On the projective dimension and the unmixed part of three cubics

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Let R be a polynomial ring over a field in an unspecified number of variables. We prove that if J ⊂ 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 ⊂ R is a three-generated ideal of height two and L ⊂ 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.