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Let $S$ be a smooth, projective surface over $\mathbb{C}$ and $L\in\mathrm{Pic}(S)$ be globally generated. Then, a general curve $C\in\vert L\vert$ is smooth. Now, let $I_\xi$ be the ideal of a (possibly non-reduced) $0$-dimensional subscheme $\xi\subset S$ and assume that $L\otimes I_\xi$ is still globally generated. Which hypotheses on $\xi$ would ensure the existence of a smooth curve in $\vert L\otimes I_\xi\vert=\mathbb{P}(H^0(S,L\otimes I_\xi)$?

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up vote 2 down vote accepted

$\xi$ has to be "curvilinear", i.e., contained in some smooth curve of $S$, or equivalently, has to be of multiplicity one. It is clear that this is necessary.

Let me now show that it is sufficient. The simplest case is when $\xi$ is reduced. Then you consider the blow up $\pi_\xi:\tilde S\rightarrow S$ at all points in $\xi$, and notice that $|L\otimes I_\xi|\cong |\pi^*L-E_\xi|$. General curves in $|\pi^*L-E_\xi|$ are smooth and intersect $E$ transversely, because $\pi^*L-E_\xi$ is globally generated, so you are done.

Now assume the curvilinear scheme $\xi$ is nonreduced but irreducible of length $k$ supported at some point $p=p_1$, and consider the blow up $\pi_1: S_1\rightarrow S$ at $p$, with exceptional divisor $E_1$. Then $\pi_1^*I_\xi=I_{E_1}\cdot I_{\xi_1}\cong I_{\xi_1}\otimes \mathcal O_{S_1}(-E_1)$, where $\xi_1$ is a 0-dimensional curvilinear subscheme of $S_1$, irreducible of length $k-1$, supported at some point $p_2\in E_1$. More important, $I_\xi=\pi_{1*}(I_{\xi_1}\otimes \mathcal O_{S_1}(-E_1))$, so $I_{\xi_1}\otimes \mathcal O_{S_1}(-E_1)$ is globally generated. Blow up $p_2$ to define $p_3$, etc, till you blow up $p_k$; denote $\pi:\tilde S \rightarrow S$ the composition of the $k$ blowups, and let $E_i$ be the pullback on $\tilde S$ of the exceptional divisor above $p_i$. Now $I_\xi=\pi_*(\mathcal O_{\tilde S}(-E_1-\dots-E_k))$ with $\mathcal O_{\tilde S}(\pi^*L-E_1-\dots-E_k)$ base point free. A general member of $|\pi^*L-E_1-\dots-E_k|$ is smooth, intersects $E_k$ transversely at a point different from the singular point of $E_1$, so its image on $S$ is nonsingular.

The above analysis can be carried over for each point on the support of $\xi$.

A local version of this, including the description of what happens for non-curvilinear subschemes (the singularity type of a general member of $|L\otimes I_\xi|$ is determined by the "resolution" of $\xi$), can be found (as "Bertini theorem") on Casas-Alvero's book on singularities of plane curves.

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@quim: what do you mean by "has to be of multiplicity 1"? – Serge Lvovski Feb 4 '13 at 10:43
For each point $p$ of the support of $\xi$ (or, for each irreducible component of $\xi$) the multiplicity of $\xi$ at $p$ is the maximal integer $m$ such that the stalk at $p$ of $I_\xi$ is contained in the $m$-th power of the maximal ideal. What I mean is that all components have to be of multiplicity 1. – quim Feb 4 '13 at 10:57

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