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I read the following fact which wasn't explained further and wonder how you exactly get it. Maybe you can give me some hint.

Start with a smooth projective variety $X$ over a $k$. Then the author said that you can choose a smooth ample hypersurface $H$ on $X$.

My question is: how do you construct that $H$, and: can it be chosen as effective?

Thanks

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This is more or less the content of Bertini theorem. By assumption, $X$ is projective, so that you can look at the linear system $|\mathcal O_X(n)|$ for $n$ big enough: it will contain (by Bertini's thm) some smooth (thus irreducible) divisor, which is what you are looking for.

In fact, your hypersurface is simply the intersection of $X$ and some hypersurface of $\mathbb P^n$ where $X\subset \mathbb P^n$.

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