Tilt Stability and Wall-Crossings for Curves in P^3 and Cone of Divisors of Curves with High Genera

In this thesis we study curves in P3. Utilizing the techniques in Bridgeland stability, conceived by Tom Bridgeland, we describe the wall and chamber structure of the tilt stability manifold for curves in P3 of arbitrary degrees and genera subject to a constraint. This result is a generalization of the results on twisted cubics by Ben Schmidt, and elliptic quartics by Gallardo, Huerta, and Schmidt. We also identify the extension classes that lie on the walls, providing geometric descriptions of the moduli space of such curves. Using the main theorem, we also provide a new example in which we completely determine all of the walls and extension classes in tilt stability for general curves of degree 3 and genus 6 in P3. Furthermore, in the case of plane curves in P3 we identify the bases that span the nef cone and the pseudoeffective cone of divisors for their Hilbert schemes. For the special case of the Hilbert schme of plane cubics, we also provide a partial description of the base locus decomposition of the cone of divisors.