zbMATH — the first resource for mathematics

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

14H10 Families, moduli of curves (algebraic)
14H20 Singularities of curves, local rings
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.