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140569741","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 : 2024-04-12DOI: 10.1017/s0017089524000089
Hsian-Hua Tseng
We consider K-theoretic Gromov-Witten theory of root constructions. We calculate some genus $0$ K-theoretic Gromov-Witten invariants of a root gerbe. We also obtain a K-theoretic relative/orbifold correspondence in genus $0$ .
我们考虑根构造的 K 理论 Gromov-Witten 理论。我们计算了一些0元属根球的K理论格罗莫夫-维滕不变式。我们还得到了 K 理论中 0$ 属的相对/双折叠对应关系。
{"title":"A note on quantum K-theory of root constructions","authors":"Hsian-Hua Tseng","doi":"10.1017/s0017089524000089","DOIUrl":"https://doi.org/10.1017/s0017089524000089","url":null,"abstract":"We consider K-theoretic Gromov-Witten theory of root constructions. We calculate some genus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000089_inline1.png\" /> <jats:tex-math> $0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> K-theoretic Gromov-Witten invariants of a root gerbe. We also obtain a K-theoretic relative/orbifold correspondence in genus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000089_inline2.png\" /> <jats:tex-math> $0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140569822","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 : 2024-04-11DOI: 10.1017/s0017089524000119
Willian Tokura, Marcelo Barboza, Elismar Batista, Priscila Kai
In this paper, we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call almost quotient Yamabe solitons as they extend quite naturally those already called quotient Yamabe solitons. We present sufficient conditions for a compact almost quotient Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact almost gradient quotient Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.
{"title":"On almost quotient Yamabe solitons","authors":"Willian Tokura, Marcelo Barboza, Elismar Batista, Priscila Kai","doi":"10.1017/s0017089524000119","DOIUrl":"https://doi.org/10.1017/s0017089524000119","url":null,"abstract":"In this paper, we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call almost quotient Yamabe solitons as they extend quite naturally those already called quotient Yamabe solitons. We present sufficient conditions for a compact almost quotient Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact almost gradient quotient Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570017","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 : 2024-04-11DOI: 10.1017/s0017089524000090
Ahmed Abouelsaad
In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to $mathrm{Bir}(mathbb{P}^2)$ . We prove that if the Galois group has order at most $3$ , it always extends to a subgroup of the Jonquières group associated with the point $P$ . Conversely, with a degree of at least $4$ , we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonquières maps with respect to $P$ . In addition, we also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.
{"title":"Galois points and Cremona transformations","authors":"Ahmed Abouelsaad","doi":"10.1017/s0017089524000090","DOIUrl":"https://doi.org/10.1017/s0017089524000090","url":null,"abstract":"In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline1.png\" /> <jats:tex-math> $mathrm{Bir}(mathbb{P}^2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove that if the Galois group has order at most <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline2.png\" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, it always extends to a subgroup of the Jonquières group associated with the point <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline3.png\" /> <jats:tex-math> $P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Conversely, with a degree of at least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline4.png\" /> <jats:tex-math> $4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonquières maps with respect to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline5.png\" /> <jats:tex-math> $P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In addition, we also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140569880","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 : 2024-03-21DOI: 10.1017/s0017089524000053
Rafael Francisco Ochoa De La Cruz, Martin Ortíz Morales, Valente Santiago Vargas
In this paper, we prove that the lower triangular matrix category $Lambda =left [ begin{smallmatrix} mathcal{T}&0 M&mathcal{U} end{smallmatrix} right ]$ , where $mathcal{T}$ and $mathcal{U}$ are $textrm{Hom}$ -finite, Krull–Schmidt $K$ -quasi-hereditary categories and $M$ is an $mathcal{U}otimes _K mathcal{T}^{op}$ -module that satisfies suitable conditions, is quasi-hereditary. This result generalizes the work of B. Zhu in his study on triangular matrix algebras over quasi-hereditary algebras. Moreover, we obtain a characterization of the category of the $_Lambda Delta$ -filtered $Lambda$ -modules.
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