Degenerations and multiplicity-free formulas for products of ψ and ω classes on M¯0,n

In this paper, we consider products of ψ and ω classes on M ¯ 0 , n + 3 . For each product, we construct a flat family of subschemes of 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 positive 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 : M ¯ 0 , n + 3 → P 1 × ⋯ × P n . In the case of the product ω 1 ⋯ 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.