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 $\vert L\otimes I\_\xi\vert=\mathbb{P}(H^0(S,L\otimes I\_\xi)$?



Smooth curves in a linear system satisfying certain conditions.
$\xi$ 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. 


