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So I know that all Grassmanians are completely covered by coordinate charts that look like affine spaces. (They are of the form $Hom_k(V,W)$.)

This seems to me an unusual property that a projective variety could have. Are there others? What about flag varieties? Is there a nice trick for putting describing charts in the variety structure of a flag variety?

Elle Najt
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  • I asked a similar question about complex manifolds here, but I have not received any answers yet. – Michael Albanese Nov 01 '15 at 20:16
  • Certainly there are many such varieties, for example, if $X$ is one such, then a vector bundle or projectivized vector bundle also have the same property. There are also others, for example, the projective plane blown up at 3 points. – Mohan Nov 01 '15 at 23:07
  • Of course such varieties have to be rational: the function field is of the form $k(x_1, \ldots, x_n)$. In fact, this is already the case when there is one open looking like affine space. – Remy Nov 02 '15 at 00:10
  • Sorry, I did not mean 3, but any number of points. For that matter, same holds for any projective space. – Mohan Nov 02 '15 at 00:59
  • @Mohan Can you explain why that fact about the blow-up is true, or provide a reference? – Elle Najt Nov 02 '15 at 13:46
  • @AreaMan Suffices to do this for blowing up one point. Then it is sufficient to show that if you blow up one point in $\mathbb{A}^n$, it can be covered by affine spaces and this is immediate using the defintion of blowing up. – Mohan Nov 02 '15 at 13:52
  • @Mohan Thanks. To make sure I understand the induction argument: you pick a copy of $A^n$ around the next point you are blowing up (which we can do because the previous steps have shown that the blow up is covered by affines), and blow up there. Then we just have to verify that the $A^n$ we replaced by the blow up of $A^n$ at one point can be covered by affine spaces, and this is the base case. When we blow up a variety at a point the idea is that we pick an affine neighborhood $U$ of that point $p$, then glue in the blow up along $Bl_p(U) \setminus Exceptional \cong U - p$? – Elle Najt Nov 02 '15 at 14:37
  • I don't follow your last step. If you want to blow up a bunch of points on $X$, which can be covered by affine spaces, you can do this one at a time. So, blowing up one point, the space can be covered by inverse images of the affine spaces covering $X$. The affine spaces not containing the point just give affine space. The ones containing the point is just blowing up a point in an affine space and they can be covered by affine spaces. – Mohan Nov 02 '15 at 15:05
  • @Mohan Thanks. I think we are saying pretty much the same thing. – Elle Najt Nov 02 '15 at 15:09

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