  We show that the $\tau$-functions of the regular KP solitons from the totally
nonnegative Grassmannians can be expressed by the Riemann theta functions on
singular curves. We explicitly write the parameters in the Riemann theta
function in terms of those of the KP soliton. We give a short remark on the
Prym theta function on a double covering of singular curves. We also discuss
the KP soliton on quasi-periodic background, which is obtained by applying the
vertex operators to the Riemann theta function.
