Séminaire Lotharingien de Combinatoire, 84B.32 (2020), 12 pp.

Renzo Cavalieri, Maria Gillespie and Leonid Monin

Projective Embeddings of M-0,n and Parking Functions

Abstract. The moduli space M-0,n may be embedded into the product of projective spaces P1 x P2 x ... x Pn-3, using a combination of the Kapranov map M-0,n -> Pn-3 and the forgetful maps π_i : M-0,i -> M-0,i-1. We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions.

This combinatorial interpretation provides a recursive formula for the generating function of the multidegree. We further show that the total degree of the embedding (thought of as the projectivization of its cone in A2 x A3 x ... x An-2) is equal to (2(n-3)-1)!!=(2n-7)(2n-9)...(5)(3)(1). As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

The following versions are available: