In the book *Discriminants, Resultants, and Multidimensional Determinants* of Andrei Zelevinsky,M.M. Kapranov and Izrail' Moiseevič Gel'fand, the authors give the following definition of degree of a hypersurface in a Grassmannian.

As they say, in generale a hypersurfaces in a projective variety is not given by the vanishing of a polynomial in its coordinate ring, but for Grassmannians this is true, since its coordinate ring is a UFD, therefore every height-one prime is principal by Krull Theorem.

However I'm stuck on the definition of degree of a hypersurface in a Grassmannian. To be more precise...I wuold prove that this definition is well posed, as in the case of projective hypersurfaces:

The set of pencils $P_{NM}$ is parametrized by the flag variety $\mathcal{Fl}(n;k-1,k+1):=\lbrace N\subset M \: : \: \dim N=k-1, \dim M=k+1 \rbrace$. I would check that there exists a nonempty open subset $U$ of $\mathcal{Fl}(n;k-1,k+1)$ such that the intersection number of $Z$ with any pencil $P_{NM}$ in $U$ is equal to the maximal number of intersection points of $Z$ with a pencil $P\not\subset Z$.

Any help?