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Abstract: I will start by reviewing the main features of Kuznetsov's Homological projective duality (HPD for short). Then I will explain how, inspired by Calabrese and Thomas' construction of derived equivalent Calabi-Yau 3-folds, one gets the idea of considering a natural HPD problem for linear systems with base loci. Inparticular, this yields a process for constructing new HP duals from old. Finally, we will explicitly apply the procedure in the example of singular cubic 4-folds. This is joint work with Zak Turcinovic (arXiv:1511.09398).