Rationality of fields of invariants for some representations of \(\text{SL}_{2} \times \text{SL}_{2}\) (Q2841772)

The main purpose in the article under review is to prove that the fields of invariants are rational for some irreducible representations of \(\mathrm{SL}_2\times\mathrm{SL}_2\). Such representations are realized as the spaces \(V_{a,b}=H^0(\mathcal O_Q(a,b))\) of biforms of bidegree \((a,b)\) on the surface \(Q=\mathbb P^1\times \mathbb P^1\). The main result of the author is that the quotient \(|\mathcal O_Q(a,b)|/\mathrm{SL}_2\times\mathrm{SL}_2\) is rational when \(a\neq b\) and \(ab\) is even.