On the Constellations of Weierstrass Points

We prove that the constellation of Weierstrass points characterizes the isomorphism-class of double coverings of curves of genus large enough. 1. Let X be a projective, irreducible, non-singular algebraic curve defined over an algebraically closed field k of characteristic p. Let n ≥ 1 be an integer and CX a canonical divisor of X. The pluricanonical linear system |nCX | defines a nondegenerate morphism πn : X → P , where N(1) = g − 1, and N(n) = (2n− 1)(g − 1)− 1 for n ≥ 2. To any P ∈ X we then associate the sequence of multiplicities {vP (π ∗ n(H)) : H hyperplane ⊆ P } = {ǫ0(P ) < ǫ1(P ) < . . . < ǫN(n)(P )}. Such a sequence is the same for all but finitely many points (cf. [F-K, III.5], [Lak, Prop.3], [S-V, §1]). These finitely many points are the so called n-Weierstrass points of X. There exists a divisor Wn on X whose support is the set of n-Weierstrass points and satisfies the property below. Let denote by vP (Wn) the coefficient ofWn in P (called the n-Weierstrass weight at P ). Then