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Admissible invariants of genus 3 curves. (English) Zbl 1339.11069

The paper under review deals with some invariants of polarized metrized graphs \((\Gamma,{\mathbf q})\) denoted by \(\phi(\Gamma)\), \(\lambda(\Gamma)\) and \(\varepsilon(\Gamma)\). These invariants are studied for their connection to the self intersection of admissible dualizing sheafs associated to a curve of genus at least 2 over a global field. In this paper, all polarized metrized graphs \((\Gamma,{\mathbf q})\) of genus 3 are determined and the invariants \(\phi(\Gamma)\), \(\lambda(\Gamma)\) and \(\varepsilon(\Gamma)\) are computed. Furthermore, if \((\Gamma,{\mathbf q})\) is of genus 3 and length \(\ell(\Gamma)\), then it is proved that \(\phi(\Gamma) \geq 17 \ell(\Gamma)/288\), \(\lambda(\Gamma) \geq 3\ell(\Gamma)/28\) and \(\varepsilon(\Gamma)\geq 2\ell(\Gamma)/9\). This improves the bound given for the effective Bogomolov conjecture in the case of genus 3 (see Theorem 2.3, Remark 5.1 in [Z. Cinkir, Invent. Math. 183, No. 3, 517–562 (2011; Zbl 1285.14029)]).

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
94C15 Applications of graph theory to circuits and networks
97H30 Equations and inequalities (educational aspects)
26D05 Inequalities for trigonometric functions and polynomials
11G50 Heights

Citations:

Zbl 1285.14029
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References:

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