This question was previously asked on Math SE.

Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \to \mathbb{CP}^3$. It follows from the degree-genus formula that the same is not true if we replace $\mathbb{CP}^3$ with $\mathbb{CP}^2$; for example, no Riemann surface of genus two can be embedded in $\mathbb{CP}^2$.

Said another way, there exists a compact complex manifold of dimension three into which every compact Riemann surface embeds (namely $\mathbb{CP}^3$), but the natural two-dimensional candidate (namely $\mathbb{CP}^2$) does not have this property. So my question is

Is there a compact complex surface into which every compact Riemann surface embeds?

The only other surface I have checked is $\mathbb{CP}^1\times\mathbb{CP}^1$. My hope was that every Riemann surface $\Sigma$ admitted an embedding $\varphi : \Sigma \to \mathbb{CP}^3$ with $\varphi(\Sigma)$ contained in the image of the Segre embedding $\mathbb{CP}^1\times\mathbb{CP}^1 \to \mathbb{CP}^3$. Many more Riemann surfaces embed in $\mathbb{CP}^1\times\mathbb{CP}^1$ than in $\mathbb{CP}^2$ - in particular, every genus can be realised. However, not all Riemann surfaces can be embedded in $\mathbb{CP}^1\times\mathbb{CP}^1$: any genus three Riemann surface which embeds in $\mathbb{CP}^1\times\mathbb{CP}^1$ must be hyperelliptic, but one can show using a dimension counting argument that there exist non-hyperelliptic genus three Riemann surfaces.