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

Remke Kloosterman

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.

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

Abstract

Let be a reduced plane curve of degree , with only nodes and ordinary cusps as singularities. Let be the ideal of the points where has a cusp. Let be a minimal resolution of . We show that . From this we obtain that the Mordell-Weil rank of the elliptic threefold equals . Using this we find an upper bound for the Mordell-Weil rank of , which is and we find an upper bound for the exponent of in the Alexander polynomial of , which is . This improves a recent bound of Cogolludo and Libgober almost by a factor 2.

Paper Structure

This paper contains 5 sections, 15 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Resolution of the locus of cusps
  3. Syzygies and MW-rank
  4. Upper bound for $\mathop{\mathrm{rank}}\nolimits \mathop{\mathrm{MW}}\nolimits(\pi)$ (strongly admissible case)
  5. Admissible case

Key Result

Proposition 2.1

Let $I$ be the ideal of finitely many distinct points in ${\mathbf{P}}^2$. Then $I$ has a free resolution such that

Theorems & Definitions (39)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • ...and 29 more