On specializations of curves. I

Abstract

Abstract.The following is proved: Given a family of projective reduced curves X -T ( T irreducible), if X, (the general curve) is integral and X0 is a special curve (having irreducible components Y,.Xr), then Z'=-, g,(Xf) « g(Y,), where g(Z) = geometric genus of Z. Conversely, if A is a reduced plane projective curve, of degree n with irreducible components Y,.Xr, and g satisfies 2'= | g,(X,) *£ g 'S .¡(n -1)(« -2), then a family of plane curves Y -7" (with 7" integral) exists, where for some tn G 7", Y, = Z and for r generic, X, is integral and has only nodes as singularities. Results of this type appear in an old paper by G. Albanese, but the exposition is rather obscure.

Genera of curves varying in a family

“…One can give a proof of (1.2) based on the results of [7], a remarkable but technical paper. But in the geometric case of interest to us one can give also a rather short and simple proof based on the semi-stable reduction theorem [as suggested in [9], (1.10)], and this is what is done in paragraph 1 of the present paper.…”

“…In general we shall use the same letter to indicate a curve and the corresponding point of P 1^. The basic facts on Severi varieties that we need are summarized in [8] (3.1), see also [10] for the details (however, here we shall use a slightly different notation).…”

“…(3.8) In case fe=l, Theorem (2.6) gives another proof of the theorem that says that any integral plane curve of degree d is a specialization of a nodal plane curve of the same degree and genus (cf. [8], §4).…”

“…[II], p. 216). Modern papers on the theory are [8] (where the case where B is reduced is discussed) and [9], where a proof of the inequality is presented, and the existence theorem is verified when r g ^ ^ niigi (where B^, . .…”

“…are the components of B with ^i>0). However, the 1=1 proof of the inequality given in [9] [Theorem (1.2) there] contains an error [Lemma (1.5) of [9] is false]. One can give a proof of (1.2) based on the results of [7], a remarkable but technical paper.…”