We classify smooth $n$-dimensional varieties $X_n\subset{\bb P}^{2n+1}$ with one apparent double point and of degree $d \leqslant 2n + 4$, showing that these are only the smooth irreducible divisors of type (2,1), (0,2) and (1,2) on the Segre manifold ${\bb P}^1\times{\bb P}^n \subset {\bb P}^{2n+1}$, 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 $X_n \subset{\bb P}^{2n+1}$ of degree $d \leqslant 2n+1$ and with ${\rm Sec}(X_n) = {\bb P}^{2n+1}$ 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 ${\bb P}^1\times{\bb P}^n\subset{\bb P}^{2n+1}$. 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 $X_n\subset{\bb P}^{2n+1}$ defined by quadratic equations and we study projections of such varieties from (subspaces of) the tangent space.