发表机构
Beijing Jiaotong University（北京交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了光滑射影复曲线上使Polishchuk--Van den Bergh猜想A失败的有限群作用，并给出最大正交例外对象数及线性嵌入分解的充要条件。
AI 中文摘要
我们构造了光滑射影复曲线上的有效有限群作用，使得Polishchuk和Van den Bergh的猜想A不成立。对于每个$h\geq1$，在$y^2=t^{12h-5}+t$的光滑射影模型上，一个显式的$S_3$作用具有亏格$h$的商，且恰好有一个分歧轨道，其惯性阶为三。三循环类的固定商由两个点组成，但其导出范畴不允许$\uCdC3$-线性精确全忠实嵌入到等变导出范畴中。更一般地，当粗商具有正亏格时，我们证明两两完全正交的例外对象的最大数量为$\sum_i\lfloor e_i/2\rfloor$，其中$e_i$是惯性阶。我们还刻画了在粗商上线性嵌入的共轭类分解：这样的分解存在当且仅当对于每个非单位共轭类和每个分支惯性群$H_i$，有$|[g]\cap H_i|\leq\lfloor e_i/2\rfloor$。充分性证明使用了$A$型彩色路径和导出反射函子，并在自然纤维积上产生核。已知的有序曲线分解仍然产生由固定商的连通分量索引的分解。
英文摘要
We construct effective finite group actions on smooth projective complex curves for which Conjecture A of Polishchuk and Van den Bergh fails. For every $h\geq1$, an explicit $S_3$ action on the smooth projective model of $y^2=t^{12h-5}+t$ has quotient of genus $h$ and exactly one ramification orbit, with inertia of order three. The fixed quotient for the three-cycle class consists of two points, but its derived category admits no $\C$-linear exact fully faithful embedding into the equivariant derived category. More generally, when the coarse quotient has positive genus, we prove that the maximum number of pairwise completely orthogonal exceptional objects is $\sum_i\lfloor e_i/2\rfloor$, where $e_i$ are the inertia orders. We also characterize conjugacy-class decompositions with embeddings linear over the coarse quotient: such a decomposition exists if and only if $|[g]\cap H_i|\leq\lfloor e_i/2\rfloor$ for every nonidentity conjugacy class and every branch inertia group $H_i$. The sufficiency proof uses coloured paths of type $A$ and derived reflection functors, and produces kernels on the natural fibre products. The known ordered curve decomposition still yields a decomposition indexed by the connected components of the fixed quotients.
CommentsAll comments are welcome!
