  We prove that the Cauchy problem of the mass-critical generalized KdV
equation is globally well-posed in Sobolev spaces $H^s(\R)$ for $s>6/13$. Of
course, we require that the mass is strictly less than that of the ground state
in the focusing case. The main approach is the "I-method" together with the
multilinear correction analysis. Moreover, we use some "partially refined"
argument to lower the upper control of the multiplier in the resonant
interactions. The result improves the previous works of Fonseca, Linares, Ponce
(2003) and Farah (2009).
