I started writing an answer about the generators and relations for the $\mathbb{Q}$-Picard group of the stack of stable maps of genus $0$ curves to an arbitrary projective homogeneous variety, but it quickly got too long. So here is an explanation of the extra divisor class relation that I was missing in my comments. This uses Lemma 5.2 of my article with A. J. de Jong.

MR3644251

de Jong, A. J.(1-CLMB); Starr, Jason(1-SUNYS)

Divisor classes and the virtual canonical bundle for genus 0 maps.

Geometry over nonclosed fields, 97–126,

Simons Symp., Springer, Cham, 2017.

https://arxiv.org/abs/math/0602642

The $\mathbb{Q}$-vector space $\text{CH}^2(\mathbb{P}^1\times \mathbb{P}^3)_{\mathbb{Q}}$ is generated by two generators $g_1=\text{pr}_1^*c_1(\mathcal{O}(1))\cdot \text{pr}_2^*c_1(\mathcal{O}(1))$ and $g_2=\text{pr}_2^* c_1(\mathcal{O}(1)^2$. Those give two divisor classes on the moduli space $\overline{\mathcal{M}}_{0,0}(\mathbb{P}^1\times \mathbb{P}^3,\beta)$. However the divisor class relation from Lemma 5.2 proves that, modulo linear combinations of boundary divisors, these two divisors are in the span of the two divisor classes that are pushforwards via $\pi$ from the universal family of stable maps, $$\pi:\mathcal{C}\to \overline{\mathcal{M}}_{0,0}(\mathbb{P}^1\times \mathbb{P}^3,\beta),\ \ f:\mathcal{C}\to \mathbb{P}^1\times \mathbb{P}^3,$$ of the codimension $2$ classes $f^*\text{pr}_1^*c_1(\mathcal{O}(1))\cdot c_1(\omega_\pi)$ and $f^*\text{pr}_2^*c_1(\mathcal{O}(1))\cdot c_1(\omega_\pi)$. Finally, when we restrict on the complement of the boundary divisors, the morphism $$(\text{pr}_1\circ f,\pi):\mathcal{C}\to \mathbb{P}^1\times \overline{\mathcal{M}}_{0,0}(\mathbb{P}^1\times \mathbb{P}^3,\beta),$$ is an isomorphism. In particular, on this open subset, the pullback $f^*\text{pr}_1^*\mathcal{O}(-2)$ is isomorphic to $\omega_\pi$. Thus, modulo the boundary, the $\pi_*$-pushforward of $f^*\text{pr}_1^*c_1(\mathcal{O}(1))\cdot c_1(\omega_\pi)$ is a $\mathbb{Q}$-multiple of the $\pi^*$ pushforward of $f^*\text{pr}_1^* c_1(\mathcal{O}(1))^2$. However, on $\mathbb{P}^1$, the square of $c_1(\mathcal{O}(1))$ equals $0$. Thus, modulo the boundary, the two cycle classes $g_1$ and $g_2$ map to divisor classes that are $\mathbb{Q}$-linearly dependent. So, together with the $3$ boundary divisor classes, this gives $4$ divisor classes, not $5$ as I wrote in the comment above.

Notes on stable maps and quantum cohomology, by Fulton-Pandharipande, in the Proceedings of the SantaCruz conference. $\endgroup$2more comments