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.

中文翻译：

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关于平面曲线节点的位置

塞维里品种$V_{d,n}$给定度数的平面曲线$d$确切地说$n$节点承认希尔伯特方案的映射$\mathbb{P}^{2[n]}$的零维子方案$\mathbb{P}^{2}$学位$n$. 该地图分配给每条曲线$C\in V_{d,n}$它的节点。对于一些$n$，我们考虑这张地图下许多已知的 Severi 品种的除数及其部分紧化的图像。我们计算这些图像的除数类$\text{图片}(\mathbb{P}^{2[n]})$并提供节点曲线的枚举数。我们还直接回答了 Diaz-Harris ['Severi 品种的几何学'，反式。阿米尔。数学。社会党。309(1988), 1-34] 关于 Severi 品种的规范类是否有效。