论文标题
相对稳定地图的模量为$ \ mathbb {p}^1 $：剪切不变式
Moduli of relative stable maps to $\mathbb{P}^1$: cut-and-paste invariants
论文作者
论文摘要
我们研究模量空间的可构造不变式$ \覆盖{\ Mathcal {m}}（\ BoldSymbol {x x}）$从零属曲线到$ \ Mathbb {p}^1 $的稳定地图，相对于$ 0 $ 0 $ and $ \ fyfty $} \ mathbb {z}^n} $。这些空间对于$ \ mathbb {p}^1 $的枚举几何形状至关重要，并提供了deligne--mumford-knudsen moduli space $ \ overline {\ mathcal {\ mathcal {m}} _ _ {0，n} $的大家族。对于向量的顺序，$ \ boldsymbol {x} $对应于与$ 0 $上的最大分支并在$ \ infty $上毫无疑问的地图，我们证明，这些空间的拓扑效率特征的生成函数满足了一个不同的方程式，可以满足其递归计算。我们还表明，在$ \ boldsymbol {x} $中，变种环中的模量空间的类是恒定的，在固定的腔室内在$ \ mathbb {z}^n $的谐振分解中变化。最后，我们在研究这些空间的研究中提出了进一步的多个方向，从而给出了（1）Euler特征的渐近行为的猜想，以及（2）Chern数字的潜在腔室结构。
We study constructible invariants of the moduli space $\overline{\mathcal{M}}(\boldsymbol{x})$ of stable maps from genus zero curves to $\mathbb{P}^1$, relative to $0$ and $\infty$, with ramification profiles specified by ${\boldsymbol{x}\in \mathbb{Z}^n}$. These spaces are central to the enumerative geometry of $\mathbb{P}^1$, and provide a large family of birational models of the Deligne--Mumford--Knudsen moduli space $\overline{\mathcal{M}}_{0,n}$. For the sequence of vectors $\boldsymbol{x}$ corresponding to maps which are maximally ramified over $0$ and unramified over $\infty$, we prove that a generating function for the topological Euler characteristics of these spaces satisfies a differential equation which allows for its recursive calculation. We also show that the class of the moduli space in the Grothendieck ring of varieties is constant as $\boldsymbol{x}$ varies within a fixed chamber in the resonance decomposition of $\mathbb{Z}^n$. We conclude by suggesting several further directions in the study of these spaces, giving conjectures on (1) the asymptotic behavior of the Euler characteristic and (2) a potential chamber structure for the Chern numbers.
