Abstract
In this paper, we consider products of \(\psi \) and \(\omega \) 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 positive combinatorial formula for the \(\kappa \) classes in terms of boundary strata. For degree-n products of \(\omega \) classes, the special fiber is a finite reduced union of (boundary) points, and its cardinality is one of the multidegrees of the corresponding embedding \(\Omega _n:\overline{M}_{0,n+3}\rightarrow \mathbb {P}^1\times \cdots \times \mathbb {P}^n\). In the case of the product \(\omega _1\cdots \omega _n\), 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.
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Acknowledgements
We thank Vance Blankers, Renzo Cavalieri, and Mark Shoemaker for several helpful discussions pertaining to this work. We also thank an anonymous referee for helpful comments.
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Gillespie, M., Griffin, S.T. & Levinson, J. Degenerations and multiplicity-free formulas for products of \(\psi \) and \(\omega \) classes on \(\overline{M}_{0,n}\). Math. Z. 304, 56 (2023). https://doi.org/10.1007/s00209-023-03313-7
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DOI: https://doi.org/10.1007/s00209-023-03313-7