Smooth affine factorial surfaces of Kodaira dimension zero with trivial units

We consider complex affine surfaces S{p,q} given in C^4 by {yx^d=z-p(x), wz^e=x-q(z)}, where p,q are polynomials of degrees d,e; p(0)=q(0)=1. Using these surfaces as a simple example, we introduce various notions in algebraic geometry and topology. First, we compute their standard boundaries, showing that S{p,q} is isomorphic to S{p',q'} if and only if {p,q}={p',q'}. Next, applying calculus of graph manifolds to tubular neighborhoods of these boundaries, we show that S{p,q} is homeomorphic to S{p',q'} if and only if {d,e}={d',e'}. In fact, we will show a topological construction of S{p,q} via a 0-surgery on a 2-bridge knot. Eventually, coming back to algebraic geometry, we will use a variant of logMMP to show that surfaces S_{p,q} exhaust all affine surfaces of Kodaira dimension zero, whose coordinate rings are factorial and have trivial units. This is a joint work with P. Raźny.