Existence of curves with prescribed topological singularities

Throughout this paper we study the existence of irreducible curves $C$ on smooth projective surfaces $\Sigma$ with singular points of prescribed topological types $\mathcal S_1,\ldots,\mathcal S_r$. There are necessary conditions for the existence of the type $\sum_{i=1}^r \mu(\mathcal S_i)\leq \alpha C^2+\beta C.K+\gamma$ for some fixed divisor $K$on $\Sigma$ and suitable coefficients $\alpha$, $\beta$ and $\gamma$, and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case $\Sigma=\mathbb P_{\mathbb C}^2$. An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up $\Sigma$ of the form $\mathcal O_{\widetilde{\Sigma}}(\pi^*D-\sum_{i=1}^rm_iE_i)$, deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in $\mathbb P_{\mathbb C}^3$, and K3-surfaces.