For each $k \geq 3$ and $1 \leq \ell \leq k-1$ we give an asymptotically best possible minimum positive codegree condition for the existence of a Hamilton $\ell$-cycle in a $k$-uniform hypergraph. This result exhibits an interesting duality with its analogue under a minimum codegree condition. The special case $\ell = k-1$ of our result establishes an asymptotic version of a recent conjecture of Illingworth, Lang, Müyesser, Parczyk and Sgueglia on tight Hamilton cycles in hypergraphs.