On the regularity index of the minimum distance function in projective nested Cartesian codes

Let $X$ be a projective nested product of fields and let ${\delta }_X(d)$ be the minimum distance in degree $d\ge 1$ of the projective nested Cartesian code $C_X(d)$. The regularity index $\mathrm{r}\mathrm{e}\mathrm{g}({\delta }_X)$ of the minimum distance function ${\delta }_X$ is the minimum integer $d_0\ge 0$ such that ${\delta }_X(d)=1$ for $d\ge d_0$. We give a formula for $\mathrm{r}\mathrm{e}\mathrm{g}({\delta }_X)$ by determining an indicator function of least degree for each point of $X$ and using the fact that $\mathrm{r}\mathrm{e}\mathrm{g}({\delta }_X)$ is the $\mathrm{v}$-number of the vanishing ideal $I_X$ of $X$. Then we give an arithmetical criterion that characterizes when $X$ is Cayley–Bacharach.