The dynamics of the Hesse derivative on the $j$-invariant

Jake Kettinger
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Abstract

The $j$-invariant of a cubic curve is an isomorphism invariant parameterized by the moduli space of elliptic curves. The Hesse derivative of a curve $V(f)$ given by the homogeneous polynomial $f$ is $V(\mathcal{H}(f))$ where $\mathcal{H}(f)$ is a the determinant of the Hesse matrix of $f$. In this paper, we compute the $j$-invariant of the Hesse derivative of a cubic curve $C$ in terms of the $j$-invariant of $C$, getting a rational function on the Riemann sphere. We then analyze the dynamics of this rational function, and investigate when a cubic curve is isomorphic to its $n$-fold Hesse derivative.
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海瑟导数在不变量 $j$ 上的动态变化
立方曲线的 $j$ 不变式是以椭圆曲线模空间为参数的同构不变式。同质多项式 $f$ 给定的曲线 $V(f)$ 的黑塞导数是 $V(\mathcal{H}(f))$,其中$\mathcal{H}(f)$ 是 $f$ 的黑塞矩阵的行列式。在本文中,我们根据立方曲线$C$的$j$不变式计算其海瑟导数的$j$不变式,从而得到黎曼球上的有理函数。然后,我们分析了这个有理函数的动力学,并研究了立方曲线何时与其 $n$ 折 Hesse 导数同构。
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