On the X-rank with respect to linearly normal curves

In this paper we study the X-rank of points with respect to smooth linearly normal curves of genus g and degree n+g. We prove that, for such a curve X, under certain circumstances, the X-rank of a general point of X-border rank equal to s is less or equal than n + 1 - s. In the particular case of g = 2 we give a complete description of the X-rank if n = 3, 4; while if n a parts per thousand yen 5 we study the X-rank of points belonging to the tangential variety of X