We show that the average size of the $2$-Selmer group of the family of Jacobians of nonhyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\mathbb{Z}∕2\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava’s orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are the construction of integral representatives using certain properties of the compactified Jacobian of the simple curve singularity of type $E_6$, and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves.

我们证明了平均大小$2$-非超椭圆属雅可比行列家族的塞尔默群-$3$具有明显有理超曲点的曲线，当按自然高度排序时，其边界为$3$. 我们通过解释来实现这一点$2$-Selmer 元素作为与稳定相关的表示的积分轨道$\mathbb{Z}∕2\mathbb{Z}$- 对类型的李代数进行分级${乙}_6$并使用 Bhargava 的轨道计数技术。我们使用这个结果来表明标记点是该族中正比例曲线的唯一合理点。主要的创新是使用类型的简单曲线奇异性的紧化雅可比行列式的某些性质来构造积分表示${乙}_6$，以及与我们的曲线族自然相关的 Mumford theta 群的表示理论解释。