Thomae's formulae for non-hyperelliptic curves and spinorial square roots of theta-constants on the moduli space of curves

Determinantal formulae for Jacobian theta functions that go back to Klein are elaborated, via an idea due to Matone and Volpato. Also, the natural square roots of theta constants on the moduli space of curves whose existence was shown by Tsuyumine are proved to have a spinorial structure.