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Curves with nodes, cusps and ordinary triple points. (English) Zbl 0589.14027

The plane algebraic curves of degree n (over an algebraically closed field of characteristic 0) are identified with points of projective space of dimension \(n(n+3)\), and W(n;d,k,t) denotes the closure of the set of points identified with curves having d nodes, k cusps and t triple points. The codimension of W is at most \(d+2k+4t\) and any component \(W_ 0\) with this codimension is said to be of good dimension. If a point c of W(n;d,k,t) also belongs to \(W'=W(n;d',k',t')\) d’\(\leq d\), \(k'+t'\leq k+t\) and t’\(\leq t\), then some of the nodes and triple points are thought of as unassigned singularities and some of the triple points as virtual cusps. The curve C is said to be virtually connected in W’ if, wherever \(C=C_ 1\cup C_ 2\) with \(C_ 1\cap C_ 2\) finite, at least one unassigned singularity or virtual cusp lies on \(C_ 1\cap C_ 2\). The main theorem states that, for a curve C which belongs to a component of W(n;d,k,t) of good dimension, C is virtually connected in W’ if, and only if, the general curve of W’ is irreducible. Several applications of the theorem are given which ensure the existence of irreducible curves with preassigned numbers of nodes, cusps and triple points.
Reviewer: D.Kirby

MSC:

14H20 Singularities of curves, local rings
14H10 Families, moduli of curves (algebraic)
14H45 Special algebraic curves and curves of low genus
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[1] A. Altman–S. Kleiman,Introduction to Grothendieck duality theory, Lectures Notes in Mathematics, 146, Springer (1970). · Zbl 0215.37201
[2] S. A. Burr–B. Grünbaum–N. J. A. Sloane,The orchard problem, Geom. Dedicata,2 (1974), pp. 397–424. · Zbl 0311.05024 · doi:10.1007/BF00147569
[3] C. Giacinti-Diébolt,Variétés des courbes projectives planes de degré et lieu singulier donnés, Math. Ann.,266 (1984), pp. 321–350. · Zbl 0504.14017 · doi:10.1007/BF01475583
[4] M. A. Gradolato, Tesi, Università di Trieste (1984).
[5] S. Iitaka,Algebraic Geometry, Springer (1982).
[6] M. Lindner,Über die Mannigfaltigkeit ebener Kurven mit Singularitäten, Arch. Math.,28 (1977), pp. 603–610. · Zbl 0329.14004 · doi:10.1007/BF01223973
[7] R. Piene,Polar classes of singular varieties, Ann. Sci. Ecole Norm. Sup., 4 sér.,11 (1978), pp. 247–276. · Zbl 0401.14007 · doi:10.24033/asens.1346
[8] B. Segre,Esistenza e dimensione di sistemi continui di curve piane algebriche con dati caratteri, Atti Accad. naz. Lincei, Rend., ser. 6,10 (1929), pp. 31–38. · JFM 55.1001.01
[9] F. Severi,Vorlesungen über algebraische Geometrie, Teubner (1921).
[10] A. Tannenbaum,Families of algebraic curves with nodes, Compositio math.,41 (1980), pp. 107–126. · Zbl 0399.14018
[11] O. Zariski,Algebraic Surfaces, 2nd supplemented edition, Springer (1971). · Zbl 0219.14020
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