# ON THE TANGENT SPACE OF A WEIGHTED HOMOGENEOUS PLANE CURVE SINGULARITY

• Accepted : 2019.10.16
• Published : 2019.12.30

#### Abstract

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]]2 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 ${\mathfrak{C}}$.

#### References

1. T. Arakawa and A. R. Linshaw, Singular support of a vertex algebra and the arc space of its associated scheme, https://arxiv.org/pdf/1804.01287.pdf.
2. T. Becker and V. Weispfenning, Grobner bases, Graduate Texts in Mathematics, 141, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-0913-3
3. D. Bourqui and M. Haiech, On the nilpotent functions at a non degenerate arc, 2018.
4. D. Bourqui and J. Sebag, Arc schemes of affine algebraic plane curves and torsion kahler differential forms, to appear in Arc Scheme and Singularities, Proceedings of the Nash conference (2017).
5. D. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. https://doi.org/10.1007/978-1-4757-2181-2
6. S. Ishii and J. Kollar, The Nash problem on arc families of singularities, Duke Math. J. 120 (2003), no. 3, 601-620. https://doi.org/10.1215/S0012-7094-03-12034-7 https://doi.org/10.1215/S0012-7094-03-12034-7
7. E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.
8. K. Kpognon and J. Sebag, Nilpotency in arc scheme of plane curves, Comm. Algebra 45 (2017), no. 5, 2195-2221. https://doi.org/10.1080/00927872.2016.1233187 https://doi.org/10.1080/00927872.2016.1233187
9. L. C. Meireles, On the classification of quasi-homogeneous curves, https://arxiv.org/pdf/1009.1664.pdf.
10. J. Nicaise and J. Sebag, Greenberg approximation and the geometry of arc spaces, Comm. Algebra 38 (2010), no. 11, 4077-4096. https://doi.org/10.1080/00927870903295398 https://doi.org/10.1080/00927870903295398
11. J. Sebag, Integration motivique sur les schemas formels, Bull. Soc. Math. France 132 (2004), no. 1, 1-54. https://doi.org/10.24033/bsmf.2458 https://doi.org/10.24033/bsmf.2458
12. J. Sebag, Arc scheme and Bernstein operators, to appear in Arc Scheme and Singularities, Proceedings of the Nash conference (2018).
13. J. Sebag, On logarithmic differential operators and equations in the plane, Illinois J. Math. 62 (2018), no. 1-4, 215-224. https://doi.org/10.1215/ijm/1552442660 https://doi.org/10.1215/ijm/1552442660