  Kempf proved that the theta divisor of a smooth projective curve C has
rational singularities. In this paper we estimate the dimensions of the jet
schemes of the theta divisor and show that all these schemes are irreducible.
In particular, we recover Kempf's theorem in this way. For general projective
smooth curves, our method also gives a formula for the log canonical threshold
of the pair (\pic^d(C), W^r_d(C)).
