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

JOURNAL OF ALGEBRA

卷:375

Article

Kloosterman, Remke

关键词: Elliptic threefolds;    Mordell-Weil rank;    Alexander polynomials of plane curves;   

DOI  :  10.1016/j.jalgebra.2012.11.015

来源: Elsevier

【 摘 要 】

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 circle plus S(-b(i)) -> circle plus S (-a(i)) -> S -> S/I be a minimal resolution of I. We show that b(i) <= 5k. From this we obtain that the Mordell-Weil rank of the elliptic threefold W: y(2) = x(3) + f equals 2#{i vertical bar b(i) = 5k}. Using this we find an upper bound for the Mordell-Weil rank of W, which is 1/18(125 + root 73 - root 2302 - 106 root 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 + root 73 - root 2302 - 106 root 73)k + l.o.t. This improves a recent bound of Cogolludo and Libgober almost by a factor 2. (C) 2012 Elsevier Inc. All rights reserved.

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