Suppose we define projective spaces over some field $k$, and consider the product $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2}$. Unlike the affine case, we have $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2} \ne \mathbb{P}^{n_1 + n_2}$. However, we can still embed the product $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2}$ into a projective space via the Segre embedding defined by the map $\sigma : \mathbb{P}^{n_1} \times \mathbb{P}^{n_2} \rightarrow \mathbb{P}^{(n_1 + 1)(n_2 + 1) - 1}$ where $\sigma: ([x_0: \cdots : x_{n_1}], [y_0: \cdots: y_{n_2}]) \mapsto [x_0 y_0 : x_0 y_1 : \cdots : x_0 y_{n_2} : \cdots : x_i y_j : \cdots : x_{n_1} y_{n_2}]$. In this setting, a (Zariski) closed set in $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2}$ is then simply a preimage of a closed set in $\mathbb{P}^{(n_1 + 1)(n_2 + 1) - 1}$ under $\sigma$.
So is this notion well-behaved in the sense that the preimage of a curve under $\sigma$ should correspond to a 'curve' in $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2}$ or variety of smaller dimension, and the same holds for subvarieties of any dimension?