Gröbner's Problem and the Geometry of Gt-Varieties
暫譯: Gröbner 問題與 Gt-變體的幾何學

Colarte-Gómez, Liena, Miró-Roig, Rosa Maria

Gröbner's Problem and the Geometry of Gt-Varieties
  • 出版商: Springer
  • 出版日期: 2024-10-03
  • 售價: $4,890
  • 貴賓價: 9.5 折 $4,646
  • 語言: 英文
  • 頁數: 154
  • 裝訂: Hardcover - also called cloth, retail trade, or trade
  • ISBN: 3031688570
  • ISBN-13: 9783031688577

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商品描述

This book presents progress on two open problems within the framework of algebraic geometry and commutative algebra: Gröbner's problem regarding the arithmetic Cohen-Macaulayness (aCM) of projections of Veronese varieties, and the problem of determining the structure of the algebra of invariants of finite groups. We endeavour to understand their unexpected connection with the weak Lefschetz properties (WLPs) of artinian ideals. In 1967, Gröbner showed that the Veronese variety is aCM and exhibited examples of aCM and nonaCM monomial projections. Motivated by this fact, he posed the problem of determining whether a monomial projection is aCM. In this book, we provide a comprehensive state of the art of Gröbner's problem and we contribute to this question with families of monomial projections parameterized by invariants of a finite abelian group called G-varieties. We present a new point of view in the study of Gröbner's problem, relating it to the WLP of Artinian ideals. GT varieties are a subclass of G varieties parameterized by invariants generating an Artinian ideal failing the WLP, called the Galois-Togliatti system. We studied the geometry of the G-varieties; we compute their Hilbert functions, a minimal set of generators of their homogeneous ideals, and the canonical module of their homogeneous coordinate rings to describe their minimal free resolutions. We also investigate the invariance of nonabelian finite groups to stress the link between projections of Veronese surfaces, the invariant theory of finite groups and the WLP. Finally, we introduce a family of smooth rational monomial projections related to G-varieties called RL-varieties. We study the geometry of this family of nonaCM monomial projections and we compute the dimension of the cohomology of the normal bundle of RL varieties. This book is intended to introduce Gröbner's problem to young researchers and provide new points of view and directions for further investigations.

商品描述(中文翻譯)

本書介紹了代數幾何和交換代數框架內的兩個開放問題的進展：Gröbner 關於 Veronese 多樣體投影的算術 Cohen-Macaulay 性質（aCM）問題，以及確定有限群不變代數結構的問題。我們努力理解它們與阿廷理想的弱 Lefschetz 性質（WLP）之間的意外聯繫。1967 年，Gröbner 顯示 Veronese 多樣體是 aCM，並展示了 aCM 和非 aCM 單項式投影的例子。受到這一事實的啟發，他提出了確定單項式投影是否為 aCM 的問題。在本書中，我們提供了 Gröbner 問題的全面現狀，並通過由有限阿貝爾群的不變量參數化的單項式投影族（稱為 G-多樣體）對這一問題作出貢獻。我們在研究 Gröbner 問題時提出了一種新的觀點，將其與阿廷理想的 WLP 相關聯。GT 多樣體是 G 多樣體的一個子類，這些多樣體由生成失敗 WLP 的阿廷理想的不變量參數化，稱為 Galois-Togliatti 系統。我們研究了 G-多樣體的幾何，計算了它們的 Hilbert 函數、同類理想的最小生成集，以及它們的同類坐標環的典範模，以描述它們的最小自由分解。我們還調查了非阿貝爾有限群的不變性，以強調 Veronese 表面的投影、有限群的不變理論和 WLP 之間的聯繫。最後，我們介紹了一個與 G-多樣體相關的平滑有理單項式投影族，稱為 RL-多樣體。我們研究了這個非 aCM 單項式投影族的幾何，並計算了 RL 多樣體的正常束的同調維度。本書旨在向年輕研究人員介紹 Gröbner 問題，並提供新的觀點和進一步研究的方向。

作者簡介

Liena Colarte-Gómez is an assistant professor at the Institute of Mathematics of the Polish Academy of Science. Her expertise comprises syzygies of projective varieties, tensor rank, and tensor rank decomposition. Liena obtained her Ph. D. and Master's in Mathematics from the University of Barcelona. She has published several articles in high-quality journals. Under international competition, she has been awarded two visiting research fellowships from ACRI and Ferran Sunyer i Balaguer foundations at the University of Genoa and the University of Napoli Federico II.

Rosa M. Miró-Roig is professoer at the University of Barcelona working in Algebraic Geometry and Comutative Algebra. She has published more than 159 articles; this includes papersin the Advances in Mathematics, Trans AMS, Mathematische Annalen, Journal für die reine und angewandte Mathematik, Compositio Mathematica and Memoirs of the AMS. She has been advisor of 12 Ph. D. Students and mentored many postdocs. She has been Managing Editor of Collectanea Mathematica (2005- 2021), and she is Associated editor of Beiträge zur Algebra und Geometrie, Journal of Commutative Algebra, Mathematics and Vietnam Journal of Mathematics. In 2007 she won the Ferran Sunyer i Balaguer prize, in 2023 she has been nominated EMS Distinguished Speaker 2023 and this year she got Premi Rosa Argelaguet i Isanta prize.

作者簡介(中文翻譯)

Liena Colarte-Gómez 是波蘭科學院數學研究所的助理教授。她的專長包括射影變數的 syzygies、張量秩和張量秩分解。Liena 在巴塞隆納大學獲得數學博士和碩士學位。她在高品質期刊上發表了多篇文章。在國際競爭中，她獲得了來自 ACRI 和 Ferran Sunyer i Balaguer 基金會的兩個訪問研究獎學金，分別在熱那亞大學和那不勒斯 Federico II 大學進行研究。

Rosa M. Miró-Roig 是巴塞隆納大學的教授，專注於代數幾何和交換代數。她已發表超過 159 篇文章，包括在《Advances in Mathematics》、《Trans AMS》、《Mathematische Annalen》、《Journal für die reine und angewandte Mathematik》、《Compositio Mathematica》和《Memoirs of the AMS》上的論文。她曾指導 12 位博士生，並輔導過許多博士後研究人員。她曾擔任《Collectanea Mathematica》的主編（2005-2021），並且是《Beiträge zur Algebra und Geometrie》、《Journal of Commutative Algebra》、《Mathematics》和《Vietnam Journal of Mathematics》的副編輯。2007 年，她獲得了 Ferran Sunyer i Balaguer 獎，2023 年被提名為 EMS 傑出演講者，並在今年獲得了 Premi Rosa Argelaguet i Isanta 獎。

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