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Theta‐characteristics on singular curves

On a smooth curve a theta‐characteristic is a line bundle L, the square of which is the canonical line bundle ω. The equivalent condition ℋom(L, ω) ≅ L generalizes well to singular curves, as applications show. More precisely, a theta‐characteristic is a torsion‐free sheaf ℱ of rank 1 with ℋom(ℱ, ω) ≅ ℱ. If the curve has non‐ADE singularities, then there are infinitely many theta‐characteristics. Therefore, theta‐characteristics are distinguished by their local type. The main purpose of this article is to compute the number of even and odd theta‐characteristics (that is ℱ with h0(C, ℱ)≡ 0 and h0(C, ℱ)≡ 1 modulo 2, respectively) in terms of the geometric genus of the curve and certain discrete invariants of a fixed local type.