论文标题
在稳定曲线的模量空间的盘子和同胞环上
On the Chow and cohomology rings of moduli spaces of stable curves
论文作者
论文摘要
在本文中，我们问：$ \ overline {\ mathcal {m}} _ {g，n} $是由$ \ overline {\ mathcal {\ mathcal {\ mathcal {\ mathcal {\ mathcal {\ mathcal {\ mathcal {g，n} $ of哪个$（g，n）$？ Keel（Chow and Coomomology Rings at All $ n $均进行了重言式）和Beloofski的属（仅当$ N \ leq 10 $时，戒指是重言式）。对于$ g \ geq 2 $，范·泽尔姆（Van Zelm）的作品表明，一旦$ 2G + n \ geq 24 $，Chow和Sopomology戒指就不是重言式，剩下有限的许多开放式案例。在这里，我们证明了$ \叠加的杂烩和同胞环{\ Mathcal {m}} _ {g，n} $是同构的，是由训to术的生成$ g = 2 $和$ n \ leq 9 $和$ n \ leq 9 $和$ 3 \ leq g \ leq g \ leq 7 $和$ 2G + n $和leq + n $ n $ $ g = 2 $和$ n \ leq + n $。对于这样的$（g，n）$，这意味着重言式戒指是Gorenstein，$ \ overline {\ Mathcal {m}} _ {g，n} $具有多项式点数。
In this paper, we ask: for which $(g, n)$ is the rational Chow or cohomology ring of $\overline{\mathcal{M}}_{g,n}$ generated by tautological classes? This question has been fully answered in genus $0$ by Keel (the Chow and cohomology rings are tautological for all $n$) and genus $1$ by Belorousski (the rings are tautological if and only if $n \leq 10$). For $g \geq 2$, work of van Zelm shows the Chow and cohomology rings are not tautological once $2g + n \geq 24$, leaving finitely many open cases. Here, we prove that the Chow and cohomology rings of $\overline{\mathcal{M}}_{g,n}$ are isomorphic and generated by tautological classes for $g = 2$ and $n \leq 9$ and for $3 \leq g \leq 7$ and $2g + n \leq 14$. For such $(g, n)$, this implies that the tautological ring is Gorenstein and $\overline{\mathcal{M}}_{g,n}$ has polynomial point count.
