The Dimension of Linear Systems on Weighted Projective Varieties.

Weighted projective varieties generalize classical projective geometry by introducing non-uniform scaling. This thesis investigates the dimension of linear systems on such varieties, with a focus on weight reduction techniques and the Veronese embedding. We explore the algebraic and geometric structures arising from weighted polynomial rings, demonstrating how weight reduction leads to well-formed spaces that simplify calculations without altering the underlying variety. A central result provides an explicit formula for the dimension of spaces of weighted homogeneous polynomials, the linear system, of a given degree — a generalization of the classical projective case. We implement this formula computationally, developing a Python-based tool set for practical computations on weighted projective varieties. The thesis culminates in applications to linear systems, illustrating how the theoretical framework connects to concrete examples and embeddings.