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On the variety of plane curves of degree d with $$\delta$$ nodes and $$\kappa$$ cusps. (English) Zbl 0702.14019
Let $${\mathbb{P}}^ N$$ be the projective space parametrizing plane curves of degree $$d,$$ and let $$V_{d,g}$$ the locally closed subset of $${\mathbb{P}}^ N$$ of reduced, irreducible plane curves of degree $$d$$ and geometric genus $$g.$$ J. Harris [Invent. Math. 84, 445-461 (1986; Zbl 0596.14017)] has proved that $$V_{d,g}$$ is irreducible. Let $$V(d,\delta,k)$$ be the subvariety of $$V_{d,g}$$ of those curves which do have $$d$$ nodes and $$k$$ ordinary cusps as their only singularities.
The author proves that $$V(d,\delta,k)$$ is irreducible for $$k\leq 3$$ except possibly when $$k=3$$ and $$d = 5$$ or 6. The main tools used in proving this result are the propositions 2.1, 4.1 and 4.2 of the paper by Harris cited above.
Reviewer: A.Del Centina

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