Interpolation is a property of vector bundles on curves closely related to slope stability. The notion is motivated by the deformation theory of curves in projective space incident to given fixed subvarieties. If the normal bundle of a projective curve satisfies interpolation, then curves in the same component of the Hilbert scheme exhibit normal behavior with respect to incident problems.
We demonstrate how to use degeneration arguments to deduce interpolation. In particular, we show that a general connected space curve of degree d and genus g satisfies interpolation for d >= g+3 unless d = 5 and g = 2. As a second application, we show that a general elliptic curve of degree d in P^n satisfies a slightly weaker notion when d >= 7, d >= n+1, and the remainder of 2d modulo n-1 lies between 3 and n-2 inclusive. We also show that interpolation is equivalent to the---a priori stricter---notion of strong interpolation.
The use of degeneration techniques to prove interpolation requires working with modifications of vector bundles. In the second part of this thesis, we develop a general theory of modifications for bundles over varieties of arbitrary dimensions. We explain how to apply this machinery when dealing with families of curves, and prove a number of results which allow us to deduce interpolation via short exact sequences.
