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.

25 pages, 7 figures

References in corpus (1)

• Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves

Cited by in corpus (1)

• When are multidegrees positive?