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Let $X$ be a smooth complex projective surface such that there is a degree two morphism $f:X\rightarrow \mathbb{P}^2$. Then I know that $X$ has to be branched over a curve of degree $2n$. I have two questions regarding $X$.

1) what is the Picard group of $X$? Is it $\mathbb{Z}$?

2) can any such $X$ be embedded in $\mathbb{P}^3$ as a closed subscheme. If this is true then $Pic\ X$ cannot be $\mathbb{Z}$ since we would need a line bundle on $X$ with four independent sections. In general what is the minimum $m$ such that $X$ can be embedded in $\mathbb{P}^m$?

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Assume the base field is $\mathbb{C}$.

If $n = 1$, $X \cong \mathbb{P}^1 \times \mathbb{P}^1$ (as Mohan said), so $Pic(X) \cong \mathbb{Z} \oplus \mathbb{Z}$ and $X$ can be embedded into $\mathbb{P}^3$.

If $n = 2$, $X$ is a del Pezzo surface of degree 2, it is isomorphic to $\mathbb{P}^2$ blown up at 7 points, $Pic(X) \cong \mathbb{Z}^8$, $X$ cannot be embedded into $\mathbb{P}^3$.

If $n = 3$, $X$ is a K3 surfaces. Generically, $Pic(X) \cong \mathbb{Z}$, but in the 19-dimensional moduli space of such covers there is a countable number of subspaces where the Picard rank jumps (up to 20).

If $n \ge 4$, $X$ is of general type, generically $Pic(X) \cong \mathbb{Z}$, but again it jumps on subvarieties of muduli space; this is governed by the Hodge theory.

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    $\begingroup$ thank you! In the $n=3$ case, you mean a generic double cover has Picard group $\mathbb{Z}$? Can you suggest a reference for this? $\endgroup$ Aug 24, 2019 at 9:52
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For 1), there is a degree 2 map from $X=\mathbb{P}^1\times\mathbb{P}^1$ to $\mathbb{P}^2$. Notice that $X$ is a quadric surface in 3-space (and a projection from a point outside $X$ gives you your degree 2 map).

For 2), remember that any $n$-dimensional smooth projective variety can be embedded in $\mathbb{P}^{2n+1}$.

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  • $\begingroup$ thank you. So 1) says it is not $Z$. Can we say what is $Pic(X)$ in general depending on the degree of branch curve? $\endgroup$ Aug 24, 2019 at 0:39
  • $\begingroup$ Also from 1) we see that, when $X$ is branched over a quadric, it can be embedded in $P^3$. So $P^5$ embedding using $f^*O(2)$ is not minimal. So i wanted to know if, again depending on degree of branch curve, whether $X$ can be embedded in $P^m$ for $m=3,4$ $\endgroup$ Aug 24, 2019 at 0:41

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