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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

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I don't understand why in the example of a node the Euler characteristic changes by 2. I would say it changes by 1: a loop in the general fiber $C_t$ is shrinked to a point in $C_0$. Indeed, $C_0\cap B_z$ is contractible, hence the characteristic is 1, while $C_t∩B_z$ is $S^1\times [0,1]$, so it has characteristic 0. –  rita Aug 16 '11 at 13:41
    
You are right, I should have probably looked at the normalization of the central fiber $C_0$ as well. now it gets separated into two discs and the characteristic is 2. I'll fix it in an edit. –  Blade Aug 16 '11 at 16:33
    
If you take normalizations I think the jump is always even.The Euler characteristic of a compact Riemann surface is even, and I have the impression that the complement of $C_t\cap B_z$ stays constant for small $t$. –  rita Aug 16 '11 at 18:51
    
One more question: what do you mean by adding ``holes''? a hole as in a torus or a boundary component? –  rita Aug 17 '11 at 9:11
    
by adding 'holes' I meant boundary components. (By the way, this now rises another issue I haven't thought of, can "holes" in the meaning of a torus appear in the deformation? I don't think so, but If you see a way these guys appear as well, please let me know) Thanks –  Blade Aug 17 '11 at 13:29

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