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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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[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
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