Homological projective duality for linear systems with base locus

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We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a HP dual pair X,Y and smooth orthogonal linear sections Xₗ,Yₗ, we prove that the blowup of X in Xₗ is naturally HP dual to Yₗ. The result does not need Y to exist as a variety, i.e. it may be "noncommutative". We extend the result to the case where the base locus Xₗ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a categorical resolution of singularities of Yₗ. Finally we give examples where, starting with a noncommutative Y, the above process nevertheless gives geometric HP duals.