Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular). Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that

  • $f$ is non constant
  • $\{f=0\}=C\cup A$, with $A$ a complex analytic set of dimension $1$
  • there is no neighborhood $V$ of $C$ and no function $g\in\mathcal{O}(V)$ such that $C=\{g=0\}$.

I would like to conclude that $C\cdot C<0$.

If $C$ is smooth, then $f$ gives me a nontrivial section of the conormal bundle: let $\{F_j\}$ be local equations for $C$, then $\{f/F_j^k\}_j$ gives (for the right natural number $k$) a non zero section of $[-C]\vert_C=N^*_{C,M}$ (the conormal bundle of $C$ in $M$), hence $\deg N_{C,M}<0$, therefore $C\cdot C<0$ (the section vanishes somewhere because of the 2nd hypothesis).

What if $C$ is singular?

NOTE: I believe this is quite a simple matter of knowing the right definitions; I tried to search around and to post this on math.stackexchange (https://math.stackexchange.com/questions/466842/negative-self-intersection-and-section-of-the-conormal-sheaf-for-a-singular-comp) but got no answers.

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    $\begingroup$ (Please add links here and on math.SE so that people can find the other question easily) $\endgroup$ – Mariano Suárez-Álvarez Aug 18 '13 at 1:07
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    $\begingroup$ Even if $C$ is singular, it is locally defined by a single equation since $M$ is a manifold, so your argument should work without any changes. $\endgroup$ – ulrich Aug 18 '13 at 7:09
  • $\begingroup$ @ulrich: if $C$ is singular, I have no definition of the normal bundle. Moreover, if $C$ is singular, I do not know if the existence of a section of $[-C]\vert_C$ implies that $C\cdot C<0$. All this I know to hold in the smooth case. $\endgroup$ – Samuele Aug 18 '13 at 9:45
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    $\begingroup$ The conormal bundle can be defined to be $I/I^2$ where $I$ is the ideal sheaf of $C$. This is locally free of rank $1$ since $I$ is locally principal. The normal bundle is defined to correspond to the dual of this and this is also equal to $O(C)|_C$. If a line bundle on any irreducible curve has a non-zero section with non-empty zero locus then its degree must be positive. $\endgroup$ – ulrich Aug 18 '13 at 12:46
  • $\begingroup$ And the degree of the conormal bundle is $C\cdot C$ (the self-intersection), even if the curve is singular? Sorry if these are dumb questions, but I've never really studied these things in depth... $\endgroup$ – Samuele Aug 18 '13 at 13:15

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