Matrix Hessenberg schemes over the minimal sheet

We study families of matrix Hessenberg schemes in the affine scheme of complex $n\times n$ matrices, each defined over a fixed sheet in the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$. It is well known that such families over the regular sheet are flat, and every regular Hessenberg scheme degenerates to a regular nilpotent Hessenberg scheme. This paper explores whether flat degenerations exist outside of the regular case. For each matrix Hessenberg scheme, we introduce a one-parameter family of matrix Hessenberg schemes that degenerates it to a specific nilpotent Hessenberg scheme. Our main theorem states that, when the family lies over the minimal sheet in $\mathfrak{gl}_n(\mathbb{C})$, this degeneration is flat. The proof leverages commutative algebra on the polynomial ring to identify the structure of the family concretely, and we explore several applications. We conjecture that flatness holds for these families over other sheets as well.