Projective Embeddings of $\overline{M}_{0,n}$ and Parking Functions

Abstract

The moduli space $\overline{M}_{0,n}$ may be embedded into the product of projective spaces $\mathbb{P}^1\times \mathbb{P}^2\times \cdots \times \mathbb{P}^{n-3}$, using a combination of the Kapranov map $|\psi_n|:\overline{M}_{0,n}\to \mathbb{P}^{n-3}$ and the forgetful maps $\pi_i:\overline{M}_{0,i}\to \overline{M}_{0,i-1}$. We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height $n-3$. We use this combinatorial interpretation to show that the total degree of the embedding (thought of as the projectivization of its cone in $\mathbb{A}^2\times \mathbb{A}^3\cdots \times \mathbb{A}^{n-2}$) is equal to $(2(n-3)-1)!!=(2n-7)(2n-9) \cdots(5)(3)(1)$. As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial.