Measurement of Singularities via the Birational Map in the Three Dimensional Minimal Model Program

This project is to study the change of singularities via a terminalthreefold ips Y 99K Y + and via a divisorial contraction f : Y ! Xthat contracts a divisor to a curve. More precisely, we would liketo try to realized how the singularities occur on the ipped curve.Also try to give algebraic generalization for increasing intersectionnumber c1(Y ):c2(Y ) via a divisorial contraction to a curve (in thecase of ?(OX) = 1). Note that this was done in the analytic case inmy project last year.