Generalized $F$-depth and graded nilpotent singularities

We address explicit constructions of new variants of $F$-nilpotent singularities. In particular, we explore how (generalized) weakly $F$-nilpotent singularities behave under gluing, Segre products, Veronese subrings, and the formation of diagonal hypersurface algebras. From these results, explicit examples are produced and we provide bounds on their Frobenius test exponents. To accomplish these tasks, we introduce the {\it generalized $F$-depth} in analogy to Lyubeznik's $F$-depth. These depth-like invariants track (generalized) weakly $F$-nilpotent singularities in a similar fashion as (generalized) depth tracks (generalized) Cohen-Macaulay singularities.