Flat Family of Fibres

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

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

My question is why and how to see that there 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?