## Robert Garbary, Pure Mathematics, University of Waterloo

### "Local Positivity of Line Bundles on Toric Surfaces"

Let $X$ be a smooth algebraic variety, and let $p \in X$. Let $\pi : \tilde{X} \to X$ denote the blow-up of $X$ at $p$ with exceptional divisor $E$. Given an effective divisor $L$ on $X$, we define $\gamma_p(L) = \sup \{ t \geq 0 : \pi^*L - tE \text{ is effective} \}$. In this thesis, we develop the theory of this number. We first prove some general results on surfaces using Riemann-Roch type estimates. We then specialize to the case of smooth, complete, toric surfaces. The main result is that, for $p$ a $T$-invariant point of $X$, we have that $\gamma_p(A+B)=\gamma_p(A)+\gamma_p(B)$ for $A,B \in Nef(X)$.