Intersection theory on moduli spaces of curves ... - User Web Pages

1.2. <strong>Intersection</strong> <strong>theory</strong> on moduli spaces 17Therefore, the family we have described is precisely the universal family C 0,4 → M 0,4 . However,as noted earlier, one may interpret this family as the forgetful map M 0,5 → M 0,4 . So wemay now conclude that M 0,5 is equal to CP 1 × CP 1 blown up at the three points (0, 0), (1, 1)and (∞, ∞).Example 1.7 (The moduli space M 1,1 ). The uncompactified moduli space M 1,1 is essentiallythe moduli space of complex elliptic curves. Every elliptic curve arises as a 2-fold branchedcovering of CP 1 doubly ramified over ∞. Topological considerations force such a covering tohave three additional ramification points which we can send to 0, 1 and λ /∈ {0, 1, ∞}. The affineequation for such a curve is y 2 = x(x − 1)(x − λ), and we let the marked point correspond tothe unique point on the curve over ∞. Note that there was some choice in normalising theramification points, so we expect an S 3 symmetry to appear. Concretely, this manifests as oneof the six transformations given by λ ↦→ λ, λ ↦→λ 1 1λ−1, λ ↦→ 1 − λ, λ ↦→1−λ, λ ↦→λandλ ↦→λ−1 λ . This symmetry can be dealt with by associating to each curve y2 = x(x − 1)(x − λ)its j-invariantj(λ) = 256 (λ2 − λ + 1) 3λ 2 (λ − 1) 2 ,which satisfies j(λ) = j(λ ′ ) if and only if λ ′ ∈ {λ,λ 1 , 1 − λ, 11−λ , λ−1λ , λλ−1}. Conversely, it iswell-known that the j-invariant distinguishes between elliptic curves which are not isomorphic.Therefore, we deduce that M 1,1∼ = C.There is precisely one stable nodal curve of genus 1 with one marked point and this correspondsprecisely to the j → ∞ limit, so M 1,1∼ = CP 1 . Our discussion up to this point has completelyignored the orbifold — or stack — structure of M 1,1 . The case of M 1,1 is exceptional since everygenus 1 curve with one marked point has at least one non-trivial automorphism — namely, theelliptic involution. Furthermore, the curve with j-invariant 0 actually has an automorphismgroup isomorphic to Z 6 while the curve with j-invariant 1728 has an automorphism groupisomorphic to Z 2 × Z 2 . Therefore, every point of M 1,1 is an orbifold point of order 2, apartfrom one point of order 4 and one point of order 6.1.2 <strong>Intersection</strong> <strong>theory</strong> on moduli spacesCharacteristic classesGiven a space with interesting topological structure, one of the first compulsions of a geometeris often to calculate associated algebraic invariants. And so it is with moduli spaces of curves,except for the fact that their homology and cohomology are notoriously intractable in general.However, a great deal of progress can be made in this direction if one is willing to adopt thefollowing two simplifications.