AI 中文总结
本文建立有理映射奇点理论，引入相关不变量，详细研究曲面情形并关联多理论，还提出高维等方向的未来研究问题。
AI 中文摘要
我们建立了有理映射的奇点理论，聚焦于映射$$ f\text{colon} X\dashrightarrow \mathbb P^n $$，即便源空间$X$具有高度奇异性，仍利用其极化图与正规化极化图进行研究。为度量映射在点$x\in X$处的奇异性（即其正则性的偏离程度），我们引入了一系列不变量，包括正规化图纤维次数$\delta_x(f)$、广义lc阈值$\lambda_x(f)$，以及度量正规化图自身奇异性的不变量。大量实例表明，所得不变量能有效刻画映射奇点的不同本质方面。我们详细研究了曲面情形：将正规化图纤维次数与重数相关联，证明了klt曲面芽的精确阈值-次数不等式，并建立了光滑与奇异曲面上线性型映射的完整理论，尤其将线性型映射的存在性与给定点处光滑曲线的存在性、局部类群及补理论相联系。最后，我们提出了关于高维情形、补与有界性、模空间、Cremona群、交换代数、曲线计数理论及正特征的问题与未来研究方向。
英文摘要
We develop a theory of singularities of rational maps, focusing on maps $$ f\colon X\dashrightarrow \mathbb P^n, $$ and using their polarised graphs and normalised polarised graphs even when the source $X$ is very singular. To measure singularities of the map at a point $x\in X$, i.e. how far it is from being regular, we introduce invariants including the normalised graph fibre degree $δ_x(f)$, a generalised lc threshold $λ_x(f)$, and invariants measuring the singularities of the normalised graph itself. Numerous examples show that the resulting invariants measure genuinely different aspects of map singularities. We investigate the surface case in detail. We relate the normalised graph fibre degree to multiplicity, prove a sharp threshold--degree inequality for klt surface germs, and develop a detailed theory of linear type maps on smooth and singular surfaces. In particular, we connect the existence of linear type maps to existence of smooth curves through the given point, and with local class groups and complement theory. We conclude with questions and future directions concerning higher dimensions, complements and boundedness, moduli, Cremona groups, commutative algebra, curve-counting theories, and positive characteristic.
Comments63 pages
