Plane quartics and heptagons

Daniele Agostini, Daniel Plaumann, Rainer Sinn, Jannik Lennart Wesner
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Abstract

Every polygon with n vertices in the complex projective plane is naturally associated with its adjoint curve of degree n-3. Hence the adjoint of a heptagon is a plane quartic. We prove that a general plane quartic is the adjoint of exactly 864 distinct complex heptagons. This number had been numerically computed by Kohn et al. We use intersection theory and the Scorza correspondence for quartics to show that 864 is an upper bound, complemented by a lower bound obtained through explicit analysis of the famous Klein quartic.
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平面四边形和七边形
在复投影面中,每个有 n 个顶点的多边形都自然地与其 n-3 度的邻接曲线相关联。因此,七边形的邻接曲线就是平面四边形。我们证明,一般的平面四边形正好是 864 个不同的复七边形的邻接曲线。我们利用交集理论和四边形的 Scorzacorrespondence 来证明 864 是一个上界,并通过对著名的克莱因四边形的明确分析得到了一个下界。
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