Grassmannians which are complete intersections

Question

I'm looking for examples of complete intersections which are U.F.D. I know that that the coordinate ring of $Gr(k,n)$ (in its Plucker embedding) is an U.F.D. I would like to know examples of some Grassmannians $Gr(k,n)$ such that the coordinate ring of $Gr(k,n)$ (in its Plucker embedding) is a complete intersection. I know $Gr(2,4)$ is a complete intersection. I would like to know other examples.

Answer 1

This is the only nontrivial example (ie not of the form $Gr(1,n)$ or $Gr(n-1,n)$). If $Gr(k,n)$ was a complete intersection in it's Plucker embedding, then the degree would be a power of two because the Plucker relations are quadratic and generate the ideal of $Gr(k,n)$. On the other hand, by a calculation involving Pieri's rule, the degree of $Gr(k,n)$ in its Plucker embedding is $$ (k(n-k))!\prod_{i=1}^k\frac{(i-1)!}{(n-k+i-1)!}$$ and I claim this is never a power of two except if $k=1$, $k=n-1$, or $(k,n)=(2,4)$.

If $1<k<n-1$ and $n>4$, then $k(n-k)\geq 2(n-2)$ while the maximum of $n-k+i-1$ is $n-1$. By Betrand's postulate, there is always a prime $p$ satisfying $n-1<p<2(n-1)-2=2(n-2)$. This prime shows up in $(k(n-k))!$ but none of $(n-k+i-1)!$, so the degree is divisible by some prime greater than two and therefore not a power of two.