03813nam 22005175 450 991089618780332120241003131210.09783031688584303168858910.1007/978-3-031-68858-4(MiAaPQ)EBC31702411(Au-PeEL)EBL31702411(CKB)36271346800041(DE-He213)978-3-031-68858-4(OCoLC)1460009920(EXLCZ)993627134680004120241003d2024 u| 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierGröbner's Problem and the Geometry of GT-Varieties /by Liena Colarte-Gómez, Rosa Maria Miró-Roig1st ed. 2024.Cham :Springer Nature Switzerland :Imprint: Springer,2024.1 online resource (161 pages)RSME Springer Series,2509-8896 ;159783031688577 3031688570 - Introduction -- Algebraic Preliminaries -- Invariants of finite abelian groups and aCM projections of Veronese varieties. Applications -- The geometry of 𝑮−varieties -- Invariants of finite groups and the weak Lefschetz property -- Normal bundle of RL-varieties.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.RSME Springer Series,2509-8896 ;15Geometry, AlgebraicAlgebraic GeometryGeometry, Algebraic.Algebraic Geometry.516.35Colarte-Gómez Liena1767339Miró-Roig Rosa Maria67000MiAaPQMiAaPQMiAaPQBOOK9910896187803321Gröbner's Problem and the Geometry of GT-Varieties4212373UNINA02447nas 2200541-a 450 991089158190332120210823213022.0(OCoLC)753956232(CKB)2550000000054892(CONSER)--2011203357(DE-599)ZDB2668433-0(EXLCZ)99255000000005489220110921a20099999 --- aengur|||||||||||txtrdacontentcrdamediacrrdacarrierJournal of the Chicago Colloquium on Digital Humanities and Computer ScienceChicago, IL Division of the Humanities, University of Chicago1 online resourceRefereed/Peer-reviewedRefereed/Peer-reviewed2163-8284 Peer reviewed articles drawn from paper and poster presentations given at the annual Chicago Colloquium on Digital Humanities and Computer Science (DHCS). The goal of the colloquium is to bring together researchers and scholars in the humanities and computer science to examine the current state of digital humanities as a field of intellectual inquiry and to identify and explore new directions and perspectives for future research.Proceedings of the Chicago Colloquium on Digital Humanities and Computer ScienceJDHCSChicago DHCS Colloquium proceedingsJournal of the Chicago Colloquium on Digital Humanities and Computer ScienceDigital humanitiesCongressesDigital humanitiesResearchCongressesHumanitiesElectronic information resourcesCongressesComputer scienceComputer sciencefast(OCoLC)fst00872451Digital humanitiesfast(OCoLC)fst00963599Conference papers and proceedings.fastDigital humanitiesDigital humanitiesResearchHumanitiesElectronic information resourcesComputer science.Computer science.Digital humanities.University of Chicago.Division of the Humanities.Chicago Colloquium on Digital Humanities and Computer Science.JOURNAL9910891581903321Journal of the Chicago Colloquium on Digital Humanities and Computer Science4261080UNINA