Flat Family of Fibres

Question

I have a question about following argument used in an example in Hartshorne's "Algebraic Geometry" (see page 259):

We have a surjective morphism $f: X \to Y$ between schemes where $X$ is integral and $Y$ a nonsingular curve.

My question is why and how to see that their conditions suffice to see that this family is flat?

I guess that in this context a flat family means just that $f$ is flat in each fiber, correct?

Especially I don't see how the condition that $f$ is surjective - a purely set theoretical condition - is used to show this algebraic condition?

Answer 1

The condition you're looking for is exactly Proposition 9.7 in the same section, which appears 2 pages before the text you've quoted.

Proposition 9.7. Let $f:X\to Y$ be a morphism of schemes, with $Y$ integral and regular of dimension $1$. Then $f$ is flat if and only if every associated point $x\in X$ maps to the generic point. In particular, if $X$ is reduced, this says that every irreducible component of $X$ dominates $Y$.

In your situation, since $X,Y$ are integral, it suffices to show that $f:X\to Y$ is dominant. As surjective implies dominant, there you go.

Further, flat family just means a morphism $f:X\to Y$ which is flat.