On the tangent space of a weighted homogeneous plane curve singularity

Mario Mor\'an Ca\~n\'on; Julien Sebag

doi:10.4134/JKMS.j180796

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2020; 57(1): 145-169

Online first article November 25, 2019 Printed January 1, 2020

https://doi.org/10.4134/JKMS.j180796

On the tangent space of a weighted homogeneous plane curve singularity

Mario Mor\'an Ca\~n\'on, Julien Sebag

Universit\'e de Rennes; Universit\'e de Rennes

Abstract

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Abstract

Let $k$ be a field of characteristic 0. Let $ \scr C=\Spec(k[x,y]/\langle f\rangle)$ be a weighted homogeneous plane curve singularity with tangent space $\pi_\scr C\colon T_{\scr C/k}\rightarrow \scr C$. In this article, we study, from a computational point of view, the Zariski closure $\scr G(\scr C)$ of the set of the 1-jets on $\scr C$ which define formal solutions (in $F[[t]]^2$ for field extensions $F$ of $k$) of the equation $f=0$. We produce Groebner bases of the ideal $\mathcal{N}_1(\scr C)$ defining $\scr G(\scr C)$ as a reduced closed subscheme of $T_{\scr C/k}$ and obtain applications in terms of logarithmic differential operators (in the plane) along $ \scr C$.

Keywords: Jet and arc scheme, derivation module, curve singularity

MSC numbers: 14E18, 32S05,13P10