A level initial ideal of the 2-minors determinantal ideal

For $\mathbb{k}$ a field, let $X$ a $m\times n$ matrix of variables and $S=\mathbb{k}[X]\mathrm{.}$ We consider the determinantal ideal $I_2\subseteq S$ generated by the $2$-minors of $X\mathrm{.}$ In this paper we find a suitable monomial order over $S$ such that $I$, the initial ideal of $I_2$ with respect to that order, is level, namely, it is Cohen-Macaulay and the socle of an Artinian reduction of the $\mathbb{N}$-graded algebra $S\mathrm{/}I$ is concentrated in only one degree. Moreover, we compare the Betti tables of $I_2$ with the tables of its initial ideals. In the last section, we prove the shellability of the simplicial complex naturally associated to $S\mathrm{/}I$ in the case $m<n\mathrm{.}$