Let $c,d<N$ be integers and consider the product of two Grassmannians $M=Gr(c,N)\times Gr(d,N)$. Define $S\subset M$ to be the set of the pairs $([A\_{c\times N}],[B\_{d\times N}])$ such that $AB^t=0$. Note that this equation is independent of the choices of $A$ and $B$ in classes $[A]$ and $[B]$. This is a subvariety of codimension $c\times d$ and its homology class can be written as 
$$ \sum\_{(a,b)\in T} \sigma\_a \times \sigma\_b,$$
over some set $T$ of pairs of Schubert cycles of $Gr(c,N)$ and $Gr(d,N)$.


Question: What is the set $T$ in terms of $c,d$ and $N$? (Note that $a$ and $b$ are non-increasing sequences of $c$ and $d$ non-negative numbers, respectively.)


For example if $c=d=1$, this just a hypersurface of bidegree $(1,1)$ an is homologous to$\sigma\_1 \times \sigma\_0+ \sigma\_0\times \sigma\_1$, where $\sigma\_0=\mathbb{P}^{N-1}$ and $\sigma\_1=\mathbb{P}^{N-2}\subset \mathbb{P}^{N-1}$.



A calculation over product of Grassmannians
Let $V$ be a vector space of dimension $N$. To be precise your factors are the Grassmannian of $c$-dimensional subspaces in $V$ and the Grassmannian of $d$-dimensional subspaces in $V^\*$. let $U\_c\subset V\otimes O$ and $U\_d \subset V^\*\otimes O$ be the tautological subbundles of rank $c$ and $d$ on these Grassmannians. Then $S$ is the zero locus of the composition
$$
U\_c \boxtimes O \to V \otimes (O \boxtimes O) \to O \boxtimes U\_d^\*
$$
on $M$, i.e. in the zero locus of the canonical global section of the vector bundle $U\_c^\*\boxtimes U\_d^\*$ on $M$. The class of the zero locus is the top Chern class
$$
\mathrm{c}\_{cd}(U\_c^\*\boxtimes U\_d^\*).
$$
You can express it as the polynomial in Chern classes of $U\_c$ and $U\_d$ and then in terms of Schubert classes. I am sure you can find an appropriate formula in Fulton's "Intersection theory".


