We increase the dimension of the set of P N hyper surfaces whose intersection with an established fixed projective variety is not integral. The increases obtained are optimal. As an application, where possible, hyper surfaces are constructed whose intersections with all the varieties of a family of positive projective varieties are intact. The degree of constructed hyper surfaces is explicit. Abstract (Bertini's theorem in family). We give upper bounds for the dimension of the set of hyper

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... ces of P N which intersection with a fixed integral projective is not integral. Our upper bounds are optimal. As an application, we construct, when possible, hypersurfaces whose intersections with all types of integral integral projective are integral. The degree of the hyper surfaces we construct is explicit. Theorem 1.1. -Let X is a P N sub variety of dimension ≥ 2. Then Fe int (X) and F e igr (X) are strict enclosures of H e .