Simple singularities of parametrized plane curves in positive characteristic

ABSTRACT

Let K be an algebraically closed field of characteristic p>0. The aim of the article is to give a classification of simple parametrized plane curve singularities over K. The idea is to give explicitly a class of families of singularities which are not simple such that almost all singularities deform to one of those and show that remaining singularities are simple.

KEYWORDS:

• Characteristic p
• parametrized plane curves
• simple singularities

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14H20
• Secondary: 14B05
• 14H50

Acknowledgments

The authors would like to thank very much the referee for his helpful comments.

Notes

¹Let $\Gamma =<{\beta }_0,\dots ,{\beta }_l>$ and $\overline{\Gamma }=<{\overline{\beta }}_0,\dots ,{\overline{\beta }}_k>$ two semigroups given by the minimal set of generators. We define $\Gamma \le \overline{\Gamma }$ iff $\Gamma =\overline{\Gamma }$ or there exists i≤ min(l,k) such that ${\beta }_0={\overline{\beta }}_0,\cdots ,{\beta }_{i-1}={\overline{\beta }}_{i-1}$ and ${\beta }_i<{\overline{\beta }}_i$. In a deformation of a parametrization with semigroup Γ the semigroup is smaller or equal to Γ.

²The conductor of a semigroup is the minimum of all c in the semigroup such that all integers greater than c are in the semigroup.

³We assume as always that n<m and n∤m.

⁴Also called sagbi normal form.

⁵According to the definition of a deformation we are allowed to choose U with the property 0∈U as small as we need.