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?
Here's a partial answer for a particular case. I suspect you probably already knew this. (So it is really just a comment, but it is too long for the comment box.)
If the polynomials are $x_0^2, x_1^2, x_2^2$, and the characteristic is not 2, then in affine coordinates the cover $\pi$ is $y_1=x_1^2, y_2=x_2^2$. So it is Galois, with Galois group $G=(\mathbb{Z}/2)^2$. You can then break up $\pi_*\mathcal{O}_{\mathbb{P}^2}$ using 4 characters of $G$, which I'll label $1, \sigma_1,\sigma_2,\sigma_1\sigma_2$. The last 3 corresponds to the involutions $(x_1\mapsto x_1)$, $( x_2\mapsto x_2)$ and, I'm guessing $(x_0\mapsto x_0)$. The invariant part of $\pi_*\mathcal{O}$ is just $\mathcal{O}$, and the remaining isotypic factors are isomorphic to $\mathcal{O}(1)$. Under the identification $$\Gamma((\pi_*\mathcal{O})(1))\cong \Gamma(\mathcal{O}(2))$$ coming from the projection formula, the sections coming from the $\mathcal{O}(1)$ factors correspond to $x_1^2,x_2^2, x_0^2$ (modulo the guess above). Let me label the factors accordingly by $1,2,0$. You presumably want the multiplication table. I'm guessing, for instance, $$\mathcal{O}(1)_{i}\otimes \mathcal{O}(1)_{i}\to \mathcal{O}$$ is $$\mathcal{O}(1)_{i}\otimes \mathcal{O}(1)_{i}\cong \mathcal{O}(2)\stackrel{x_i^2}{\to} \mathcal{O}$$ Etc.

$\begingroup$ Thanks, Donu. Yes, I was more or less aware of this particular Galois case. I am intrigued by the general case... $\endgroup$ – abx Jun 20 at 16:46

$\begingroup$ I imagine you can do something similar if only one of the quadratic forms is a square. $\endgroup$ – Piotr Achinger Jun 22 at 17:47
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.

$\begingroup$ Thank you @Francesco Polizzi, but I am not sure this helps for my problem. According to this paper, my $\pi$ corresponds to a certain section of $\bigwedge^2 \mathsf{S}^2 \mathscr{0}_{\mathbb{P}^2}(1)^3\otimes \mathscr{0}_{\mathbb{P}^2}(3)$, that is, $\mathscr{0}_{\mathbb{P}^2}(1)^{15}$. How do I find such a section starting from 3 quadratic forms $P,Q,R$ in 3 variables? $\endgroup$ – abx Jun 24 at 6:01