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Brill--Noether theory studies the different projective embeddings that an algebraic curve admits. For a curve with a given projective embedding, we study the question of what other projective embeddings the curve can admit. Our techniques use curves on K3 surfaces. Lazarsfeld's proof of the Gieseker--Petri theorem solidified the role of K3 surfaces in the Brill--Noether theory of curves. In this thesis, we further the study of the Brill--Noether theory of curves on K3 surfaces.
We prove results concerning lifting line bundles from curves to K3 surfaces. Via an analysis of the stability of Lazarsfeld--Mukai bundles, we deduce a bounded version of a conjecture of Donagi--Morrison concerning when a Brill--Noether special line bundle of rank 3 on a curve on a polarized K3 surface lifts to a line bundle on the K3 surface. In joint work with Asher Auel, we also present a strategy for distinguishing Brill--Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill--Noether loci. Using our new lifting results, we prove cases of the maximal Brill--Noether loci conjecture. We also investigate the Brill--Noether theory of K3 surfaces and verify cases of a conjecture of Knutsen and Mukai concerning the Picard groups of K3 surfaces with Brill--Noether special curves.
Haburcak, Richard, "Brill--Noether theory via K3 surfaces" (2023). Dartmouth College Ph.D Dissertations. 197.