ON THE POSITION OF NODES OF PLANE CURVES
The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^{2}$ of degree $n$ . This map assigns to every curve $C\in V_{d,n}$ its nodes. For some $n$ , we consider the image under this map of many known divisors of the Severi variety and its partial compactification. We compute the divisor classes of such images in $\text{Pic}(\mathbb{P}^{2[n]})$ and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz–Harris [‘Geometry of the Severi variety’, Trans. Amer. Math. Soc.309 (1988), 1–34] about whether the canonical class of the Severi variety is effective.
2015 ◽
Vol 152
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pp. 115-151
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2019 ◽
Vol 113
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pp. 483-487
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2012 ◽
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2010 ◽
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2011 ◽
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2012 ◽
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2019 ◽
Vol 2019
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pp. 161-200
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