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In other words, you want the normalization of $X$ to be smooth, plus a condition on the embedding: the pull back of $\mathcal{O}_X(1)$ to $\tilde{X} $ should be very ample. The latter property depends on the embedding and is not always true, even for curves: see this postthis post and the answers there.

In other words, you want the normalization of $X$ to be smooth, plus a condition on the embedding: the pull back of $\mathcal{O}_X(1)$ to $\tilde{X} $ should be very ample. The latter property depends on the embedding and is not always true, even for curves: see this post and the answers there.

In other words, you want the normalization of $X$ to be smooth, plus a condition on the embedding: the pull back of $\mathcal{O}_X(1)$ to $\tilde{X} $ should be very ample. The latter property depends on the embedding and is not always true, even for curves: see this post and the answers there.

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In other words, you want the normalization of $X$ to be smooth, plus a condition on the embedding: the pull back of $\mathcal{O}_X(1)$ to $\tilde{X} $ should be very ample. The latter property depends on the embedding and is not always true, even for curves: see this post and the answers there.