Lazy tournaments and multidegrees of a projective embedding of $\overline{M}_{0,n}$

Abstract

We provide a new geometric interpretation of the multidegrees of the (iterated) Kapranov embedding Φn : M 0,n+3 → P 1 × P 2 × • • • × P n , where M 0,n+3 is the moduli space of stable genus 0 curves with n + 3 marked points. We enumerate the multidegrees by disjoint sets of boundary points of M 0,n+3 via a combinatorial algorithm on trivalent trees that we call a lazy tournament. These sets are compatible with the forgetting maps used to derive the recursion for the multidegrees proven in 2020 by Gillespie, Cavalieri, and Monin.The lazy tournament points are easily seen to total (2n − 1)!! = (2n − 1)giving a natural proof of the fact that the total degree of Φn is the odd double factorial. This fact was first proven using an insertion algorithm on certain parking functions, and we additionally give a bijection to those parking functions.

Degenerations and multiplicity-free formulas for products of $ψ$ and $ω$ classes on $\overline{M}_{0,n}$

“…We conclude with some further observations and avenues for future research, both in combinatorial directions (Sections 6.1 through 6.3) and geometric (Sections 6.4 through 6.7). The first follows from a bijection with column-restricted parking functions [2,7] which naturally satisfy the asymmetric string recursion. The second follows from counting intersection points with parametrized hyperplanes, and has the inductive structure of the slide rule.…”

“…It remains to be seen whether all tournament points admit such a geometric realization. A hint toward achieving this goal is [7,Theorem 1.8], which states that the coordinates of the points Tour(k 1 , . .…”

“…A recursive formula for them was previously given in [2], as well as a combinatorial interpretation via parking functions. In [7], it was also shown that a different set of boundary points called Tour(k) also enumerates the multidegrees deg k (Ω n ). These points are defined combinatorially via an algorithm called a lazy tournament, and we will recall the definition in Section 5 below.…”

“…The first map is the combined or total Kapranov map given by the psi classes, while the second map, sometimes called the iterated Kapranov map (see [2,7,14,16]), is an embedding and is given by the omega classes. Hyperplane sections of these maps represent the intersection products (1.1) in A • (M 0,n+3 ) above.…”

“…When k i = n, it is well-known that the product of psi classes ψ k is the multinomial coefficient n k 1 ,...,kn times the class of a point. The product of omega classes ω k is the so-called asymmetric multinomial coefficient n k 1 ,...,kn times the class of a point [2,7].…”