## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, December 4, 2014 — 10:00 AM EST

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 $A,B \in Nef(X)$.

Location

M3 - Mathematics 3

2134

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.