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

TL;DR

This paper establishes bounds on the Mordell-Weil rank of elliptic threefolds associated with certain plane curves, using syzygies of cusp ideals, and improves previous bounds significantly.

Contribution

It introduces a new bound on the Mordell-Weil rank based on syzygies of cusp points, refining earlier estimates and connecting algebraic geometry with topological invariants.

Findings

Bound on the degrees of syzygies of cusp ideals: b_i ≤ 5k.

Explicit upper bound on Mordell-Weil rank involving algebraic constants.

Improved the previous bound on Alexander polynomial exponents by nearly a factor of 2.

Abstract

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