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I ran into an exercise saying that

Any projective variety is isomorphic to the intersection of a Veronese variety with a linear space.

But I don't understand the definition of the term "linear space" in a projective space. In fact, I don't even think $\mathbb{P}^n$ is a vector space.

Can anyone tell me what a linear space is?

hxhxhx88
  • 5,257
  • If $\mathbf{P}^n$ is the projective space corresponding to a vector space $V$ of dimension $n+1$, then a linear space in $\mathbf{P}^n$ just means the closed subset corresponding to a (nonzero) vector subspace $W \subseteq V$. You're quite correct that $\mathbf{P^n}$ is not a vector space. –  Oct 27 '13 at 16:04
  • @AsalBeagDubh, I understand now, thanks! – hxhxhx88 Oct 27 '13 at 16:30
  • You're welcome! –  Oct 27 '13 at 16:38
  • @AsalBeagDubh, by the way, if we say a "line" in a projective space, does it mean the subset corresponding to a 2-dim vector subspace of the affine space $V$ (because in projective space, it has dim 1)? – hxhxhx88 Oct 27 '13 at 16:59
  • Dear hxhxhx88: no. Remember that we go from $V$ to $\mathbf{P}^n$ by taking the quotient of $V \setminus {0}$ by the action of $k^\times$. So the 1-dimensional subspaces of $V$ correspond exactly to the points of $\mathbf{P}^n$, and a line (meaning a 1-dimensional linear subspace) in $\mathbf{P}^n$ corresponds to a 2-dimensional subspace of $V$. –  Oct 27 '13 at 17:04
  • Yeah, I realized and have edited by question. But I guess you have been answering already..Anyway, thanks again! – hxhxhx88 Oct 27 '13 at 17:11

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