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
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-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.
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