We classify smooth n-dimensional varieties X n ⊂ P 2n+1 with one apparent double point and of degree d 2n + 4, showing that these are only the smooth irreducible divisors of type (2, 1), (0, 2) and (1, 2) on the Segre manifold P 1 × P n ⊂ P 2n+1 , a 3-fold of degree 8 and two Mukai manifolds, the first one of dimension 4 and degree 12, the second one of dimension 6 and degree 16. We also prove that a linearly normal variety X n ⊂ P 2n+1 of degree d 2n + 1 and with Sec(X n ) = P 2n+1 is regular and simply connected, that it has one apparent double point and hence it is a divisor of type (2, 1), (0, 2) or (1, 2) on the Segre manifold P 1 × P n ⊂ P 2n+1 . To this aim we study linear systems of quadrics on projective space whose base locus is a smooth irreducible variety and we look for conditions assuring that they are (completely) subhomaloidal; we also show some new properties of varieties X n ⊂ P 2n+1 defined by quadratic equations and we study projections of such varieties from (subspaces of ) the tangent space.
Abstract. Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.
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