Characterizing Curves by Their Odd Theta-characteristics

Consider, for every g ≥ 4, a non-singular, complex, canonical curve of genus g, that is, a curve X embedded in P by its complete canonical series |ωX |. Each such a curve possesses 2 2g line bundles L of degree g − 1 such that L = ωX , the theta-characteristics of X, and precisely Ng := ( 2 2 ) of them are odd (i.e. h(X,L) is odd). To a non-zero section σ of a theta-characteristic one associates a “half canonical” divisor D = (σ), whose double 2D is cut on X by a hyperplane H in P (whose scheme-theoretic intersection with X is, of course, everywhere non reduced). For obvious reasons, such a hyperplane H will be called a theta-hyperplane of X. Assume that X is general. Then all odd theta-characteristics L satisfy h(X,L) = 1, while the even ones have no non-zero sections. Therefore X has exactly Ng theta-hyperplanes; the set of such hyperplanes will be denoted θ(X), and considered as an element of Symg (P). This said, the main result of this paper is (Theorem 6.1.1)