Componentwise linear powers and the $x$-condition

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gröbner basis of the defining ideal $J$ of $A$ we give a condition, called the $x$-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $\mathcal{R}(I)$ of a graded ideal or the symmetric algebra $\textrm{Sym}(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.