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 $\oplus S(-b_i)\to \oplus S(-a_i)\to S\to S\mathrm{/}I$ be a minimal resolution of $I$. We show that $b_i\le 5k$. From this we obtain that the Mordell-Weil rank of the elliptic threefold $W:y^2=x^3+f$ equals $2#{i\mid b_i=5k}$\mathrm{.}UsingthiswefindanupperboundfortheMordell-Weilrankof$W$, which is $1\mathrm{/}18(125+\sqrt{73}-\sqrt{2302-106\sqrt{73}})k+l\mathrm{.}o\mathrm{.}t\mathrm{.}$ and we find an upper bound for the exponent of $(t^2-t+1)$ in the Alexander polynomial of $C$, which is $1\mathrm{/}36(125+\sqrt{73}-\sqrt{2302-106\sqrt{73}})k+l\mathrm{.}o\mathrm{.}t\mathrm{.}$. This improves a recent bound of Cogolludo and Libgober almost by a factor 2.