Space curves, X-ranks and cuspidal projections

Abstract

Let Xsubset mathbb {P}^3 be an integral and non-degenerate curve. We say that qin mathbb {P}^3setminus X has X-rank 3 if there is no line Lsubset mathbb {P}^3 such that qin L and #(Lcap X)ge 2. We prove that for all hyperelliptic curves of genus gge 5 there is a degree g+3 embedding Xsubset mathbb {P}^3 with exactly 2g+2 points with X-rank 3 and another embedding without points with X-rank 3 but with exactly 2g+2 points qin mathbb {P}^3 such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed qin mathbb {P}^3setminus X. For all integers gge 5, xge 0 we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, deg (X) = x+g+3 and having at least x+2g+2 points of X-rank 3.