A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{O}_{X}} $. Such a map $ \phi $ is called a splitting. A closed sub-scheme $ Y $ of $ X $ is compatibly split if there exists a splitting $ \phi $ such that $ \phi(F_{\ast}(\mathcal{I}_{Y})) \subseteq \mathcal{I}_{Y} $. Note that $ \mathcal{I}_{Y} \subseteq \phi(F_{\ast}(\mathcal{I}_{Y})) $ already.
If $ X $ is a normal variety, then $ X $ if $ F $-regular if for all effective Weil divisors $ D $ there is an $ e \in \mathbb{N} $ such that $ \mathcal{O}_{X} \to F^{e}_{\ast}(\mathcal{O}_{X}) \to F^{e}_{\ast}(\mathcal{O}_{X}(D)) $ splits.
All toric varieties are $ F $-regular. Suppose that $ X_{\Sigma} $ is a simplicial, projective, toric variety and that $ Y $ is equal to $ \operatorname{Bl}_{e}(X_{\Sigma}) $ where $ e $ is a smooth point of the torus (we may take it to be the identity of the torus). If $ e $ splits compatibly with $ X_{\Sigma} $, then it follows that $ \operatorname{Bl}_{e}(X_{\Sigma}) $ should be $ F $-regular. When is a smooth point of a simplicial, projective, toric variety compatibly split?
