LEADER 03813nam 22005175 450 001 9910896187803321 005 20241003131210.0 010 $a9783031688584 010 $a3031688589 024 7 $a10.1007/978-3-031-68858-4 035 $a(MiAaPQ)EBC31702411 035 $a(Au-PeEL)EBL31702411 035 $a(CKB)36271346800041 035 $a(DE-He213)978-3-031-68858-4 035 $a(OCoLC)1460009920 035 $a(EXLCZ)9936271346800041 100 $a20241003d2024 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aGröbner's Problem and the Geometry of GT-Varieties /$fby Liena Colarte-Gómez, Rosa Maria Miró-Roig 205 $a1st ed. 2024. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2024. 215 $a1 online resource (161 pages) 225 1 $aRSME Springer Series,$x2509-8896 ;$v15 311 08$a9783031688577 311 08$a3031688570 327 $a- Introduction -- Algebraic Preliminaries -- Invariants of finite abelian groups and aCM projections of Veronese varieties. Applications -- The geometry of ??varieties -- Invariants of finite groups and the weak Lefschetz property -- Normal bundle of RL-varieties. 330 $aThis book presents progress on two open problems within the framework of algebraic geometry and commutative algebra: Gröbner's problem regarding the arithmetic Cohen-Macaulayness (aCM) of projections of Veronese varieties, and the problem of determining the structure of the algebra of invariants of finite groups. We endeavour to understand their unexpected connection with the weak Lefschetz properties (WLPs) of artinian ideals. In 1967, Gröbner showed that the Veronese variety is aCM and exhibited examples of aCM and nonaCM monomial projections. Motivated by this fact, he posed the problem of determining whether a monomial projection is aCM. In this book, we provide a comprehensive state of the art of Gröbner?s problem and we contribute to this question with families of monomial projections parameterized by invariants of a finite abelian group called G-varieties. We present a new point of view in the study of Gröbner?s problem, relating it to the WLP of Artinian ideals. GT varieties are a subclass of G varieties parameterized by invariants generating an Artinian ideal failing the WLP, called the Galois-Togliatti system. We studied the geometry of the G-varieties; we compute their Hilbert functions, a minimal set of generators of their homogeneous ideals, and the canonical module of their homogeneous coordinate rings to describe their minimal free resolutions. We also investigate the invariance of nonabelian finite groups to stress the link between projections of Veronese surfaces, the invariant theory of finite groups and the WLP. Finally, we introduce a family of smooth rational monomial projections related to G-varieties called RL-varieties. We study the geometry of this family of nonaCM monomial projections and we compute the dimension of the cohomology of the normal bundle of RL varieties. This book is intended to introduce Gröbner?s problem to young researchers and provide new points of view and directions for further investigations. 410 0$aRSME Springer Series,$x2509-8896 ;$v15 606 $aGeometry, Algebraic 606 $aAlgebraic Geometry 615 0$aGeometry, Algebraic. 615 14$aAlgebraic Geometry. 676 $a516.35 700 $aColarte-Gómez$b Liena$01767339 701 $aMiró-Roig$b Rosa Maria$067000 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910896187803321 996 $aGröbner's Problem and the Geometry of GT-Varieties$94212373 997 $aUNINA