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 $2$.



Picard groups of determinantal varieties
There is a natural morphism
$$
Y\_k \to Gr(k,4) \times Gr(k,4)
$$
associating to a matrix its image and coimage. Moreover, this morphism factors as the composition
$$
Y\_k \hookrightarrow \mathbb{P}\_{Gr(k,4) \times Gr(k,4)}(U\_1^\vee \otimes U\_2) \to Gr(k,4) \times Gr(k,4),
$$
where $U\_i$ are the tautological vector bundles of rank $k$ on the Grassmannians, the first morphism is an open embedding, and the comlement image is the hypersurface
$$
Z\_k \subset \mathbb{P}\_{Gr(k,4) \times Gr(k,4)}(U\_1^\vee \otimes U\_2)
$$
of degenerate matrices. If $H\_1,H\_2$ are the generators of the Picard groups of the Grassmannians and $H$ is the hyperplane class of the projective bundle, the class of $Z\_k$ is equal to
$$
[Z\_k] = kH + H\_1 - H\_2.
$$
Therefore,
$$
\mathrm{Pic}(Y\_k) = (\mathbb{Z}H \oplus \mathbb{Z}H\_1 \oplus \mathbb{Z}H\_2)/\mathbb{Z}(kH + H\_1 - H\_2) \cong \mathbb{Z}^2.
$$


