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On nodal plane curves. (English) Zbl 0644.14009
Denote by \(V(d,g)\) the family of all irreducible nodal algebraic plane curves of degree \(d\) and genus \(g.\) A well known conjecture of Severi, recently proved by J. Harris [Invent. Math. 84, 445-461 (1986; Zbl 0596.14017)], claims the irreducibility of V(d,g). The author gives here a new proof of this fact by degenerating the projective plane \({\mathbb{P}}^ 2\) to a union \({\tilde {\mathbb{P}}}^ 2_ 1\cup...\cup {\tilde {\mathbb{P}}}^ 2_{d-1}\cup {\mathbb{P}}^ 2_ d\) where each \({\tilde {\mathbb{P}}}^ 2_ i\) is the blow up of a projective plane at a point. The limit of an arbitrary component V of V(d,g) is shown to contain a suitable union of lines and rulings on the \({\tilde {\mathbb{P}}}^ 2_ i\), and this in turn implies that V itself contains a union of general lines. This last fact gives the claim because it is well known that there is a single component of V(d,g) containing a union of general lines [cf. W. Fulton, in Algebraic geometry - Open problems, Proc. Conf., Ravello/Italy 1982, Lect. Notes Math. 997, 146-155 (1983; Zbl 0514.14012)].
Reviewer: E.Casas-Alvero

MSC:
14H10 Families, moduli of curves (algebraic)
14H20 Singularities of curves, local rings
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References:
[1] Arbarello, E., Cornalba, M.: Su una proprieta notevole dei morfismi di una curva a moduli generali in uno spazio proiettivo. Rend. Semin. Mat. Torino38, 87-99 (1980) · Zbl 0478.14016
[2] Arbarello, E., Cornalba, M.: A few remarks. Ann. Sci. Ec. Norm. Super16, 467-488 (1983) · Zbl 0553.14009
[3] Fulton, W.: On nodal curves. In: Algebraic geometry: open problems. Springer Lect. Notes Math.997
[4] Harris, J.: On the Severi problem. Invent. Math.84, 445-461 (1986)* · Zbl 0596.14017 · doi:10.1007/BF01388741
[5] Ran, Z.: Degeneration of linear systems (Preprint)
[6] Severi, F.: Vorlesungen über Algebraische Geometrie (Anhang F), Leipzig: Teubner 1921 · JFM 48.0687.01
[7] Zariski, O.: Algebraic systems of plane curves. Am. J. Math.104, 209-226 (1982) · Zbl 0516.14023 · doi:10.2307/2374074
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