On specialization of curves. II
[For part I see Trans. Am. Math. Soc. 282, 739-748 (1984; Zbl 0537.14019).] The following inequality is proved: let \(p: X\to T\) be a flat morphism of complex algebraic varieties, where T is integral (i.e., irreducible and reduced) and the fibers are projective, one dimensional schemes (a family of curves). Assume the geometric general fiber A is an integral curve of geometric genus g and, for \(0\in T\), the fiber \(X_ 0\) has associated cycle \(\sum^{s}_{i=1}k_ iC_ i,\) where the geometric genus \(g_ i\) of \(C_ i\) is \(\neq 0\) if and only if \(i\leq r\leq s\) for a suitable number r, \(0\leq r\leq s\). Then, \(g\geq \sum^{s}_{i=1}(k_ ig_ i-k_ i+1).\) In the paper it is assumed that the fibers are Cohen-Macaulay curves, but this hypothesis is never used in the proof. Also, lemma 1.5 is incorrect as stated: one must assume that ''D is connected''. Then the given proof, with minor changes, works. The new assumption holds where the lemma is used. The paper also contains a proof of a partial converse. Given a plane curve \(X_ 0\) of degree n, having associated cycle \(\sum k_ iC_ i\) and given an integer g, \(\sum k_ ig_ i\leq g\leq (n-1)(n- 2)\) (where \(g_ i\) is the geometric genus of \(C_ i)\), then there is a family of plane curves \(X\to T\), whose general member is an integral curve of geometric genus g, having only nodes as singularities. There are also some remarks about the ''missing'' cases, i.e., \(\sum k_ ig_ i- (k_ i-1)\leq g<k_ ig_ i,\quad the\) general problem being reduced to a special situation. - These results appear in an old paper of Albanese, the present article gives a modern presentation of the theory.
scientific article; zbMATH DE number 3952942