Exact minimum co-degree conditions for $\mathrm{\ell }$-Hamiltonicity in hypergraphs

Suppose $1\le \mathrm{\ell }<k$ such that $(k-\mathrm{\ell })\nmid k$. Given an $n$-vertex $k$-uniform hypergraph $\mathcal{H}$, for all $k\mathrm{/}2<\mathrm{\ell }<3k\mathrm{/}4$ and sufficiently large $n\in (k-\mathrm{\ell })\mathbb{N}$, we prove that if $\mathcal{H}$ has minimum co-degree at least $\frac{n}{\lceil \frac{k}{k-\mathrm{\ell }}\rceil (k-\mathrm{\ell })}$, then $\mathcal{H}$ contains a Hamilton $\mathrm{\ell }$-cycle, which partially verifies a conjecture of Han and Zhao and (partially) resolves a problem of Rödl and Ruciński. Moreover, we show that assuming minimum co-degree $\frac{n}{\lceil \frac{k}{k-\mathrm{\ell }}\rceil (k-\mathrm{\ell })}+\frac{k^2}{2}$ is enough for all $\mathrm{\ell }$.