AN AFFIRMATIVE ANSWER TO A QUESTION OF CILIBERTO

We give an example of a nondegenerate n-dimensional smooth projective variety X in P-2n+1 with the canonical bundle ample a variety X whose tangent variety TanX has dimension less than 2n over an algebraically closed field of any characteristic when n greater than or equal to 9. This variety X is not ruled by lines and the embedded tangent space at a general point of X intersects X at some other points, so that this yields an affirmative answer to a question of Ciliberto.