On Syzygies of Projected Algebraic Curves

Let C ⊂ ℙ^r be a linearly normal projective integral curve of arithmetic genus g ≥ 1 and degree d = 2g + 1 + p for some p ≥ 1. It is well known that C is cut out by quadric and satisfies Green-Lazarsfeld's property N_p. Recently it is known that for any q ∈ ℙ^r\C such that the linear projection π_q: C → ℙ^r-1 of C from q is an embedding, the projected image C_q:= π_q(C) ⊂ ℙ^r-1 is 3-regular, and hence its homogeneous ideal is generated by quadratic and cubic equations. In this article we study the problem when C_q is still cut out by quadrics. Our main result in this article shows that if the relative location of q with respect to C is general then the homogeneous ideal of C _q is still generated by quadrics and the syzygies among them are generated by linear syzygies for the first a few steps.