Consider the set of all complex plane curves $C\subset \mathbb{P}^2$ of degree $d$. This is in fact a projective space, say $\mathbb{P}^N$. Let $X\subset \mathbb{P}^2\times \mathbb{P}^N$ denote the universal curve. Then $p:X\rightarrow \mathbb{P}^N$ is a faithfully flat family of curves.
Is $p$ a locally trivial fibration? If not if we restrict to some open subset of the base do we get a locally trivial fibration?
In particular what does it mean to say something is a locally trivial fibration in this context? One consequence of locally trivial fibration is that the fibres can be deformed into another without changing topological data. Is that right? Does it happen here?
Best Answer
A locally trivial fibration $f:X\to Y$ with fiber $F$ means that there's an open covering of $Y$ given by $U_i$ so that for each $U_i$ we have $f^{-1}(U_i)\cong U_i\times F$. In particular, this means that we should have that all fibers are isomorphic if we're working over a connected base. Saying that you're interested in this as a locally trivial fibration of schemes means that you want this isomorphism to be an isomorphism of schemes, which is what I'll assume for the remainder of the post. (If you're instead asking about a topological fibration with the $\Bbb C$ points, please say so in the comments.)
This won't happen often. As soon as you come to degree 2 or higher equations, you have to deal with singular curves versus smooth curves - this already shows that the map cannot be a locally trivial fibration except possibly in the degree-one case. And even then, restricting to the open set of smooth curves won't fix things if you're asking this to be a locally trivial fibration in the world of schemes/varieties. Consider the case of elliptic curves, given by degree three equations - the $j$-invariant associated to the fiber won't be constant on any open set, which is a problem because it's an isomorphism invariant of elliptic curves (which you'd need to be constant on any open set you're hoping for a locally trivial fibration on).
Things won't get better with higher-degree equations - there will be at least $3g-3$ invariants hanging around that you'll need to check in the genus $g>1$ cases, which will let you play the same game we did with the $j$-invariant above. (Why $3g-3$? This is the dimension of the moduli space of curves of genus $g>1$.)