On smooth curves spin structures (or theta characteristics) are square roots of the canonical bundle. The algebraic rank h^0(C,L) of spin structures L on a curve C is a Z_2-quadratic form. For tropical curves the combinatorial rank provides an analogue of h^0(C,L), which will be recalled during the talk. This invariant is not Z_2-quadratic. I will discuss the case of hyperelliptic curves, where the claim still holds naturally and turns out to be quite interestingly related to the moduli functor and its stratification.

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