AI 中文总结
该研究为曲线上的商概型引入乔伊斯枚举不变量，结合顶点代数框架证明其Virasoro约束，并用于计算商概型上$f$-类的虚相交数。
AI 中文摘要
设$C$是复域$\boldsymbol{C}$上的光滑射影曲线，$E$是$C$上的向量丛，记$\text{Quot}_{r,d}(E)$为参数化$E$的秩为$r$、次数为$d$的商的商概型。遵循Joyce的方法[Joy21]，我们为商概型$\text{Quot}_{r,d}(E)$引入乔伊斯枚举不变量，该不变量可视为商概型虚基本循环的推广。我们通过显式计算$\text{Quot}_{\text{rank}(E)-1,d}(E)$的不变量，来评估该商概型上的相交配对。遵循[BLM24]中层论Virasoro约束在乔伊斯顶点代数框架下的重新表述，我们给出商概型$\text{Quot}_{r,d}(E)$的Virasoro约束的证明。借助这些约束，我们计算商概型上$f$-类的（虚）相交数。
英文摘要
Let $C$ be a smooth projective curve over $\mathbb C$ and let $E$ be a vector bundle over $C$. Let $\text{Quot}_{r,d}(E)$ denote the Quot scheme which parametrizes quotients of $E$ of rank $r$ and degree $d$. Following Joyce's recipe [Joy21], we introduce Joyce's enumerative invariant for the Quot scheme $\text{Quot}_{r,d}(E)$. The invariant can be viewed as a generalization of the virtual fundamental cycle of the Quot scheme. We evaluate intersection pairings on the Quot scheme $\text{Quot}_{\text{rank}(E)-1,d}(E)$ by computing its invariant explicitly. Following the reformulation of sheaf-theoretic Virasoro constraints in terms of Joyce's vertex algebra framework in [BLM24], we give a proof of the Virasoro constraints for the Quot scheme $\text{Quot}_{r,d}(E)$. With the help of these constraints, we compute the (virtual) intersection numbers of $f$-classes on the Quot schemes.
CommentsComments are welcome
