AI 中文总结
探讨加权射影空间有理点稀疏观点对雅可比猜想的意义，证明分次符号模式决定一切，如权重全正等变凯勒映射是自同构，二维任意符号模式也成立，还阐述了凯勒条件在商空间情况及相关轨迹情况，提出按权重签名分类等变多项式映射。
AI 中文摘要
加权射影空间上的有理点是稀疏的：仅当在每个素数处库默条件成立时，\(\mathbb{P}^n(\mathbb{Q})\)中的点才会沿着维罗内塞态射提升（arXiv:2509.02319）。我们探讨此观点对雅可比猜想的意义。最近被宣布为反例的映射对于分次\(\mathrm{wt}(x,y,z)=(1,-1,-2)\)是等变的，我们证明这种分次的符号模式决定一切。若权重全为正，在加权射影空间的设定下，等变凯勒映射总是自同构，所以不存在这种分次的反例。在二维中，对任何符号模式都成立。凯勒条件本身下降到商空间，表明商映射的雅可比沿着收缩轨迹消失到二阶。在\(\mathbb{Q}\)上，反例的像为薄集。但沿着群作用具有稳定子\(\mu_2\)的线，当\(-a\)是平方数时，收缩轨迹上的两个原像恰好是有理的，所以正权重理论的库默条件在层状层上再次出现。我们提出按权重签名对等变多项式映射进行分类，以加权射影几何为正例情况。
英文摘要
Rational points on weighted projective spaces are sparse: a point of $\mathbb{P}^n(\mathbb{Q})$ lifts along the Veronese morphism only when a Kummer condition holds at every prime (arXiv:2509.02319). We ask what this viewpoint says about the Jacobian Conjecture. The map recently announced as a counterexample is equivariant for the grading $\mathrm{wt}(x,y,z)=(1,-1,-2)$, and we show that the sign pattern of such a grading decides everything. If the weights are all positive, the setting of weighted projective spaces, an equivariant Keller map is always an automorphism, so no counterexample can be graded that way. In dimension two the same holds for every sign pattern. The Keller condition itself descends to the quotient, where it says that the Jacobian of the quotient map vanishes to order two along the contracted locus. Over $\mathbb{Q}$ the image of the counterexample is a thin set. But along the line where the group action has stabilizer $μ_2$, the two preimages on the contracted locus are rational exactly when $-a$ is a square, so the Kummer condition of the positive-weight theory reappears on the stacky stratum. We propose a classification of equivariant polynomial maps by the signature of the weights, with weighted projective geometry as the positive case.
