Gröbner's problem and the geometry of GT-varieties
[eng] Within the framework of algebraic geometry and commutative algebra, this thesis makes advances in (1) the Gröbner's longstanding problem of determining whether a monomial projection of the Veronese variety Vn,d is an aCM variety; (2) it contributes to the fundamental problem of describing...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/180691 |
| Acceso en línea: | https://hdl.handle.net/2445/180691 http://hdl.handle.net/10803/672633 |
| Access Level: | acceso abierto |
| Palabra clave: | Àlgebra commutativa Geometria algebraica Mòduls de Cohen-Macaulay Commutative algebra Algebraic geometry Cohen-Macaulay modules |
| Sumario: | [eng] Within the framework of algebraic geometry and commutative algebra, this thesis makes advances in (1) the Gröbner's longstanding problem of determining whether a monomial projection of the Veronese variety Vn,d is an aCM variety; (2) it contributes to the fundamental problem of describing the internal structure of the ring of invariants of a finite subgroup of GL(n+1,K). Our approach towards these subjects involves combinatorics with an application to the Lefschetz properties of artinian ideals. The heart of this dissertation is expounded in four chapters with an introductory Chapter 1 collecting all the basic notions and results needed onwards; and an Appendix A containing two algorithms and implementations with the software Wolfram Mathematica. In Chapter 2, we treat Gröbner's problem and we study the invariants of the cyclic extension bG of a finite diagonal abelian group G in GL(n+1,K)$ of order d. We prove that the set B1 of monomial invariants of G of degree d minimally generates the ring RbG of invariants. We establish that B1 parameterizes an aCM monomial projection Xd of Vn,d, which we call a bG-variety with group G. They form a family of aCM monomial projections of Vn,d blending commutative algebra, algebraic geometry, combinatorics and the Lefschetz properties. In Chapter 2, we study the geometry of bG-varieties Xd with group G. We investigate their Hilbert function and series from the perspectives of invariant theory and combinatorics. We prove that their homogeneous ideals I(Xd) are generated by binomials of degree at most 3 and we exhibit examples reaching this bound. We identify the canonical module ωXd of Xd with an ideal I(relint(HA)) of RbG and we prove that it is generated by monomial of degree d and 2d. We characterize the Castelnuovo-Mumford regularity of Xd in terms of ωXd. In Chapter 3, we investigate the invariants of finite supgroups of SL(3,K) and we relate them to the weak Lefschetz property. We consider the cyclic extension of a representation in SL(n+1,K) of the dihedral group D2d of order 2d. We prove that its ring of invariants is minimally generated by a set of monomials and binomials of degree 2d which generates a non monomial GT-system with group D2d and parameterizes an aCM projection SD2d of V2,d. We describe a minimal graded free resolution of SD2d and we compute a minimal set of generators of I(SD2d) of degree 2. In Chapter 4, we introduce RL-varieties Xd: a family of smooth rational non aCM monomial projections of Vn,d related to bG-varieties Xd with group G. They are parameterized by a set of monomials of degree d determined by ωXd which defines an embedding of Pn. These properties allow us to describe their normal bundles NXd and to contribute to the classical problem of computing the dimension of the cohomology of the normal bundle of projective varieties. |
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