We study the symbolic powers of determinantal ideals of generic, generic symmetric, and Hankel matrices of variables, and of Pfaffians of generic skew-symmetric matrices, in prime characteristic. Specifically, we show that the limit $\lim\limits_{ n\to\infty} \frac{ \textrm{ reg}(I^{ (n)})}{ n}$ exists and that $\textrm{ depth}(R/I^{ (n)})$ stabilizes for $n\gg 0$. Furthermore, we give explicit formulas for the stable value of $\textrm{ depth}(R/I^{ (n)})$ in the generic and skew-symmetric cases. In order to show these results, we introduce the notion of symbolic $F$-purity of ideals which is satisfied by the classes of ideals mentioned above. Moreover, we find several properties satisfied by symbolic $F$-pure ideals. For example, we show that their symbolic Rees algebras and symbolic associated graded algebras are $F$-pure. As a consequence, their $a$-invariants and depths present good behaviors. In addition, we provide a Fedder's-like Criterion for symbolic $F$-purity.
Categories: math.AC