On the geometric realization of Albanese's inequality (Q1814108)

Consider a flat family of projective schemes with general fiber an integral curve of geometric genus \(g\) and with a special fiber whose underlying 1-cycle is \(\sum m_ iB_ i\). Albanese's inequality is \(g\leq\sum\varepsilon_ i(m_ ig_ i-m_ i+1)\), where \(g_ i\) is the geometric genus of \(B_ i\) and \(\varepsilon_ i=\min(1,g_ i)\). The author examines the ``converse'' to this inequality, i.e. given an effective 1-cycle \(\sum m_ iB_ i\) in \(\mathbb{P}^ n\) and a number \(g\) which satisfies Albanese's bound, does there exist a flat family of curves as above with general fibre of geometric genus \(g\) and special fiber with underlying 1-cycle \(\sum m_ iB_ i\)? Partial answers are given under additional hypotheses involving the graph of the curve \(\bigcup B_ i\) and the vanishing of certain cohomology groups. A weaker statement is also obtained: if \(g\), \(g_ i\) and \(m_ i\) are given satisfying Albanese's inequality, then there exists a flat family of curves with smooth general fiber of genus \(g\) and special fiber with underlying 1-cycle \(\sum m_ iB_ i\), where \(B_ i\) is a projective curve of geometric genus \(g_ i\).