We investigate the existence of complete intersection threefolds X ⊂ n with only isolated, ordinary multiple points and we provide some sufficient conditions for their factoriality.

We study some special systems of generators on finite groups, introduced in the previous work by the first author and called diagonal double Kodaira structures, in order to study non-abelian, finite quotients of the pure braid group P2(Σ b ), where Σ b is a closed Riemann surface of genus b. Our main result is that, if G is such a quotient, then |G| ≥ 32, and equality holds if and only if G is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two 3-dimensional families of double Kodaira fibrations having signature 16.

Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of δ 2 ordinary double points, δ 3 ordinary triple points and if δ 2 + 4δ 3 < (k − 1) 2 , then any smooth surface contained in V is a complete intersection on V .Refining an argument of Severi (see [8]), Ciliberto and Di Gennaro prove in [1] that any surface contained in a hypersurface of P 4 with few ordinary double points is a complete intersection.In what follows we will discuss the possibility of extending the argument in [1] when the singularities involved are not only ordinary double points. More precisely we want to prove the following statement: Theorem 1. Let V ⊂ P 4 be a reduced and irreducible hypersurface of degree k 3. If the singular locus of V consists of δ 2 ordinary double points, δ 3 ordinary triple points and if δ 2 + 4δ 3 < (k − 1) 2 , then any smooth projective surface contained in V is a complete intersection on V .If the singularities involved are of order greater then three we obtain a weaker result: Proposition 2. Let V ⊂ P 4 be a reduced and irreducible hypersurface of degree k 3. Suppose that the singular locus of V consists of δ ordinary points of order at mostr and let δ r be the number of singular points of order r, 2 r r. If r r=2 (r − 1) 2 δ r < (k − 1) 2

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Denote by ν m (d) the maximal integer for which there exists for d 0 a threefold X ⊂ P 5 complete intersection of hypersurfaces of degree respectively d and d − 1 such that X has only ordinary singularities of order m and |Sing(X )| = ν m (d). We prove that, ν m (d) ≥ ϕ(d) where ϕ(d) ∼ d 5 asymptotically. This result extends (Di Gennaro and Franco in Commun Contemp Math 10(5):745-764, 2008, Corollary 2.10).