Free divisors and rational cuspidal plane curves

A characterization of freeness for plane curves in terms of the Hilbert function of the associated Milnor algebra is given as well as many new examples of rational cuspidal curves which are free. Some stronger properties are stated as conjectures.Comment: version 4: a misprint in the conjecture added yesterday is correcte

L^2-Betti numbers of hypersurface complements

In \cite{DJL07} it was shown that if \scra is an affine hyperplane arrangement in \C^n, then at most one of the $L^2$--Betti numbers b_i^{(2)}(\C^n\sm \scra,\id) is non--zero. In this note we prove an analogous statement for complements of complex affine hyperurfaces in general position at infinity. Furthermore, we recast and extend to this higher-dimensional setting results of \cite{FLM,LM06} about $L^2$--Betti numbers of plane curve complements.Comment: 10 pages; comments are very welcome; v2: minor clarifications added in the tex