Let $C subset mathbb{P}^3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic zero. For a line $l$, we consider the projection $pi_lcolon C rightarrow mathbb{P}^1$ with center $l$ and the extension of the function fields $pi^_lcolon k(mathbb{P}^1) hookrightarrow k(C)$. A line $l$ is referred to as a cyclic line if the extension $k(C)/pi_l^(k(mathbb{P}^1))$ is cyclic. A line $l subset mathbb{P}^3$ is said to be skew if $C cap l = emptyset$. We prove that the number of skew cyclic lines is equal to $0,1,3$ or $9$. We determine curves that have nine skew cyclic lines.
设$C subset mathbb{P}^3$为特征为零的代数闭域$k$上的属$4$的正则曲线。对于直线$l$，我们考虑以$l$为中心的投影$pi_lcolon C rightarrow mathbb{P}^1$和函数域$pi^_lcolon k(mathbb{P}^1) hookrightarrow k(C)$的扩展。如果扩展$k(C)/pi_l^(k(mathbb{P}^1))$是循环的，则将行$l$称为循环线。如果是$C cap l = emptyset$，就说一条线$l subset mathbb{P}^3$是倾斜的。证明了歪斜环状线的个数等于$0,1,3$或$9$。我们确定有九条歪斜循环线的曲线。
{"title":"Galois lines for a canonical curve of genus 4, II: Skew cyclic lines","authors":"Jiryo Komeda, Takeshi Takahashi","doi":"10.4171/rsmup/141","DOIUrl":"https://doi.org/10.4171/rsmup/141","url":null,"abstract":"Let $C subset mathbb{P}^3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic zero. For a line $l$, we consider the projection $pi_lcolon C rightarrow mathbb{P}^1$ with center $l$ and the extension of the function fields $pi^_lcolon k(mathbb{P}^1) hookrightarrow k(C)$. A line $l$ is referred to as a cyclic line if the extension $k(C)/pi_l^(k(mathbb{P}^1))$ is cyclic. A line $l subset mathbb{P}^3$ is said to be skew if $C cap l = emptyset$. We prove that the number of skew cyclic lines is equal to $0,1,3$ or $9$. We determine curves that have nine skew cyclic lines.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135829630","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}
A natural place to study the Chow ring of the classifying space BG, for G a linear algebraic group, is Voevodsky's triangulated category of motives, inside which Morel and Voevodsky, and Totaro have defined motives M(BG) and M^c(BG), respectively. We show that, for any group G of order p^3 over a field of characteristic not p which contains a primitive p^2-th root of unity, the motive M(BG) is a mixed Tate motive. We also show that, for a finite group G over a field of characteristic zero, M(BG) is a mixed Tate motive if and only M^c(BG) is a mixed Tate motive.
{"title":"Groups of order $p^3$ are mixed Tate","authors":"Tudor Pădurariu","doi":"10.4171/rsmup/132","DOIUrl":"https://doi.org/10.4171/rsmup/132","url":null,"abstract":"A natural place to study the Chow ring of the classifying space BG, for G a linear algebraic group, is Voevodsky's triangulated category of motives, inside which Morel and Voevodsky, and Totaro have defined motives M(BG) and M^c(BG), respectively. We show that, for any group G of order p^3 over a field of characteristic not p which contains a primitive p^2-th root of unity, the motive M(BG) is a mixed Tate motive. We also show that, for a finite group G over a field of characteristic zero, M(BG) is a mixed Tate motive if and only M^c(BG) is a mixed Tate motive.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135939153","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 𝑅 be a commutative noetherian ring. Denote by mod 𝑅 the category of finitely generated 𝑅 -modules. In the present paper, we introduce the notion of solid subcategories of mod 𝑅 and investigate it. The main result of this paper not only recovers results of Schoutens, Krause and Stevenson, and Takahashi on thick subcategories, but also unifies and extends them to solid subcategories. Moreover, it provides some contributions to the study of the question asking when a thick subcategory is Serre.
{"title":"Solid generators in module categories and applications","authors":"Ryo Takahashi","doi":"10.4171/rsmup/138","DOIUrl":"https://doi.org/10.4171/rsmup/138","url":null,"abstract":"– Let 𝑅 be a commutative noetherian ring. Denote by mod 𝑅 the category of finitely generated 𝑅 -modules. In the present paper, we introduce the notion of solid subcategories of mod 𝑅 and investigate it. The main result of this paper not only recovers results of Schoutens, Krause and Stevenson, and Takahashi on thick subcategories, but also unifies and extends them to solid subcategories. Moreover, it provides some contributions to the study of the question asking when a thick subcategory is Serre.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135981901","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 prove that, assuming Vopvenka's principle, every small projectivity class in a locally presentable category is accessible.
{"title":"Fibrantly generated weak factorization systems","authors":"Sean Cox, Jiří Rosický","doi":"10.4171/rsmup/135","DOIUrl":"https://doi.org/10.4171/rsmup/135","url":null,"abstract":"We prove that, assuming Vopvenka's principle, every small projectivity class in a locally presentable category is accessible.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135983534","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}
Phùng Hô Hai, João Pedro dos Santos, Pham Thanh Tâm
This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over $C((x))$ and $mathbb P^1_Csmallsetminus{0,infty}$, where $C$ is algebraically closed and of characteristic zero. It aims at reading the existing classification results as an equivalence between regular-singular systems and representations of the group $mathbb Z$. In the second part, we deal with regular-singular connections over $R((x))$ and $mathbb P_R^1smallsetminus{0,infty}$, where $R=C[[t_1,ldots,t_r]]/I$. The picture we offer shows that regular-singular connections are equivalent to representations of $mathbb Z$, now over $R$.
{"title":"Algebraic theory of formal regular-singular connections with parameters","authors":"Phùng Hô Hai, João Pedro dos Santos, Pham Thanh Tâm","doi":"10.4171/rsmup/134","DOIUrl":"https://doi.org/10.4171/rsmup/134","url":null,"abstract":"This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over $C((x))$ and $mathbb P^1_Csmallsetminus{0,infty}$, where $C$ is algebraically closed and of characteristic zero. It aims at reading the existing classification results as an equivalence between regular-singular systems and representations of the group $mathbb Z$. In the second part, we deal with regular-singular connections over $R((x))$ and $mathbb P_R^1smallsetminus{0,infty}$, where $R=C[[t_1,ldots,t_r]]/I$. The picture we offer shows that regular-singular connections are equivalent to representations of $mathbb Z$, now over $R$.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135938449","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 $b ge 2$ and $ell ge 1$ be integers. We establish that there is an absolute real number $K$ such that all the partial quotients of the rational number $$ prod_{h = 0}^ell , (1 - b^{-2^h}), $$ of denominator $b^{2^{ell+1} - 1}$, do not exceed $exp(K (log b)^2 sqrt{ell} 2^{ell/2})$.
设$b ge 2$和$ell ge 1$为整数。我们建立了一个绝对实数$K$，使得分母$b^{2^{ell+1} - 1}$的有理数$$ prod_{h = 0}^ell , (1 - b^{-2^h}), $$的所有偏商都不超过$exp(K (log b)^2 sqrt{ell} 2^{ell/2})$。
{"title":"On the rational approximation to Thue–Morse rational numbers","authors":"Yann Bugeaud, Guo-Niu Han","doi":"10.4171/rsmup/133","DOIUrl":"https://doi.org/10.4171/rsmup/133","url":null,"abstract":"Let $b ge 2$ and $ell ge 1$ be integers. We establish that there is an absolute real number $K$ such that all the partial quotients of the rational number $$ prod_{h = 0}^ell , (1 - b^{-2^h}), $$ of denominator $b^{2^{ell+1} - 1}$, do not exceed $exp(K (log b)^2 sqrt{ell} 2^{ell/2})$.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135938452","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":"$G$-equivariance of formal models of flag varieties","authors":"Andrés Sarrazola Alzate","doi":"10.4171/rsmup/104","DOIUrl":"https://doi.org/10.4171/rsmup/104","url":null,"abstract":"","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80001229","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 prove that if 𝐺 is a finite group, then the 𝑘 -th term of the lower central series is nilpotent if and only if for every 𝛾 𝑘 -values 𝑥, 𝑦 ∈ 𝐺 with coprime orders, either 𝜋 ( 𝑜 ( 𝑥 ) 𝑜 ( 𝑦 )) ⊆ 𝜋 ( 𝑜 ( 𝑥𝑦 )) or 𝑜 ( 𝑥 ) 𝑜 ( 𝑦 ) ≤ 𝑜 ( 𝑥𝑦 ) . We obtain an analogous version for the derived series of finite solvable groups, but replacing 𝛾 𝑘 -values by 𝛿 𝑘 -values. We will also discuss the existence of normal Sylow subgroups in the derived subgroup in terms of the order of the product of certain elements.
{"title":"Orders of products of elements and nilpotency of terms in the lower central series and the derived series","authors":"J. Martínez","doi":"10.4171/rsmup/130","DOIUrl":"https://doi.org/10.4171/rsmup/130","url":null,"abstract":"– In this paper we prove that if 𝐺 is a finite group, then the 𝑘 -th term of the lower central series is nilpotent if and only if for every 𝛾 𝑘 -values 𝑥, 𝑦 ∈ 𝐺 with coprime orders, either 𝜋 ( 𝑜 ( 𝑥 ) 𝑜 ( 𝑦 )) ⊆ 𝜋 ( 𝑜 ( 𝑥𝑦 )) or 𝑜 ( 𝑥 ) 𝑜 ( 𝑦 ) ≤ 𝑜 ( 𝑥𝑦 ) . We obtain an analogous version for the derived series of finite solvable groups, but replacing 𝛾 𝑘 -values by 𝛿 𝑘 -values. We will also discuss the existence of normal Sylow subgroups in the derived subgroup in terms of the order of the product of certain elements.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75469623","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}
This paper deals with the isoperimetric problem for clusters in a Euclidean space with double density. In particular, we show that a limit of an isoperimetric minimizing sequence of clusters with volumes V is always isoperimetric for its own volumes (which may be smaller than V). In particular, if it is strictly smaller, we provide an explicit formula. Mathematics Subject Classification (2020). Primary: 49Q10. Secondary: 49Q20, 58B20
{"title":"A formula for the minimal perimeter of clusters with density","authors":"Vincenzo Scattaglia","doi":"10.4171/rsmup/120","DOIUrl":"https://doi.org/10.4171/rsmup/120","url":null,"abstract":"This paper deals with the isoperimetric problem for clusters in a Euclidean space with double density. In particular, we show that a limit of an isoperimetric minimizing sequence of clusters with volumes V is always isoperimetric for its own volumes (which may be smaller than V). In particular, if it is strictly smaller, we provide an explicit formula. Mathematics Subject Classification (2020). Primary: 49Q10. Secondary: 49Q20, 58B20","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"1984 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82234267","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}
Given a prime number p let Cp be the topological completion of the algebraic closure of the field of p-adic numbers. Let O(T ) be the Galois orbit of a transcendental element T of Cp with respect to the absolute Galois group. Our aim is to study the class of Galois equivariant functions defined on O(T ) with values in Cp. We show that each function from this class is continuous and we characterize the class of Lipschitz functions, respectively the class of differentiable functions, with respect to a new orthonormal basis. Then we discuss some aspects related to analytic continuation for the functions of this class. Mathematics Subject Classification (2020). Primary: 11S99; Secondary: 11S20, 11S80.
{"title":"Galois equivariant functions on Galois orbits in large $p$-adic fields","authors":"V. Alexandru, M. Vâjâitu","doi":"10.4171/rsmup/127","DOIUrl":"https://doi.org/10.4171/rsmup/127","url":null,"abstract":"Given a prime number p let Cp be the topological completion of the algebraic closure of the field of p-adic numbers. Let O(T ) be the Galois orbit of a transcendental element T of Cp with respect to the absolute Galois group. Our aim is to study the class of Galois equivariant functions defined on O(T ) with values in Cp. We show that each function from this class is continuous and we characterize the class of Lipschitz functions, respectively the class of differentiable functions, with respect to a new orthonormal basis. Then we discuss some aspects related to analytic continuation for the functions of this class. Mathematics Subject Classification (2020). Primary: 11S99; Secondary: 11S20, 11S80.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87401714","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：