The present thesis consists of two parts. The first part of the thesis is related to the study of unprojection theory and the construction of three codimension 6 families of Fano 3-folds anticanonically embedded in weighted projective space. The second part concerns the study of the Lefschetz and anisotropy properties of the Stanley-Reisner ring of simplicial spheres. In Chapter 2, we introduce some preliminary notions and known results from Commutative Algebra, Algebraic Geometry and

more »

... ial Algebra, with a particular emphasis to Gorenstein rings, Fano 3-folds and simplicial complexes. In Chapter 3, we recall some existing results related to unprojection theory. Unprojection theory, which is due to Miles Reid, uses ideas from birational geometry to construct more complicated commutative rings starting from simpler data. It is our main tool for the geometric applications. In Chapter 4, we develop a new parallel unprojection format, for which we give the name Tom & Jerry triples format. We use the format to prove, starting from codimension 3, the existence of two codimension 6 families of Fano 3-folds. In Chapter 5, we develop a second parallel unprojection format, which we call the 4-intersection format. We use the format to prove, starting from codimension 2, the existence of a codimension 6 family of Fano 3-folds. In Chapter 6, which is joint work with Stavros Argyrios Papadakis, we introduce the notion of generic anisotropy of a simplicial sphere. We prove that a simplicial sphere is generically anisotropic over any field of characteristic 2, and that a 1-dimensional simplicial sphere is generically anisotropic over any field. As an application, we give a second proof of McMullen's g-conjecture for simplicial spheres.