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I am wondering how to find the Zariski Cover of the Grassmannian over $\mathbb{C}$. I was hoping some reference would go through this calculation. I was recently told that it was not too hard, so I wondered if it was written somewhere.

Any reference that does this, or at least gives an introduction for how to calculate Zariski Covers of specific schemes, would be much appreciated.

Thank you!

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What do you mean by "the Zariski cover"? – Dan Petersen Dec 10 '10 at 11:00
@Dan, presumably, a cover by affine Zariski sets (?) – Mariano Suárez-Alvarez Dec 10 '10 at 14:09
I have this written up in detail (for a problem set a while back). If you'd like to see that, feel free to shoot me an email. – Akhil Mathew Dec 10 '10 at 14:22
@Akhil, thanks for the offer. It is my first name Bryan then a period, then my second name Bischof at the google mail. – BBischof Dec 10 '10 at 15:00

Consider Grassmannian of $d$-dimensional subspaces of $n$-dimensional vector space $V$. Then you have Pluker inbeding of grassmanian $G=Gr(d,V)$ of d-planes of $V$ in $\mathbb P(\Lambda ^d V)=\mathbb P^N, N=\binom{n}{d}-1$, as follows. For every $d-plane$ D chose base $v_1,...,v_d$ of $D$ and consider exterior product $\omega = v_1\wedge...\wedge v_d \in \wedge ^d V$. Pluker inbeding send $D$ to line: Span $\omega \in \mathbb P(\Lambda ^d V)$ .

For find Zariski covering, fix base $e_1,...,e_n$ of $V$ and receive base $e_I=e_{i_1}\wedge...\wedge e_{i_d}$ of $\wedge ^d V $ ($I$ is increasing sequence $I=(1\leq i_1 \lt ...\lt i_d \leq n$)). Zariski covering consist of $U_I$ formed by these $d-$ planes $D$ whose Pluker image in $\mathbb P(\wedge ^d V)$ has coordinate $z_I\neq 0$. This mean that when you chose base $v_1,...,v_d$ of $D$ and developp $\omega = v_1\wedge...\wedge v_d \in \wedge ^d V$ in base $e_I$ of $\Lambda ^d V$, then coefficient of $e_I$ is non-null. This Zariski cover $U_I$ of grasmanian $Gr(d,V)$ possesses $N+1=\binom{n}{d}$ open sets (Attention: $\mathbb P^N$ has $N+1$ coordinates, not $N$)

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Thanks, I will have a look later and respond. – BBischof Dec 10 '10 at 15:00

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