Consider the two types of Grassmannians Gr(2,7) and Gr(3,6) having their plucker embeddings in $\mathbb P^{20}$ and $\mathbb P^{19}$ respectivley. The first one is 10-dimensional and latter is 9-dimensional, so each having codimension 10. We can easily compute their defining equations and both of them are defined by 35 equations.


By using computer algebra system(such as Macaulay 2 or Magma) we can also compute their Hilbert series and so the Hilbert numerator. If we do the computation then both varieties have basically same numerator, which suggests that both idels have the same(kind of) free resolution. 


This raises the question whether Gr(3,6) is a linear section of Gr(2,7) or not.? 



How Gr(2,7) and Gr(3,6) are related?
I don't know whether the ideals have the same kind of free resolutions, but $Gr(3,6)$ is definitely not a hyperplane section of $Gr(2,7)$. Otherwise, their $H^{\leq 8}$ would be the same by the Lefschetz hyperplane theorem. However, $H^6(Gr(3,6))$ is 3-dimensional and is spanned by $c\_1^3, c\_1c\_2$ and $c\_3$, and $H^6(Gr(2,7))$ is 2-dimensional: it is spanned by $c\_1^3$ and $c\_1c\_2$. Here $c\_i$'s are the Chern classes of the respective tautological bundles.


