Special subhomaloidal systems of quadrics and varieties with one apparent double point

We classify smooth n-dimensional varieties X-n subset of P2n+1 with one apparent double point and of degree d less than or equal to 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 x P-n subset of P2n+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 subset of P2n+1 of degree d less than or equal to 2n + 1 and with Sec(X-n) = P2n+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 x P-n subset of P2n+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 Xn subset of P2n+1 defined by quadratic equations and we study projections of such varieties from (subspaces of) the tangent space.