Hello all. Let's say we have a one parameter (complex) family of complex curves on a family of corresponding surfaces. i.e, think of the whole thing living inside some threefold, fibered along corresponding surfaces.

Let's look on the central fiber, we have a curve $C_0$ on a surface, and assume we have some isolated singularity there at point $z$ . pick a small closed ball around the singularity.

Topologically $\(C_0 \cap B_z \)$ is a bouquet of discs. now, lets vary the parameter and look what happens to this fragment of the curve. Look at the Normalization of $\(C_t \cap B_z \)$. it's a collection of surfaces with some holes and handles. I want to compare the Euler Characteristic of this normalization with the Euler Characteristic of the normalization of the corresponding fragment of the central fiber, $C_0$.

Let's say I know that the Euler Characteristic (Topological) has changed in the deformation by at most 2.

Now what can it mean? Either some branches joined by a handle. or we could have added holes. EDIT: (by a handle I mean a tube, topologically $S^1\times\[0,1\]$ joining the two branches, like in the deformation $x^2 + y^2 = t$ : for $t=0$ we have bouquet of two discs and the normalization separates them to two disjoint discs, and for $t\ne0$ they join by a handle). It still decreases Euler Characteristic by 2 because it cuts out two small discs)

I find it hard to understand what does adding holes mean, or even whether it is entirely possible that holes add up in the deformation. can the Euler characteristic jump at an increment of 1 and not 2 (adding 1 hole)? Can I get two holes instead of a handle? and at what conditions of the arrangement of branches of the central fiber?

I hope I made my question clear, If not, please tell me and I'll try to explain better. for now, that's the best I could do, well... since this whole thing is not very clear to me, It's also hard to be fluent and precise when asking the question.

Thank you all in advance