# Picard group of degree two cover of the plane

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 $$\mathbb{P}^m$$?

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.

• 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? Aug 24, 2019 at 9:52

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 $$\mathbb{P}^{2n+1}$$.

• 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? Aug 24, 2019 at 0:39
• 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 $m=3,4$ Aug 24, 2019 at 0:41