東華大學圖書館 |

語系: 繁體中文

說明(常見問題)

回圖書館首頁

手機版館藏查詢

登入

回首頁

切換: 標籤 | MARC模式 | ISBD

Gröbner's problem and the geometry ...

Colarte-Gómez, Liena.

FindBook

Google Book

Amazon

博客來

Gröbner's problem and the geometry of GT-varieties

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Gröbner's problem and the geometry of GT-varieties/ by Liena Colarte-Gómez, Rosa Maria Miró-Roig.
作者:	Colarte-Gómez, Liena.
其他作者:	Miró-Roig, Rosa M.
出版者:	Cham :Springer Nature Switzerland : : 2024.,
面頁冊數:	xii, 154 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
標題:	Geometry, Algebraic. -
電子資源:	https://doi.org/10.1007/978-3-031-68858-4
ISBN:	9783031688584

Gröbner's problem and the geometry of GT-varieties
Colarte-Gómez, Liena.

Gröbner's problem and the geometry of GT-varieties[electronic resource] /by Liena Colarte-Gómez, Rosa Maria Miró-Roig. - Cham :Springer Nature Switzerland :2024. - xii, 154 p. :ill., digital ;24 cm. - RSME Springer series,v. 152509-8896 ;. - RSME Springer series ;v. 15..

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.

ISBN: 9783031688584

Standard No.: 10.1007/978-3-031-68858-4doiSubjects--Topical Terms:

532048
Geometry, Algebraic.

LC Class. No.: QA564

Dewey Class. No.: 516.35

Gröbner's problem and the geometry of GT-varieties
LDR:03022nmm a2200325 a 4500 001 2375318
003 DE-He213
005 20241003131210.0
006 m d
007 cr nn 008maaau
008 241231s2024 sz s 0 eng d
020 $a 9783031688584 $q (electronic bk.)
020 $a 9783031688577 $q (paper)
024 7 $a 10.1007/978-3-031-68858-4 $2 doi
035 $a 978-3-031-68858-4
040 $a GP $c GP
041 0 $a eng
050 4 $a QA564
072 7 $a PBMW $2 bicssc
072 7 $a MAT012010 $2 bisacsh
072 7 $a PBMW $2 thema
082 0 4 $a 516.35 $2 23
090 $a QA564 $b .C683 2024
100 1 $a Colarte-Gómez, Liena. $3 3724761
245 1 0 $a Gröbner's problem and the geometry of GT-varieties $h [electronic resource] / $c by Liena Colarte-Gómez, Rosa Maria Miró-Roig.
260 $a Cham : $b Springer Nature Switzerland : $b Imprint: Springer, $c 2024.
300 $a xii, 154 p. : $b ill., digital ; $c 24 cm.
490 1 $a RSME Springer series, $x 2509-8896 ; $v v. 15
520 $a 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.
650 0 $a Geometry, Algebraic. $3 532048
650 0 $a Commutative algebra. $3 604140
650 1 4 $a Algebraic Geometry. $3 893861
700 1 $a Miró-Roig, Rosa M. $3 3724762
710 2 $a SpringerLink (Online service) $3 836513
773 0 $t Springer Nature eBook
830 0 $a RSME Springer series ; $v v. 15. $3 3724763
856 4 0 $u https://doi.org/10.1007/978-3-031-68858-4
950 $a Mathematics and Statistics (SpringerNature-11649)