In the book Discriminants, Resultants, and Multidimensional Determinants of Andrei Zelevinsky 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.
For doing this I think it's enough to check:
- The maximum number of intersection points of $Z$ with a flag is finite, say equals to $d\geq 0$;
- There exist two Zariski-open $U\subset G(k-1,n)$ and $V\subset G(k+1,n)$, such that for every flag with $N\in U$ and $M\in V$, the cardinality of $P_{NM}\cap Z$ is equals to $d$.
Any help or reference it's well accepted.