Consider a general $4\times 4$ matrix: $$ X:=\left( \begin{array}{cccc} X_0 & X_1 & X_2 & X_3 \\ X_4 & X_5 & X_6 & X_7 \\ X_8 & X_9 & X_{10} & X_{11} \\ X_{12} & X_{13} & X_{14} & X_{15} \end{array} \right) $$ and let $Y_k\subset\mathbb{P}^{15}$ the variety of matrices of rank equal to $k$. What is the Picard group of $Y_k$?

For instance, for $k = 1$ we have that $Y_1\cong\mathbb{P}^3\times\mathbb{P}^3$ so that $\text{Pic}(Y_1)\cong\mathbb{Z}\times \mathbb{Z}$. The variety $Y_3$ is the hypersurface given by $\det(X) = 0$ from which we remove the locus of matrices of rank less than or equal to $2$.