Suppose $X$ is a smooth projective variety, and $H_1,H_2$ are smooth divisors. Does $H_1\cap H_2$ satisfy Serre $S_2$ criterion: for every prime $p$ the local ring $R_p$ has depth at least $\min\{2,\, \dim(R_p)\}$?
How about the special case $X=\mathbb{P^2}_{[X_0,X_1,X_2]}\times\mathbb{P^2}_{[Y_0,Y_1,Y_2]}\times\mathbb{P}^{15}$ (the last space has coordinates $a_{d_0d_1d_2}$ with ${\sum d_i=4} $, in other words, they parameterize degree $4$ plane curves) and $H_1=\sum a_{d_0d_1d_2}X_0^{d_0}X_1^{d_1}X_2^{d_2}$, $H_2=\sum a_{d_0d_1d_2}Y_0^{d_0}Y_1^{d_1}Y_2^{d_2}$. In this case, the singular locus has codimension at least $2$, I hope it would be normal, but I am not clear how to check the $S_2$ criterion.