Rational curves and maximal rank in multiprojective spaces

We study the multigraded Hilbert function of general rational curves of prescribed multidegree contained in a multiprojective space Y. We have strong negative results, e.g. for almost all line bundles L on Y there is a multidegree such that maximal rank with respect to L fails for all smooth rational curves C of that multidegree. This is different from the case of general rational curves in projective spaces. We also have positive results, most of them being for the two-factor case with one factor of dimension 1.