THE SIGMA FUNCTION OVER A FAMILY OF CURVES WITH A SINGULAR FIBER

In this paper we investigate the behavior of the sigma function over the family of cyclic trigonal curves X-s defined by the equation y(3) = x(x - s)(x - b(1))(x - b(2)) in the affine (x, y) plane, for s is an element of D epsilon := {s is an element of C parallel to s vertical bar < epsilon}. We compare the sigma function over the punctured disc D-epsilon*= D-epsilon \ {0} with the extension over s = 0 that specializes to the sigma function of the normalization X-<(0)over cap> 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. We demonstrate, using modular properties, that sigma, unlike the theta function, has a limit. In particular, we obtain the limit of the theta characteristics and an explicit description of the theta divisor translated by the Riemann constant.