Moduli Spaces of semistable rank 2 co-Higgs bundles over $\mathbb{P}^1 \times \mathbb{P}^1$

It has been observed, by S. Rayan, that the complex projective surfaces that potentially admit non-trivial examples of semistable co-Higgs bundles must be found at the lower end of the Enriques-Kodaira classification. Motivated by this remark, it is natural to study the geometry of these objects over $\mathbb{P}^1 \times \mathbb{P}^1$. In this paper, we present necessary and sufficient conditions on the Chern classes $c_1,c_2$ of a bundle that guarantee the non-emptiness of the moduli space $\mathcal{M^{\text{co}}}(c_1,c_2)$ of rank 2 semistable co-Higgs bundles over $\mathbb{P}^1 \times \mathbb{P}^1$ (we do this with respect to the standard polarization). We also give an explicit description of the moduli spaces for certain choices of $c_1$ and $c_2$.