UC Berkeley

This thesis is devoted to the study of two central objects in algebraic geometry - algebraic curves and vector bundles - through the unifying lens of liaison theory. The liaison theory of curves originated in the work of M. Noether in the late XIX century, and has since become an instrumental tool in the classification of space curves and the study of Hilbert schemes in general. In Chapter I, we develop a biliaison theory of sheaves and prove several main results in analogy to those from the liaison theory of codimension two subvarieties. In Chapter II, we use the liaison of curves on surfaces with ordinary singularities to prove multiple results about homological invariants of general projections of curves into projective three-space. In Chapter III, we apply biliaison of sheaves to the study of vector bundles. A general program is outlined to describe the moduli of bundles on a projective variety of irregularity zero through a stratification approach. We take the first steps by classifying bundles in the biliaison class of the zero sheaf on projective spaces and describing the moduli. In particular, our results give a description of the moduli of bundles on the projective plane.