Consider the Cremona transformation $\zeta:\mathbb{P}^2\to\mathbb{P}^2$, taking $(x:y:z)$ to $(\frac{1}{x}:\frac{1}{y}:\frac{1}{z})$. It is undefined at the three coordinate points $p_1=(1:0:0),\ p_2=(0:1:0),\ p_3=(0:0:1).$ If a smooth curve $X\subset \mathbb{P}^2$ is of degree $d$, then its image $\zeta(X)$ has degree $2d-\sum_{i=1}^3\mu_i,$ where $\mu_i$ is the multiplicity of $X$ at $p_i$. This helps a lot with classifying the curves $X$ which are invariant under $\zeta$.
I am currently interested in curves $X\subset\mathbb{P}^2\times\mathbb{P}^2$ invariant under $\tau=\zeta\times\zeta$. Is there a way to compute some invariants of the image of curve, like its degree or genus, similarly to how it's done for the case of a plane curve?
I tried to apply the idea of the Standard Cremona Involution question. Let $\alpha,\beta$ the generators of Chow ring of $\mathbb{P}^2\times\mathbb{P}^2$ corrseponding to hyperplanes in the factors. Then if we have $[X]=A\alpha\beta^2 + B\alpha^2\beta$, it is tempting to write $[\tau(X)]=A'\alpha\beta^2+B'\alpha^2\beta$ with $$A'=2a-\sum \text{mult}_p X,$$ $$B'=2b-\sum \text{mult}_p X,$$ where the sum is taken over the points $p$ where $\tau$ is undefined, which are of the form $p_i\times q$, $q\times p_i.$ However, something is bugging me about this answer, I expected that the multiplicities of these points to affect $A'$ and $B'$ differently. Is this correct?
I am also interested in general results of this sort, like what can be said about the image of a projective variety under a Cremona transformation of $\mathbb{P}^n$ or even more general rational transformations, and would love to see any references for this topic.