On projective curves of next to maximal regularity

It is known that if a rational curve C⊂Pr (r≥4) of degree d≥r+4 fails to be (d−r)-regular then C admits a unique (d−r+1)-secant line L and the arithmetic genus of C is at most 1. In this paper, we study the effect of such a secant line on algebraic and geometric properties of the curve C. We show that if the singular locus of C does not lie on L then C is obtained by a simple linear projection of a curve of maximal regularity. Also we show that if d≤2r−2 then C∪L is 3-regular which enables us to estimate the Hartshorne–Rao module and the graded Betti numbers of C.