Very ample line bundles on reducible curves

Here we study the very ample line bundles and the birationally very ample line bundles on a reduced but reducible projective curve X (e.g. classify all stable curves of compact type on which every very ample line bundle induces an arithmetically Cohen–Macaulay embedding and the genus g stable curves without a very ample line bundle of degree at most 3g−1). We also give a condition implying that two morphisms f :X→P1 and h : X→P1 factor through the same involution.