Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$ and $\{x_i\}_{i \in I}$ be a finite set of closed points in $X$. Let $X'$ be the blow up of $X$ at these points. Then,
$1)$ Is there a natural/canonical choice of a very ample line bundle on $X'$?
$2)$ What are the possible degrees of $X'$ (depending on the embedding into a projective space)?
If someone could suggest a good reference on blow up of smooth projective surfaces and their embeddings into projective space, that would be very helpful.
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I had a similar question once, and at least part of your first question is addressed in this paper.