关于平面曲线的节点位置
On the position of nodes of plane curves
摘要：
给定次数$d$的平面曲线的Severi簇$V_{d,n}$和精确的$n$节点允许到次数为$n$的$\mathbb{P}^2$的零维子方案的Hilbert方案$\mathbb{P}^{2[n]}$的映射。该贴图为每条曲线$C\in V_{d,n}$指定其节点。对于某些$n$，我们考虑了许多已知的Severi簇的因子在这个映射下的象及其部分紧化。我们在Pic $(\mathbb{P}^{2[n]})$中计算这些图像的除数类，并提供节点曲线的枚举数。我们还直接回答了迪亚兹-哈里斯的一个问题，关于是否规范类的塞维品种是有效的。
英文摘要：
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 Pic$(\mathbb{P}^{2[n]})$ and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz-Harris about whether the canonical class of the Severi variety is effective.
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