We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over $\mathbb{C}$ using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on $\mathbb{N}$-graded Lie algebras of maximal class. As shown by $\text{A}$. Fialowski there are only three isomorphism types of $\mathbb{N}$-graded Lie algebras $L\,=\,\oplus _{i=1}^{\infty }\,{{L}_{i}}$ of maximal class generated by ${{L}_{i}}$ and ${{L}_{2}}$, $L\,=\,\left\langle {{L}_{1}},\,{{L}_{2}} \right\rangle$. Vergne described the structure of these algebras with the property $L\,=\,\left\langle {{L}_{1}} \right\rangle$. In this paper we study those generated by the first and $q$-th components where $q\,>\,2$, $L\,=\,\left\langle {{L}_{1}},\,{{L}_{q}} \right\rangle$. Under some technical condition, there can only be one isomorphism type of such algebras. For $q=\,3$ we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

Let R be a d-dimensional regular local ring of characteristic p > 0 with maximal ideal $\mathfrak m$, let I be an ideal of R and let A = R/I. We describe some properties of the local cohomology module HiI(R), in particular its vanishing, in terms of the Frobenius action on the local cohomology module $H^{d-i}_{\mathfrak m}(A)$.