Cuspidal plane curves, syzygies and a bound on the MW-rank
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2011
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| _version_ | 1866916443236335616 |
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| author | Kloosterman, Remke |
| author_facet | Kloosterman, Remke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1107_2043 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Cuspidal plane curves, syzygies and a bound on the MW-rank Kloosterman, Remke Algebraic Geometry Commutative Algebra 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. |
| title | Cuspidal plane curves, syzygies and a bound on the MW-rank |
| topic | Algebraic Geometry Commutative Algebra |
| url | https://arxiv.org/abs/1107.2043 |