Componentwise Linearity Under Square-Free Gröbner Degenerations

Abstract

Using the recent results on square-free Gröbner degenerations by Conca and Varbaro, we proved that if a homogeneous ideal $I$ of a polynomial ring is such that its initial ideal ${\mathrm{in}}_{<}(I)$ is square-free and ${\beta }_0(I)={\beta }_0({\mathrm{in}}_{<}(I))$, then $I$ is a componentwise linear ideal if and only if ${\mathrm{in}}_{<}(I)$ is a componentwise linear ideal. In particular, if furthermore one of $I$ and ${\mathrm{in}}_{<}(I)$ is componentwise linear, then their graded Betti numbers coincide.