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I think the concept of flatness is very theoretical and I find hard to work with it. Hence, I would like to understand some easy examples before going deeper in its study.

For instance, I read that the projection $$ (t Y-(X_0X_2-X_1^2)=0)\subset \mathbb{P}(1,1,1,2)\times \mathbb{A}^1\rightarrow \mathbb{A}^1 $$ is flat. Nevertheless, I do not know how to prove it.

Notice that this morphism has fibers isomorphic to $\mathbb{P}^2$ for $t\neq 0$ and isomorphic to $\mathbb{P}(1,1,4)$ for $t=0$.

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In this case you can use so-called miracle flatness (Hartshorne Exercise III.10.9):

if $f: Y \rightarrow X$ is a morphism with $Y$ Cohen-Macaulay, $X$ regular, and every fibre of the same dimension, then $f$ is flat.

In your case, the only thing you need to check is that your variety is Cohen-Macaulay; in fact, it is nonsingular, as you can verify by compting partial derivatives.

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