Vector bundle techniques in the geometry of projective varieties

Abstract

This thesis discusses on various vector bundle techniqeus and their applications, particularly on Ulrich bundles on algebraic surfaces. Ulrich bundles have several different motivations from linear algebra, commutative algebra and projective geometry. Eisenbud and Schreyer made a comprehensive study on Ulrich bundles, and even suggested a conjecture that every projective scheme supports such bundles. The main results of this thesis are existence theorems on some rational surfaces and general Enriques surfaces. We also construct new surfaces which supports rank 2 Ulrich bundles. The main ingredient is the Lazarsfeld-Mukai bundle, which plays an essential role in Brill-Noether theory. Since rank 2 Ulrich bundles of suitable first Chern class on algebraic surfaces are Lazarsfeld-Mukai bundles, we investigate properties of Lazarsfeld-Mukai bundles on those surfaces and study the space of Lazarsfeld-Mukai bundles of suitable Chern classes. In particular, we find Brill-Noether type conditions on rational and Enriques surfaces so that surfaces satisfying these conditions must carry rank 2 Ulrich bundles. Under these conditions, we first estimate the dimension of the space of Lazarsfeld-Mukai bundles. Then we find the parametrization of the failure locus to be Ulrich. We conclude that Lazarsfeld-Mukai bundles which are not Ulrich cannot fill the whole space. We also make an extensive study on Brill-Noether type conditions for rational surfaces with an anticanonical pencil. A parameter count by Lelli-Chiesa implies the analogue of Harris-Mumford conjecture for rational surfaces with an anticanonical pencil, so that a minimal pencil of a curve lying on such surfaces extends to a line bundle on these surfaces. We construct some rational surfaces which satisfy those Brill-Noether conditions. On the other hand, we show the existence of Ulrich bundles on a general Enriques surface by constructing an Ulrich bundle on its K3 cover and taking push-forward. We show that a general Enriques surface of degree 4 carries a rank 4 Ulrich bundle from the geometry of Kummer surfaces. In contrast to the case of rational surfaces, Lazarsfeld-Mukai bundles on Enriques surfaces behave not very good. We use a sharper estimation and suggest a promising approach for the existence of Ulrich bundles via Lazarsfeld-Mukai bundle techniques. Finally, we introduce a few more applications of Ulrich bundles. The most important application is that these constructions of Ulrich bundles provide a systematic way to compute Cayley-Chow forms of those surfaces. Also we compute syzygies of some 0-cycles on curves and surfaces defined by sections of a twist of Ulrich bundles of rank $=$ dimension. We show that sections of suitable twist of Ulrich line bundles on curves correspond to the case of the general set of points in the viewpoint of the minimal resolution conjecture. On the other hand, syzygies of 0-cycles on surfaces do not coincide with the general set of points but still have an interesting symmetry.