Homological Projective Duality for Linear Systems with Base Locus

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 homological projective (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 also holds true when Y is a noncommutative variety or just a category. 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 weakly crepant categorical resolution of singularities of YL. Finally we give examples where, starting with a noncommutative Y, the above process nevertheless gives geometric HP duals.