1 results
ON THE POSITION OF NODES OF PLANE CURVES
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 103 / Issue 1 / February 2021
- Published online by Cambridge University Press:
- 01 June 2020, pp. 62-68
- Print publication:
- February 2021
-
- Article
- Export citation
-
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.
