论文标题
$ \ overline {m} _ {0，n} $的$ψ$和$ω$类产品的退化和不含多重公式
Degenerations and multiplicity-free formulas for products of $ψ$ and $ω$ classes on $\overline{M}_{0,n}$
论文作者
论文摘要
我们考虑$ \ Overline {M} _ {0，N+3} $的$ψ$类和$ω$类的产品的产品。对于每种产品，我们构建了$ \叠加{M} _ {0，n+3} $的纯种族，其一般纤维是代表产品的完整交叉点，其特殊纤维是边界层的一般减少的结合。我们的构造是使用超平面截面的明确的参数化集合作为一组参数退化的序列构建的。合并，我们的构造将每种产品表示为无数性的边界层类别总和。这些由我们称为“滑动标签”的树上的组合算法给出。作为推论，我们就边界层获得了$κ$类别的组合公式。 For degree-$n$ products of $ω$ classes, the special fiber is a finite reduced union of (boundary) points, and its cardinality is one of the multidegrees of the corresponding embedding $Ω_n: \overline{M}_{0,n+3}\to \mathbb{P}^1\times \cdots \times \mathbb{P}^n$.对于产品$ω_1\ cdotsω_n$，这些点与置换模式避免相关。最后，我们表明，在某些情况下，还可以通过退化来获得对多视频的事先解释。
We consider products of $ψ$ classes and products of $ω$ classes on $\overline{M}_{0,n+3}$. For each product, we construct a flat family of subschemes of $\overline{M}_{0,n+3}$ whose general fiber is a complete intersection representing the product, and whose special fiber is a generically reduced union of boundary strata. Our construction is built up inductively as a sequence of one-parameter degenerations, using an explicit parametrized collection of hyperplane sections. Combinatorially, our construction expresses each product as a positive, multiplicity-free sum of classes of boundary strata. These are given by a combinatorial algorithm on trees we call 'slide labeling'. As a corollary, we obtain a combinatorial formula for the $κ$ classes in terms of boundary strata. For degree-$n$ products of $ω$ classes, the special fiber is a finite reduced union of (boundary) points, and its cardinality is one of the multidegrees of the corresponding embedding $Ω_n: \overline{M}_{0,n+3}\to \mathbb{P}^1\times \cdots \times \mathbb{P}^n$. In the case of the product $ω_1\cdots ω_n$, these points exhibit a connection to permutation pattern avoidance. Finally, we show that in certain cases, a prior interpretation of the multidegrees via tournaments can also be obtained by degenerations.
