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A note on the variety of plane curves with nodes and cusps. (English) Zbl 0702.14020
Let \(V(d,\delta,k)\) be the variety parametrizing all the reduced, irreducible plane curves of degree \(d\) with \(\delta\) nodes and k ordinary cups as their only singularities. S. Diaz and J. Harris [Trans. Am. Math. Soc. 309, No.1, 1-34 (1988; Zbl 0677.14003)] showed that if \(k\leq 2d-1\) then \(V(d,\delta,k)\) is a smooth variety of dimension \(N-\delta-2k\) where N is the dimension of the projective space parametrizing all plane curves of degree \(d.\) In the present paper the following theorem is proved: If \(k\leq (d+1)/2-g\) then \(V(d,\delta,k)\) is irreducible.
To give an idea of the method of proof we need some notation: Let \(G^ 2_{d,k}(C)\) the set of all \(g^ 2_ d\)’s on a smooth curve C the associated maps \(\phi: C\to {\mathbb{P}}^ 2\) of which give rise to plane curves of degree \(d\) with \(k\) cups, and let G parametrize pairs (C,V), where \(C\in {\mathcal M}_ g\) (the moduli space of genus g curves) and \(V\in G^ 2_{d,k}(C).\)
Then the theorem follows by showing that G is irreducible and its general numbers give rise to plane curves in \(V(d,\delta,k)\).
In another paper the author has shown that \(V(d,\delta,k)\) is irreducible if \(k\leq 3\) except possibly when \(k=3\) and \(d=5\) or 6 [Trans. Am. Math. Soc. 316, 165-192 (1989; Zbl 0702.14019; see the preceding review)].
Reviewer: A.Del Centina

14H10 Families, moduli of curves (algebraic)
14H45 Special algebraic curves and curves of low genus
Full Text: DOI
[1] Steven Diaz and Joe Harris, Ideals associated to deformations of singular plane curves, Trans. Amer. Math. Soc. 309 (1988), no. 2, 433 – 468. · Zbl 0707.14022
[2] D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), no. 3, 371 – 418. · Zbl 0527.14022 · doi:10.1007/BF01394242 · doi.org
[3] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001
[4] Joe Harris, On the Severi problem, Invent. Math. 84 (1986), no. 3, 445 – 461. · Zbl 0596.14017 · doi:10.1007/BF01388741 · doi.org
[5] P. Kang, On the variety of plane curves of degree \( d\) with \( \delta \) nodes and \( \kappa \) cusps, Trans. Amer. Math. Soc. (to appear). · Zbl 0702.14019
[6] Ziv Ran, On nodal plane curves, Invent. Math. 86 (1986), no. 3, 529 – 534. · Zbl 0644.14009 · doi:10.1007/BF01389266 · doi.org
[7] Oscar Zariski, Algebraic surfaces, Second supplemented edition, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 61. · Zbl 0085.36202
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