A Zariski-Nagata Theorem for Smooth Toric Surfaces

The Affine Zariski-Nagata theorem is a classical result in commutative algebra that gives an expression for the nth symbolic power of a radical ideal I in a polynomial ring over a field in terms of the nth ordinary powers of the maximal ideals in [affine variety]max(I). In this thesis we discuss a well-known projective analog of Zariski-Nagata and provide the necessary background on toric varieties to present a generalization of this result to toric surfaces. We conclude with a brief discussion about work toward characterizing which abstract toric varieties have smooth point fibers with the aim of understanding the symbolic powers of higher dimensional toric varieties. Advisor: Jack Jeffries