General cyclic covers and their Thomae formula

Let $X$ be a general cyclic cover of ${\mathbb{C}\mathbb{P}}^1$ ramified at $m$ points, ${\lambda }_1\mathrm{.}\mathrm{.}\mathrm{.}{\lambda }_m\mathrm{.}$ we define a class of non positive divisors on $X$ of degree $g-1$ supported in the pre images of the branch points on $X$, such that the the standard theta function doesn’t vanish on their image in $J(X)\mathrm{.}$ These divisors generalize the divisors introduced in [BR] and [Na]. Generalizing the results of [BR],[Na] and [EG] we show that up to a certain determinant of the non standard periods of $X$, the value of the theta functions at these divisors is a polynomial in the branch point of the curve $X\mathrm{.}$ Our treatment is based on a generalization of Accola’s results of the 3 cyclic sheeted cover [Ac1] and a straightforward generalization of Nakayashiki’s approach explained in [Na] in the non singular case for any singular cyclic cover.