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.
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.
