Tjurina number of a generic irreducible curve singularity
Let \(\gamma\) be an irredcible germ of plane analytic curve with equation \(f=0\) and let \(\tau(\gamma)= \dim_\mathbb{C} \mathbb{C}\{x,y\}/(f, f_y',f_x')\) be the Tjurina number of \(\gamma\). Various algorithms are known to compute Tjurina numbers of germs of plane curves having particular semigroups. In the present paper the author presents an algorithm that, starting from the semigroup of \(\gamma\), computes \(\tau(\gamma)\) and allows her to determine a \(\mathbb{C}\)-basis of \(\mathbb{C}\{x,y\}/(f, f_y',f_x')\). Moreover, by the upper semicontinuity of \(\tau (\gamma)\), the author obtains the minimum Tjurina number in the equisingularity class of \(\gamma\); this gives the dimension of the generic component of the moduli space of irreducible curves with fixed semigroup. An implementation of the algorithm (written by the author and \textit{J. Elias}) in a Mathematica package is also available.
scientific article; zbMATH DE number 1089117