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ON THE TANGENT SPACE OF A WEIGHTED HOMOGENEOUS PLANE CURVE SINGULARITY
Canon, Mario Moran,Sebag, Julien Korean Mathematical Society 2020 대한수학회지 Vol.57 No.1
Let k be a field of characteristic 0. Let ${\mathfrak{C}}=Spec(k[x,y]/{\langle}f{\rangle})$ be a weighted homogeneous plane curve singularity with tangent space ${\pi}_{\mathfrak{C}}:T_{{\mathfrak{C}}/k}{\rightarrow}{\mathfrak{C}$. In this article, we study, from a computational point of view, the Zariski closure ${\mathfrak{G}}({\mathfrak{C}})$ of the set of the 1-jets on ${\mathfrak{C}}$ which define formal solutions (in F[[t]]<sup>2</sup> for field extensions F of k) of the equation f = 0. We produce Groebner bases of the ideal ${\mathcal{N}}_1({\mathfrak{C}})$ defining ${\mathfrak{G}}({\mathfrak{C}})$ as a reduced closed subscheme of $T_{{\mathfrak{C}}/k}$ and obtain applications in terms of logarithmic differential operators (in the plane) along ${\mathfrak{C}}$.
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On the tangent space of a weighted homogeneous plane curve singularity
Mario Canon Mario,Julien Sebag 대한수학회 2020 대한수학회지 Vol.57 No.1
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$.
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