射影直线上四次有理映射的不变环
The invariant ring of degree-four rational maps on the projective line
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中文总结 AI 辅助
本文确定射影直线四次有理映射对应联合不变环$\boldsymbol{\text{R}}_{5,3}$的极小生成集，据此描述零锥、构造模空间$\boldsymbol{\text{M}}_4^1$的绝对不变量，给出判定映射共轭性的有效准则。
中文摘要 AI 辅助
设k为特征零的代数闭域，射影直线$\boldsymbol{\text{P}}^1$上的四次有理映射在共轭意义下对应二元形式对$(F,G)\text{∈}V_5\bigoplus V_3$，其关联不变环为联合不变环$\boldsymbol{\text{R}}_{5,3}=k[V_5\bigoplus V_3]^{\boldsymbol{\text{SL}}_2}$。本文确定$\boldsymbol{\text{R}}_{5,3}$的极小生成集，包含50个次数不超过18的显式联合转置量；作为推论，描述$V_5\bigoplus V_3$的零锥，构造生成模空间$\boldsymbol{\text{M}}_4^1$函数域的6个显式绝对不变量，并给出判定四次有理映射共轭性的有效准则。
英文摘要
Let $k$ be an algebraically closed field of characteristic zero. Degree-four rational maps on $\mathbb{P}^1$, up to conjugation, correspond to pairs of binary forms $(F,G)\in V_5\oplus V_3$. The associated invariant ring is the joint invariant ring $\mathcal{R}_{5,3}=k[V_5\oplus V_3]^{\mathrm{SL}_2}$. We determine a minimal generating set for $\mathcal{R}_{5,3}$, consisting of fifty explicit joint transvectants of degrees at most $18$. As consequences we describe the null cone of $V_5\oplus V_3$, construct six explicit absolute invariants that generate the function field of the moduli space $\mathcal{M}_4^1$, and give an effective criterion for determining conjugacy of degree-four rational maps.
