Combinatorial Theory

We consider the (iterated) Kapranov embedding \(\Omega_n:\overline{M}_{0,n+3} \hookrightarrow \mathbb{P}^1 \times \cdots \times \mathbb{P}^n\), where \(\overline{M}_{0,n+3}\) 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 \(\Omega_n\) and showed, using two combinatorial insertion algorithms on certain parking functions, that the total degree of \(\Omega_n\) is \((2n-1)!!=(2n-1)\cdot (2n-3) \cdots 5 \cdot 3 \cdot 1\). In this paper, we give a new proof of this fact by enumerating each multidegree by a set of boundary points of \(\overline{M}_{0,n+3}\), 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 \(\overline{M}_{0,n+2}\), of which there are \((2n-1)!!\), giving an immediate proof of the total degree formula.

Mathematics Subject Classifications: 05E14, 14N10, 05C05, 14H10, 05A19, 05C85

Keywords: Moduli spaces of curves, projective embeddings, multidegrees, trivalent trees