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I'm looking for references on a very classical question: Let $X$ be a compact surface and let $L \to X$ be an ample line bundle. We assume that $L$ has nonzero sections. Then the linear system $|L|$ defines a family of curves $\mathcal C \to |L|$ on the surface $X$, most of whom are nonsingular of genus $1 + \frac 12 L \cdot (L + K_X)$. I'm very interested in the singular curves in this family.

What kind of singularities can arise here? For example, can we say (under some conditions on $X$ and $L$) that some curve in the system will have only nodal singularities and predict the number of nodes? Is there any way of predicting what the "worst" singular curve in this system looks like?

I apologize for the vague question, but like I said, this is more of a reference hunt than a precise question.

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    I guess this question is a bit old, but something that might be of interest here is Göttsche's conjecture: see for instance http://arxiv.org/abs/1010.3211.2013-10-10

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Even in the simplest case $X=\mathbb P^2$ and $L=O_X(d)$, you can't say much about the singularities (except that they are plane singularities). The number of singular points is bounded by the arithmetic genus (given by your adjunction formula) for irreducible curves in the family.

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    Yes there should be some global constraints on the curves when $X$ is not rational. I am not convinced that there are constraints of local nature.2012-12-13