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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

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14H45 Special algebraic curves and curves of low genus
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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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