TITLE: Direct image of the structure sheaf by an endomorphism of $\mathbb{P}^2$
QUESTION [14 upvotes]: Let us take 3 quadratic forms on $\mathbb{P}^2$ with no common zero; they define a map $\pi : \mathbb{P}^2\rightarrow \mathbb{P}^2$ of degree 4. It is not difficult to see that $\pi _*\mathscr{O}_{\mathbb{P}^2}\cong \mathscr{O}_{\mathbb{P}^2}\oplus \mathscr{O}_{\mathbb{P}^2}(-1)^3$. Does anyone know how to write down the algebra stucture of $\pi _*\mathscr{O}_{\mathbb{P}^2}$ in terms of this decomposition?

REPLY [2 votes]: In general, if $f \colon X \to Y$ is a quadruple cover of algebraic varieties, the structure of the algebra $f_* \mathcal{O}_X$ is studied in the paper
D. Hahn, R. Miranda: Quadruple Covers in Algebraic Geometry, Journal of Algebraic Geometry 8 (1999).
You can download the paper from the webpage of the second author.