On lengths of modules over certain Artinian complete intersections

Let $(Q,\mathfrak{n})$ be a regular local ring of dimension $c\ge 2$ with algebraically closed residue field $k=Q\mathrm{/}\mathfrak{n}$. Let $f_1,f_2,\dots f_{c-1},g$ be a regular sequence in $Q$ such that $f_i\in {\mathfrak{n}}^2$ for all $i$ and $g\in \mathfrak{n}$. Set $A=Q\mathrm{/}(f_1,\dots ,f_{c-1},g^r)$ with $r\ge 2$. Notice $A$ is an Artinian complete intersection of codimension $c$. We show that there exists ${\alpha }_A\in {\mathbb{P}}^{c-1}(k)$ such that there exists integer $m_A\ge 2$ (depending only on $A$) with $m_A$ dividing $\mathrm{\ell }(M)$ for every finitely generated $A$-module $M$ with ${\alpha }_A\mathrm{\notin }\mathcal{V}(M)$ (here $\mathrm{\ell }(M)$ denotes the length of $M$ and $\mathcal{V}(M)$ denotes the support variety of $M$). As an application we prove that if $k$ be a field and $R=k[X_1,\dots ,X_c]\mathrm{/}(X_1^{a_1},\dots ,X_c^{a_c})$ with $a_i\ge 2$ and $c\ge 2$. Let $p$ be a prime number and assume $p$ divides two of the $a_i$. Then $p$ divides $\mathrm{\ell }(E)$ for any $A$-module with bounded betti numbers.