Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O objetivo deste trabalho à desenvolver a teoria bÃsica de esquemas e mostrar que duas variedades projetivas birracionalmente equivalentes e nÃo-singulares sobre um corpo algebricamente fechado possuem um mesmo gÃnero geomÃtrico. Um resultado relacionado permite determinar se uma hipersuperfÃcie nÃo-singular de grau d em um espaÃo projetivo Pn à uma variedade nÃo-racional. / This work aims to develop basic scheme theory and show that two projective, non-singular and birationally equivalent varieties over an algebraically closed field have same geometric genus. A related result allows to check whether a non-singular hipersurface of degree d in a projective space Pn is a non-rational variety.

This thesis consists of six papers. In Paper I, we give an algorithm for merging sorted lists of monomials and together with a projection technique, we obtain a new complexity bound for the Buchberger-Möller algorithm and the FGLM algorithm. In Paper II, we discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give complexity bounds. As an application we drastically improve the computational algebra approach to the reverse engineering of gene regulatory networks. In Paper III, we introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero. In Paper IV, we consider a subset of projective space over a finite field and give a geometric description of the minimal degree of a non-vanishing form with respect to this subset. We also give bounds on the minimal degree in terms of the cardinality of the subset. In Paper V, we study an associative version of an algorithm constructed to compute the Hilbert series for graded Lie algebras. In the commutative case we use Gotzmann's persistence theorem to show that the algorithm terminates in finite time. In Paper VI, we connect the commutative version of the algorithm in Paper V with the Buchberger algorithm. / At the time of doctoral defence, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript. Paper 5: Manuscript. Paper 6: Manuscript