Topological Bounds on Certain Open Subvarieties of the compactfied Moduli Space of Curves.

다국어 초록 (Multilingual Abstract)

Let M⩽kg⊂M g be the moduli of stable curves with at most k rational components, and H⩽g=Hg ∩M⩽kg . We study the affine stratification number (asn) of H⩽kg . Our main result here is asn H⩽kg ≤ g − 1 + k. Conjecturally this bound holds also for M⩽kg . We then investigate the cohomological excess (ce) of H⩽0g , and show that ce( H⩽0g ) ≥ g − 1. We prove this by demonstrating a natural constructible sheaf on H⩽kg , which has non-vanishing cohomology in the correct degree. Our result in turn shows that the upper bound we showed for asn H⩽kg is sharp when k = 0. Our result also shows that ce( M⩽0g ) ≥ g − 1.
The fact that Hg is the quotient of M0,2 g+2 by a symmetric group, is used throughout. We give a technique to relate the strata of M 0,2g+2, to that of Hg. The proof of the main theorem involves analysing a spectral sequence in compactly supported cohomology, which has a natural symmetric group action on it. We study this spectral sequence using the operad structure on the cohomology of M 0,2g+2 and investigating the symmetric group action which yields the result. The associated operads are the gravity and hypercommutative operads which were shown to be Koszul duals of each other in Getzler. We utilise that and give an alternative proof of the fact in this thesis.