We consider a pointed curve (X, P ) which is given by the Weierstrass normal form,where x is an affine coordinate on P 1 , the point ∞ on X is mapped to x = ∞, and each A j is a polynomial in x of degree ≤ js/r for a certain coprime positive integers r and s (r < s) so that its Weierstrass non-gap sequence at ∞ is a numerical semigroup. It is a natural generalization of Weierstrass' equation in the Weierstrass elliptic function theory. We investigate such a curve and show the Jacobi inversion formulae of the strata of its Jacobian using the result of Jorgenson [Jo].
In this paper we investigate the behavior of the sigma function over the family of cyclic trigonal curves X s defined by the equationin the affine (x, y) plane, for s ∈ D ε := {s ∈ C||s| < ε}. We compare the sigma function over the punctured disc D * ε := D ε \ {0} with the extension over s = 0 that specializes to the sigma function of the normalization X 0 of the singular curve X s=0 by investigating explicitly the behavior of a basis of the first algebraic de Rham cohomology group and its period integrals. Since the sigma function of X s determines the structure of the line bundle L Js (corresponding to its zero divisor, a translate of the theta divisor) on its Jacobian J s , and that of X 0 also describes the corresponding line bundle L J 0 , our work gives an explicit description of the line bundle over the Jacobians, extended over the disc D ε .
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