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The cohomology of Kontesevich’s stack of stable maps to Pⁿ, the case of conics O’Halloran, Anne Fionnuala

Abstract

In this thesis we consider the singular cohomology of M₀,₀(Pⁿ,2), the coarse moduli space associated to Kontsevich's stack of degree two stable maps to Pⁿ, M₀,₀(Pⁿ,2). We show that the cohomology ring is generated by a divisor d which corresponds to the locus of pairs (C,g) with C reducible, and the first and second Chern classes, c₁ and c₂, of the canonical rank three vector bundle E = π۰f0(1) on M₀,₀(Pⁿ,2), where π is the canonical projection associated to the universal curve C and f is the universal map. We give the cohomology ring as a quotient of a polynomial ring in these generators. The relations are in degrees n, n + 1 and n + 2. We also give a representation of the cohomology ring in terms of the Chern roots of E. The results are conjectural for n >> 0.

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