Why the heck should it imply that $f$ is flat? Did you forget a hypothesis?

link.springer.com/chapter/10.1007%2F978-0-8176-4576-2_10

Locally trivial meaning locally a product? This is an extremely restrictive notion; usually one uses the étale topology. Also, if you want an analogue of the topological notion of fiber bundle even having isomorphic fibers may be too strong. For example, a smooth projective morphism is topologically a fiber bundle, but does not have isomorphic fibers in general.

What's your definition of a fiber bundle?

Sorry, I misread your post, I thought you were saying that you map was defined as the suspension of the Hopf map.

I am no topologist, but I think that it is standard that the Freudenthal suspension map $\pi_3(S^2) \to \pi_4(S^3)$ is surjective. Since the Hopf map generates $\pi_3(S^2)$, its suspension generates $\pi_4(S^3)$.

This is explained in en.wikipedia.org/wiki/Unitary_group . I voted to close as "too localized".

If you can't get a position after a four years post-doc, I don't think that the problem can be where you got your PhD.

The question is whether the Segre embedding is linearly normal, not projectively normal. Anyway, this follows from the Künneth formula for coherent sheaves.

$X_H$ is an open subset of $X^H$. I voted to close as "too localized".

For one thing, the dimension is wrong. And, have you tried computing an example?

There might not exist a lifting of $X$ to $A/p^2$, let alone a canonical one.

To Noam: ok, I replaced the $-$ with a long dash. They look identical to me.