Ru, Min2019-09-142019-09-14May 20172017-05May 2017Portions of this document appear in: C. Mills and M. Ru. An Improved Defect Relation for Holomorphic Curvesin Projective Varieties. Complex Analysis and Dynamical Systems VII, theIsrael Mathematics Conference Proceedings (IMCP), Contemporary Math-ematics (AMS)https://hdl.handle.net/10657/4601In this dissertation we improve Min Ru's defect relation (as well as the Second Main Theorem) for holomorphic curves $f: {\Bbb C}\rightarrow X$ intersecting $D:=D_1+\cdots +D_q$, where $D$ is a divisor of equi-degree, and $D_1, \dots, D_q$ are big, nef, and have no components in common. Our results will decrease the number of divisors $D_i$ that $f$ is needed to omit in order to conclude that $f$ is degenerate. The corresponding arithmetic results are also obtained.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Holomorphic curvesProjectiveVariety2019-09-14Thesisborn digital