For general curves, the set {ζ in G | H^0(C, L ⊗ ζ) ≠ 0} is empty if and only if L ⊗ ζ_G is even, where ζ_G is the nontrivial 2-torsion point of G.
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Normal Functions, Even Theta Characteristics and the Theta Divisor
For general curves, the set {ζ in G | H^0(C, L ⊗ ζ) ≠ 0} is empty if and only if L ⊗ ζ_G is even, where ζ_G is the nontrivial 2-torsion point of G.