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On Free and Nearly Free Conjecture of Rational Unibranched Projective Plane Curves
Abstract
Let C⊂ PP2 be a reduced irreducible complex plane curve with complement U. In this paper we prove that χ(U)≥ ν(C), where ν(C) is the freeness defect of C defined in \cite{Dim24}. When C is a rational curve with unibranched singularities, this inequality proves the conjecture of Dimca and Sticlaru, that a rational unibranched plane curve is either free or nearly free.
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BibTeX
@misc{zhang2026free,
title = {{On Free and Nearly Free Conjecture of Rational Unibranched Projective Plane Curves}},
author = {Xiping Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24810},
eprint = {2609.24810},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24810},
}
RIS
TY - UNPB
TI - On Free and Nearly Free Conjecture of Rational Unibranched Projective Plane Curves
AU - Xiping Zhang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24810
UR - https://arxiv.org/abs/2609.24810
AB - Let C⊂ PP^2 be a reduced irreducible complex plane curve with complement U. In this paper we prove that χ(U)≥ ν(C), where ν(C) is the freeness defect of C defined in \cite{Dim24}. When C is a rational curve with unibranched singularities, this inequality proves the conjecture of Dimca and Sticlaru, that a rational unibranched plane curve is either free or nearly free.
ER -