Cuspidal plane curves, syzygies and a bound on the MW-rank

Let C=Z(f) be a reduced plane curve of degree 6k, with only nodes and ordinary cusps as singularities. Let I be the ideal of the points where C has a cusp. Let ⊕S(−bi)→⊕S(−ai)→S→S/I be a minimal resolution of I. We show that bi⩽5k. From this we obtain that the Mordell–Weil rank of the elliptic threefold W:y2=x3+f equals 2#{i|bi=5k}. Using this we find an upper bound for the Mordell–Weil rank of W, which is and we find an upper bound for the exponent of (t2−t+1) in the Alexander polynomial of C, which is . This improves a recent bound of Cogolludo and Libgober almost by a factor 2.