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Let $S$ be a complex surface with ample canonical class. Let $C_1$ and $C_2$ be smooth complex curves in $S$ that intersect transversally at $n $ points. Furthermore, assume that the self-intersection of the homology class $[C_1] + [C_2]$ is positive. Is it true that the intersections of $C_1$ and $C_2$ can be smoothed holomorphically to obtain a smooth complex curve in the homology class $[C_1] + [C_2]$?

Any reference to the proof or a counterexample is appreciated. I tried to use the Bertini's Theorem, but could not find a suitable version of it in the literature.

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  • $\begingroup$ If I understand correctly, what you want is not a resolution, but rather a smoothening. $\endgroup$ – Sasha Mar 27 '15 at 5:02
  • $\begingroup$ @Sasha Yes, I am considering a smoothening (locally replace two intersecting disks in the 4-ball with an annulus). I revised my question. $\endgroup$ – user69766 Mar 27 '15 at 5:17
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The answer is no in general, as the following example shows.

There exist complex surfaces $S$ with ample canonical class, $$p_g(S):=h^0(S, \, K_S)=2, \quad q(S):=h^1(S, \, K_S)=0$$ and $|K_S|$ composed with a pencil. This means that $$|K_S|=M + |F|,$$ where $M$ is a fixed curve and $F$ is a divisor with $h^0(S, \, F)=2$. It is no difficult to construct examples where $M$ and the general curve in $|F|$ are smooth and intersect transversally. For a general discussion on such surfaces, see [1].

Since $K_S$ is ample, we have $(M+F)^2 >0$. Moreover, by a result of Bombieri (see [2]) the canonical divisor is numerically $2$-connected, hence $MF \geq 2$. Finally, since $q(S)=0$, by the exponential sequence the group $\textrm{Pic}(S)=H^1(S, \, \mathcal{O}_S^*)$ injects into $H^2(S, \, \mathbb{Z})$, hence homologically equivalent divisors are linearly equivalent.

Being $K_S$ ample, $S$ is a projective variety. It follows by Chow Lemma that any closed analytic subvariety of $S$ is algebraic, i.e. all compact holomorphic curves in $S$ are actually effective divisors.

Now, $S$ contains no smooth divisor linearly equivalent to $K_S$, since all divisors in the complete linear system $|K_S|$ are of the form $M+F$. In other words, the curves in the homology class of $K_S$ cannot be smoothed holomorphically in $S$.

References.

[1] R. Pignatelli: On surfaces with a canonical pencil, Matematische Zeitschrift 270, 1 (2012), 403-422.

[2] E. Bombieri: Canonical models of surfaces of general type, Publications Mathématiques de l'Institut des Hautes Études Scientifiques 42, Issue 1, pp 171-219 (1973).

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  • $\begingroup$ do you know some additional conditions on $S$ or the configuration of curves for which complex smoothening works? For example, what I mentioned above works for the configuration of lines (in a general position) in $\mathbb{CP}^2$. $\endgroup$ – user69766 Mar 27 '15 at 13:20
  • $\begingroup$ It is hard to give an answer in general. Of course, by Bertini Theorem what you want is true if the divisor $C_1+C_2$ is very ample (or, more generally, base-point free). Your example of a configuration of lines in $\mathbb{CP}^2$ falls into this situation (by the way, the general position assumption is not necessary here). $\endgroup$ – Francesco Polizzi Mar 27 '15 at 13:43

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