The Variety of Lie Algebras of Maximal Class

We present an explicit description of the affine variety $M_\mathrm{Fil}$ of Lie algebras of maximal class (filiform Lie algebras): the formulas of polynomial equations that determine this variety are written down. The affine variety $M_\mathrm{Fil}$ can be considered as the base of the nilpotent versal deformation of the $\mathbb N$-graded Lie algebra $\mathfrak m_0$.