Ru, Min2018-11-302018-11-30August 2012016-08August 201http://hdl.handle.net/10657/3550In this dissertation, we describe a paper that improves on the conditions that imply holomorphic curves and integral points are degenerate or not Zariski-dense. Specifically, we show that for a holomorphic curve into a projective variety of dimension n intersecting q divisors in subgeneral position whose sum is equidegreelizable, if q is greater than or equal to n 2 , then the curve is degenerate. This is an improvement from 2n 2 under the same conditions in paper. To achieve this result, we borrow methods from that combine divisors in pairs and uses a joint filtration result from linear algebra. Lastly, a pointwise filtration approach, first considered by Corvaja, Levin, and Zannier, is used to give further improvements such that if q is greater than or equal to n 2 − n, then the curve is degenerate. This pointwise filtration may be constructed by using linear algebra on the power series locally representing the sections.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. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Nevanlinna theoryDiophantine approximationDefect relationEquidegreeSubgeneral position2018-11-30Thesisborn digital