There are several examples in the literature where compactness properties of a cardinal $\kappa$ imply "bad" behavior of certain generic ultrapowers with critical point $\kappa$; particularly generic ultrapowers associated with tower forcings (Woodin's stationary tower forcing is an example of a tower forcing). I will discuss instances of this phenomenon due to Burke, Foreman-Magidor, and Cox-Viale.

I will discuss the Diagonal Reflection Principle (DRP), which is a highly simultaneous form of stationary set reflection that follows from strong forcing axioms like $PFA^{+\omega_1}$. DRP can be viewed as a weaker version of the statement "there is a normal ideal with completeness $\omega_2$ whose associated poset is proper (i.e. preserves stationary subsets of $[X]^\omega$ for all $X$)". In the presence of sufficiently absolute partitions of $cof(\omega)$, such ideals yield generic embeddings $j: V \to_G ult(V,G)$ with critical point $\omega_2$ such that large portions of $j$ are visible to $V$.