I wanted to get generating ideal of an elliptic normal curve of degree d and the syzygy of it, so I tried to read Eisenbud's book.
In Eisenbud's "The Geometry of Syzygies" Proposition 6.20, he says
Proposition 6.20. Let $Y$ be the surface defined by zero locus of $2\times2$ minors of $M$ below. The divisor class group of $Y$ is $Pic Y = \mathbf{Z}H ⊕ \mathbf{Z}F$, where $H$ is the class of a hyperplane section and $F$ is the class of a line defined by the vanishing of one of the rows of the matrix $M=\mathcal{M}(\mathcal{O}_X(D),\mathcal{L}(−D))$ used to define $Y$. The intersection numbers of these classes are $F · F = 0, F · H = 1$, and $H · H = r − 1.$
$M=\begin{pmatrix} x_0 & x_1 && \cdots && x_{n-1} && y_0 & y_1 && \cdots && y_{m-1} \\ x_1 & x_2 && \cdots && x_{n} && y_1 & y_2 && \cdots && y_{m} \end{pmatrix}$
Then does "$F$ is the class of a line defined by the vanishing of one of the rows of the matrix M" means $F=[Z(x_0,\cdots,y_{m-1})]$?
I did not study algebraic geometry much, so I am confused about this.