AI 中文总结
该研究证明了紧致型模空间$\u039c_{g,n}^{\mathrm{ct}}$在特定参数下重言环非Gorenstein，并验证了Pixton猜想的多个新案例，相关结果是主极化阿贝尔簇模空间非重言闭链研究的关键基础。
AI 中文摘要
我们证明，当g≥2且2g+n≥12时，$\u039c_{g,n}^{\mathrm{ct}}$的重言环不是Gorenstein的。我们证明了Pixton猜想的新情形，即3-自旋关系是重言环的一组完全关系，包括$\u039c_{6}^{\mathrm{ct}}$、$\u039c_{5,2}^{\mathrm{ct}}$和$\u039c_7^{\mathrm{ct}}$。这些是Pixton猜想成立但重言环非Gorenstein的首批已知情形。这些结果也是关于主极化阿贝尔簇模空间上非重言闭链的近期研究的关键组成部分。
英文摘要
We show that the tautological ring of $\mathcal{M}_{g,n}^{\mathrm{ct}}$ is not Gorenstein for $g\geq 2$ and $2g+n\geq 12$. We prove new cases of Pixton's conjecture that the $3$-spin relations are a complete set of relations for the tautological ring, including $\mathcal{M}_{6}^{\mathrm{ct}}$, $\mathcal{M}_{5,2}^{\mathrm{ct}}$, and $\mathcal{M}_7^{\mathrm{ct}}$. These are the first known cases where Pixton's conjecture is true, but the tautological ring is not Gorenstein. These results are also a key ingredient in recent work on non-tautological cycles on the moduli space of principally polarized abelian varieties.
Commentsv2: preliminary version. Will be updated with results of ongoing computer calculations. Fixed numbers we transcribed in error in Theorem 35 and some other typos. Results unchanged
