Let $X$ be a projective surface embedded into $\mathbb{P}^n_{\mathbb{C}}$ having at most rational singularities. Let $\tilde{X} \to X$ be the minimal resolution of $X$. Is it possible to embed $\tilde{X}$ into $\mathbb{P}^n$ for the same integer $n$? If not, is there any additional condition we can impose on $X$ so that this is possible?

## 1 Answer

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Certainly not. The simplest example is the quadratic cone in $\mathbb{P}^3$; the minimal resolution is the surface $\mathbb{F}_2$, which is not isomorphic to a smooth surface in $\mathbb{P}^3$. The same can be done with any rational double point. On the other hand any smooth surface can be embedded in $\mathbb{P}^5$, so the answer is positive for $n\geq 5$. But I do not see any relation between embeddings of $X$ and those of $\tilde{X} $.