Pullback of a linear system in a ramified cover (through an example)

Question

I am trying to understand the pullback of sections of a line bundle, in the situation of a ramified cover $\pi:X\rightarrow Y$ of algebraic varieties.

I am trying to work out some nice concrete examples first, as the following one with a K3 surface $X$.

Let $\pi:X\rightarrow\Bbb{P}^2$ be a double cover of the plane branched along a smooth sextic. Then $$H^0(\pi^\ast O(3))=\pi^\ast H^0(O(3))\oplus V,$$ where $V$ is the subspace of sections vanishing on the ramification locus.

It is (at least intuitively) quite clear to me that in the linear system $|\pi^\ast O(3)|$ we must get the system of double covers of the plane cubics. But I don't really understand the " $\oplus V$ " part. If somebody could elaborate on this, I would highly appreciate.

Answer 1

In general for a degree-$d$ cyclic cover $\pi: X \rightarrow Y$ branched over a smooth section of a line bundle $L^{\otimes d}$, we have the formula

$$ \pi_* O_X = O_Y \oplus L^\vee \oplus \cdots \oplus (L^\vee)^{\otimes d-1}$$

For any line bundle $M$ on $Y$, the projection formula then gives

$$\pi_* \pi^* M = M \oplus (M \otimes L^\vee) \oplus \cdots \oplus (M \otimes (L^\vee)^{\otimes d-1}).$$

Since $H^0(\pi^*M) = H^0(\pi_* \pi^* M)$, this allows us to calculate the global sections of the pullback.

In the special case of a double cover, putting $d=2$ we get

$$H^0(\pi^*M) = H^0(M) \oplus H^0(M \otimes L^\vee);$$

in your case we have $M=L=O(3)$, so we get

$$H^0(\pi^*M) = H^0(M) \oplus V$$ where $V$ is the space of constant functions on $\mathbf P^2$.

Then the question becomes: how do we identify sections of $M \otimes L^\vee$ (i.e. constant functions) on $\mathbf P^2$ with sections of $\pi^*M$ which vanish along the ramification locus? Briefly, the idea is as follows: the double cover $X$ is locally defined by an equation of the form

$$y^2=f(x_0,x_1,x_2)$$ where $f$ is a sextic, so the ramification divisor is $\{y=0\}$. For any local regular function $f$ on $\mathbf P^2$, we can take the function $y \cdot (f \circ \pi)$ on $X$; this gives us a local regular function on $X$ vanishing along the ramification divisor. One can check that everything patches together in the right way, to see that this gives a correspondence between sections of $M \otimes L^\vee$ on $\mathbf P^2$ and sections of $\pi^*M$ which vanish along the ramification locus.