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 $d\geq 1$ of the projective nested Cartesian code $C_X(d)$. The regularity index ${\rm reg}(\delta_X)$ of the minimum distance function $\delta_X$ is the minimum integer $d_0\geq 0$ such that $\delta_X(d)=1$ for $d\geq d_0$. We give a formula for ${\rm reg}(\delta_X)$ by determining an indicator function of least degree for each point of $X$ and using the fact that ${\rm reg}(\delta_X)$ is the ${\rm v}$-number of the vanishing ideal $I_X$ of $X$. Then we give an arithmetical criterion that characterizes when $X$ is Cayley--Bacharach.