A note on the variety of plane curves with nodes and cusps.

*(English)*Zbl 0702.14020Let \(V(d,\delta,k)\) 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)].

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

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14H45 | Special algebraic curves and curves of low genus |

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\textit{P.-L. Kang}, Proc. Am. Math. Soc. 106, No. 2, 309--312 (1989; Zbl 0702.14020)

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##### References:

[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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