Stability in the homology of Deligne-Mumford compactifications

Abstract

Using the the theory of FS op modules, we study the asymptotic behavior of the homology of M g,n , the Deligne-Mumford compactification of the moduli space of curves, for n ≫ 0. An FS op module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via maps that glue on marked P 1 's, we give the homology of M g,n the structure of an FS op module and bound its degree of generation. As a consequence, we prove that the generating function n dim(H i (M g,n ))t n is rational, and its denominator has roots in the set {1, 1/2, . . . , 1/p(g, i)} where p(g, i) is a polynomial of order O(g 2 i 2 ). We also obtain restrictions on the decomposition of the homology of M g,n into irreducible S n representations.

Categorifications of rational Hilbert series and characters of $FS^{op}$ modules

“…For i, g ∈ N, the sequence of S n representation n → H i (M g,n , Q), where M g,n is the moduli space of marked stable curves, can be extended to an FS op module. In [28] we showed that this FS op module is a subquotient of one that is finitely generated in degree ≤ 8g 2 i 2 + 29g 2 i + 16gi 2 + 21g 2 + 10gi.…”

“…Proudfoot-Young used FS op to study the Kazhdan-Lusztig polynomial of the braid matroid, by constructing an action of FS op on the intersection homology of the dual reciprocal plane of the braid arrangement [17]. In [28], we constructed an FS op module structure on the homology of the moduli space of stable curves. And Proudfoot-Ramos constructed FS op modules in the cohomology of the resonance arrangement [20].…”