Deformation equivalence of affine normal surfaces

Two varieties $X_1$, $X_2$ over $\mathbb{C}$ are called deformation equivalent, if there is a family $(\mathcal{X}(s))_{s\in S}$ over a connected base $S$ such that $X_i=\mathcal{X}(s_i)$, $i=1,2$, for some points $s_1, s_2\in S$. If for the class of varieties considered there is a moduli space then the varieties belonging to a connected component of this moduli space form just deformation equivalent varieties. In this talk we consider deformation equivalence for the class say $\mathcal{C}$ of affine normal surfaces, which admit an $\mathbb{A}^1$-fibration. A family of surfaces in $\mathcal{C}$ consists in a completable flat morphism $p:\mathcal{V}\to S$ such that every fiber is a surface in $\mathcal{C}$. Here the morphism $p$ is called completable if it is the restriction of some proper flat map $\bar{p}\colon\bar{\mathcal{V}}\to S$ to an open subset $\mathcal{V}\subset \bar{\mathcal{V}}$ such that the boundary $\mathcal{D}=\bar{\mathcal{V}}\setminus \mathcal{V}$ is a family of normal crossing divisors with constant dual graph. We note that except for a few exceptional cases one cannot expect for this class a moduli space. We characterize in this talk as to when two surfaces in $\mathcal{C}$ are deformation equivalent. This characterization is given in purely combinatorial terms using the extended divisor of a surface with a $\mathbb{C}_+$-action. (Joint with S. Kaliman and M. Zaidenberg.)