Let $S$ be a connected scheme. We say that $S$ is simply connected if every smooth and proper morphism $X \to S$ of relative dimension $0$ has a section. This is equivalent to the standard definition, since finite etale morphisms are smooth and proper morphisms of relative dimension $0$, and, after reducing to the connected case, havinga section is equivalent to being trivial.

Now say that $S$ is $d$-trivial if every smooth and proper morphism $X \to S$ of relative dimension $d$ has a section. So being $0$-trivial is the same as being simply connected.

We might extend the definition to call $\infty$-trivial schemes that are $d$-trivial for all $d$.

For what numbers $d>0$ can we say something about which schemes are $d$-trivial?

This question was originally inspired by this excellent question in which it is proven that $\operatorname {Spec} \mathbb Z$ is $1$-trivial but not $6$-trivial. The apparent difficulty of further progress on $\operatorname{Spec} \mathbb Z$ led me to wonder if other cases, such as varieties over $\mathbb C$, might be easier. In particular:

For what $d$ is $\mathbb P^1_\mathbb C$ $d$-trivial? $\mathbb P^n_\mathbb C$?