A General Defect Relation and Height Inequality for Divisors in Subgeneral Position



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In 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.



Nevanlinna theory, Diophantine approximation, Defect relation, Equidegree, Subgeneral position