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Questions about K3 surfaces, which are smooth complex surfaces $X$ with trivial canonical bundle and vanishing $H^1(O_X)$. They are examples of Calabi-Yau varieties of dimension $2$.

5 votes

cuspidal curves in K3 surfaces

Only now I notice your question. In case you are still interested in the problem, I think that the paper http://arxiv.org/abs/1107.4568 completely answers to your question.
Concettina Galati's user avatar