Rational curves of Castelnuovo–Mumford regularity d−r+1

The purpose here is to show that for r⩾4 and d⩾r+4, a nondegenerate rational curve of degree d in Pr has a unique (d−r+1)-secant line if and only if X fails to be (d−r)-regular. This is a next case to a result of Gruson, Lazarsfeld and Peskine, which asserts that for r⩾3 and d⩾r+2, a nondegenerate curve X of degree d in Pr has a (d−r+2)-secant line if and only if X fails to be (d−r+1)-regular.