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

Abstract

We consider the (iterated) Kapranov embedding Ω n : M 0,n+3 → P 1 × • • • × P n , where M 0,n+3 is the moduli space of stable genus 0 curves with n + 3 marked points. In 2020, Gillespie, Cavalieri, and Monin gave a recursion satisfied by the multidegrees of Ω n and showed, using two combinatorial insertion algorithms on certain parking functions, that the total degree of Ω n is (2n − 1)!! = (2n − 1)In this paper, we give a new proof of this fact by enumerating each multidegree by a set of boundary points of M 0,n+3 , via an algorithm on trivalent trees that we call a lazy tournament. The advantages of this new interpretation are twofold: first, these sets project to one another under the forgetting maps used to derive the multidegree recursion. Second, these sets naturally partition the complete set of boundary points on M 0,n+2 , of which there are (2n − 1)!!, giving an immediate proof of the total degree formula.