Weakly defective varieties

Abstract

A projective variety X X is ‘ k k -weakly defective’ when its intersection with a general ( k + 1 ) (k+1) -tangent hyperplane has no isolated singularities at the k + 1 k+1 points of tangency. If X X is k k -defective, i.e. if the k k -secant variety of X X has dimension smaller than expected, then X X is also k 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 k -defective surfaces.