Horizontal vector subsepace of electromagnetic connection in Minkowskian spacetime

For Minkowskian space-time $M$, the principal bundle for electromagnetism is $(M times U(1), M, proj_{1}, U(1))$. I imagine there is a global gauge potential (since I can choose a global section, because the bundle is trivial) of the shape,

$$mathcal{A}(q) = (-phi(q), vec{A}(q)) in T^{*}_{q} M,$$

where $vec{A}$ is the magnetic vector potential and $phi$ is the electrical potential.

So, if I try to identify the horizontal subspace of such a connection I would have to make $mathcal{A}(q)(X) = 0$. But then I get to

$$-phi(q)X_{0} + vec{A}(q) cdot vec{X} = 0.$$

Is this right? For it is not clear to me the shape of the horizontal subspace.

I thank you.