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In this paper, we compute the genus of the variety of linear series of rank r and degree d on a general curve of genus g, with ramification at least α and β at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve. [ABSTRACT FROM AUTHOR]

Shows the existence of stable vector bundles of rank two with a preassigned number of sections. Theorems; Review of some technical results; Case of even degree and even number of sections.