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In this paper, we first introduce geometric operations for linear categories,and as a consequence generalize Orlov's blow up formula [O04] to possiblysingular local complete intersection centres. Second, we introduce refinedblowing up of linear category along base--locus, and show that this operationis dual to taking linear section. Finally, as an application we produceexamples of Calabi--Yau manifolds which admits Calabi--Yau categoriesfibrations over projective spaces.

Blowing up linear categories, refinements, and homological projective duality with base locus