# Hilbert series of special linear sections of Grassmannian $Gr(2,n)$

Consider the Grassmannian $$\operatorname{Gr}(2,n)$$. I want to know Hilbert series of $$H_1 \cap H_2 \dots \cap H_m \cap \operatorname{Gr}(2,n)$$ in the Plücker embedding of $$\operatorname{Gr}(2,n)$$, here $$H_1, H_2, \dots ,H_m$$ are very special hyperplanes, that is, each defined by vanishing of one Plücker coordinate.

• The answer depends on he choice of $H_i$. For instance, for $n = 4$, $m = 3$, if you consider the Plucker equations $x_{12} = x_{13} = x_{14} = 0$, the corresponding linear section is $\mathbb{P}^2$, and if instead you consider $x_{12} = x_{23} = x_{34} = 0$, the corresponding linear section is a reducible conic. They have different Hilbert polynomials. – Sasha May 15 at 18:33
• Thanks Sasha. I want to know if there are some results (with some conditions on coordinates, if needed) for Gr(2,n). Or any progress in this direction. – SAG May 15 at 19:14
• if the linear section is dimensionally transverse (i.e., its dimension is equal to $2(n-2) - m$), then the Hilbert polynomial is equal to $\sum (-1)^i\binom{m}{i} h(t - i)$, where $h(t)$ is the Hilbert polynomial of $Gr(2,n)$. – Sasha May 15 at 19:54
• Thanks Sasha. But I am interested in knowing the case when the hyperplanes are very special. – SAG May 15 at 20:12
• Then the answer depends on how special they are. Please, try to be more precise. – Sasha May 16 at 4:30