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Suppose you have a surface (of general type if it helps), and it's embedded in a 3-fold. Naively, the intersection of divisors on the 3 fold and the surface, will be a divisor (a curve) in a surface. I want to know whether this curve is reducible or not, and if it is, then how should I write it in terms of the reducible ones on the surface.

Does anyone can help me to find some reference for it? Usually the textbooks assume that we have reduced irreducible curve, but they don't say how to check that in general. Is there any criterion for that in terms of intersection numbers etc?

Thanks very much. Mohsen

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  • $\begingroup$ I assume everything is smooth and projective (the latter is not absolutely necessary). If $X\subset Y$ are the surface and threefold and $H$ is a divisor on $Y$, then, $H\cdot X$ can be written as $\sum_{i=1}^r n_iD_i$, where $D_i$s are irreducible curves on $X$ and $n_i\in\mathbb{Z}$. For the latter question, search for Bertini's theorems- intersection numbers do not guarantee irreducibility. $\endgroup$ – Mohan Nov 1 '16 at 22:39
  • $\begingroup$ Thanks Mohan, actually my question is about the $n_i$s and $r$ and the intersection number between the irreducible curves. Bertini's theorem tells me (if I'm right) that if $H$ is amble the $H\cdot X$ is irreducible, what if it's not irreducible? Are there anything else to give more information? $\endgroup$ – Mohsen Karkheiran Nov 1 '16 at 23:52
  • $\begingroup$ No, Bertini does not say what you say. If $H$ is `general' then $H\cdot X$ is irreducible. If it is not irreducible (if $H$ is not general), then the intersection can be complicated, so it depends on what you want to get. Both $n_i,r$ can be as large in general as is allowed. $\endgroup$ – Mohan Nov 2 '16 at 1:14

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