Round 6: Tossup 15

Given a value of this quantity, Deligne (“duh-LEEN”) and Mumford used “stacks” to prove the irreducibility of moduli spaces of algebraic curves. When this quantity is N for an object, the space of “marked complex structures” has dimension 3N minus 3 and is named for Teichmüller. A dimension term equals the degree of a divisor minus this quantity plus one by the Riemann–Roch (10[1])theorem. (10[3])For an irreducible plane curve of degree d, this quantity equals “d minus 1 times (10[1])d minus 2, all over 2,” and as such is 1 for elliptic curves. The formula (10[1])“3 times this quantity plus the number of boundaries minus 3” (-5[1])determines a compact surface’s decomposition into “pairs of pants.” Two minus twice this (10[1]-5[1])quantity (-5[1])equals (10[1])a closed orientable object’s Euler characteristic. (10[2]-5[1])For 10 points, name (10[1])this (10[1])topological invariant (10[1])that characterizes how many “holes” (10[2])an object has. ■END■ (10[1]0[4])

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