  Let be a general curve of genus g embedded via a general linear series of
degree d in P^r. The well-known Maximal Rank Conjecture asserts that the
restriction maps H^0(O_{P^r}(m)) \to H^0(O_C(m) are of maximal rank; if known,
this conjecture would determine the Hilbert function of C. In this paper, we
prove an analogous statement for the hyperplane sections of unions general
curves. More specifically, if H is a general hyperplane, we show that
H^0(O_H(m)) \to H^0(O_{(C_1 \cup C_2 \cup \cdots \cup C_n) \cap H}(m)) is of
maximal rank, except for some counterexamples when m = 2.
  As explained in arXiv:1809.05980, this result plays a key role in the
author's proof of the Maximal Rank Conjecture.
