Weakly defective varieties

A projective variety X is ‘k-weakly defective’ when its intersection with a general (k + 1)-tangent hyperplane has no isolated singularities at the k + 1 points of tangency. If X is k-defective, i.e. if the k-secant variety of X has dimension smaller than expected, then X is also k-weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini’s classification of k-defective surfaces.