Infinitely many genus two and three curves of small fixed positive rank

We prove that there exist infinitely many genus two curves over $\mathbb{Q}$ whose Jacobians have ranks between $0$ and $11$, and genus three curves over $\mathbb{Q}$ whose Jacobians have ranks $0$, $1$, and $2$. Furthermore, we consider a generalisation of these results over number fields satisfying some conditions.