  Let $R$ be a polynomial ring over a field in an unspecified number of
variables. We prove that if $J \subset R$ is an ideal generated by three cubic
forms, and the unmixed part of $J$ contains a quadric, then the projective
dimension of $R/J$ is at most 4. To this end, we show that if $K \subset R$ is
a three-generated ideal of height two and $L \subset R$ an ideal linked to the
unmixed part of $K$, then the projective dimension of $R/K$ is bounded above by
the projective dimension of $R/L$ plus one.
