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My question: Why can a projective variety of dimension $n$ be covered by $n+1$ affine open subsets?

I can see the result holds when the variety is $\mathbb P^n$ or a hypersurface $X$ in $\mathbb P^{n+1}$. The second is true because you can suppose the point $[0:\cdots:0:1] \not \in X$ and then cover $X$ with $A_0, \ldots,A_n$, where $$A_i = \{[x_0: \cdots:x_{n+1}]: x_i \neq 0\}.$$

Obs.: The reason why I want to prove this fact is to prove that the Čech cohomology $H^p(X,F)$ of a projective variety $X$ of dimension $n$ is $0$ for $p>n$, where $F$ is a quasicoherent sheaf.

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Let $X_d\subseteq \mathbb{P}^n$ be a projective variety of dimension $d$. Pick a hyperplane $H_d$ which does not contain any irreducible component of $X_d$. Then $X_{d-1}=X_d\cap H_d$ has dimension $d-1$. We can then repeat this process and pick a hyperplane $H_{d-1}$ which does not contain any irreducible component of $X_{d-1}$, and define $X_{d-2}=X_{d-1}\cap H_{d-1}$. We repeat this until we reach $X_0$ which is $0$-dimensional and finally have $H_0$ which is disjoint from $X_0$. We then see that every point of $X_d$ is not in $H_i$ for some $i=0,\dots,d$ so $X_d$ is covered by the $d+1$ affine open subvarieties $X_d\setminus H_i$.

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