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ON THE POSITION OF NODES OF PLANE CURVES

Part of: Curves Cycles and subschemes

Published online by Cambridge University Press:  01 June 2020

CÉSAR LOZANO HUERTA Open the ORCID record for CÉSAR LOZANO HUERTA [Opens in a new window]  and
TIM RYAN Open the ORCID record for TIM RYAN [Opens in a new window]
CÉSAR LOZANO HUERTA*
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Oaxaca, Mexico email lozano@im.unam.mx
TIM RYAN
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, USA email rtimothy@umich.edu
*
✉lozano@im.unam.mx
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Abstract

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.

Keywords

Hilbert scheme of pointsSeveri varietynodal plane curves

MSC classification

Primary: 14H10: Families, moduli (algebraic)
Secondary: 14C22: Picard groups 14H50: Plane and space curves
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 62 - 68
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

During the preparation of this article the first named author was partly supported by the CONACYT grant CB-2015/253061; he is currently a CONACYT Research Fellow in Mathematics, project no. 1036.

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