Numerical Restrictions on Seshadri Curves, with Applications to P^1 x P^1
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We study r-point Seshadri constants on smooth surfaces. The Seshadri constant is bounded above by the Nagata bound, and in cases where it is not maximal this fact is witnessed by a set of curves which we call Nagata curves. In Chapter 2, we study the properties of Nagata curves, especially the set of necessary conditions on the numerical class of a Nagata curve found in Theorem 2.6.2. In Chapter 3, we make a quick study of surfaces which are fibrations where the class of a fibre contains a Nagata curve. In Chapter 4, we develop tools to approximate Seshadri constants which allow us to improve upon known bounds in several situations. Finally, in Chapter 5, we make an in-depth study when the surface is P^1 x P^1. We compute Seshadri constants (for all values of r) for a class of line bundles, which we call outer bundles. The remaining line bundles, which we call inner bundles, are more mysterious, but we do show that their Seshadri constants are not always maximal. We then use our results about Seshadri constants for outer bundles to completely solve the symplectic packing problem for P^1 x P}^1. We end with some particular calculations for the line bundle L = (1,1).