  In this paper, we consider the bilinear Strichartz estimates for the periodic
KdV equation. We give a concrete counterexample to the false $L^p$ Strichartz
estimates for $p=8$, at least for a subset of the range of Lebesgue exponents
$p$.Moreover, we prove the bilinear Strichartz estimate in different frequency
regions, which represents a kind of a smoothing effects. When the period tends
to infinity, the frequency localized version of Strichartz estimate on the
torus matches to the one in the real line setting. In addition, the quartic
generalized KdV equation is considered as an application.
