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Is there any example (or more ambitiously, classification) of $X$ with following properties?

  • $X$ is a variety over $\mathbb{C}$;
  • $X$ is projective and normal;
  • $\rho(X) = 1$;
  • $X$ is birational to $\mathbb{P}^n$.

Also, I want to hear a result after adding a singularity condition: How about when $X$ is $\mathbb{Q}$-factorial? How about $X$ is non-singular?

I can't find an example which is not isomorphic to $\mathbb{P}^n$. The only result I know in this direction is Mori's theorem: If a nonsingular variety $X$ has ample tangent bundle, then $X$ is isomorphic to $\mathbb{P}^n$.

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There a plenty of nonsingular examples: quadrics, Grassmanians... – Donu Arapura Jan 5 '12 at 17:11
There are also non-homogeneous examples, e.g., an intersection of two generic quadrics in $\mathbb{P}^5$, certain cubic hypersurfaces in $\mathh{P}^{2n+1}$ with $n>1$, moduli spaces of rank $r$ vector bundles on a fixed genus $g>1$ curve with fixed determinant of degree $d$ relatively prime to $r$, ... – Jason Starr Jan 5 '12 at 18:55
Oh, I thought it too difficult way... Thank you for comments! – Moon Jan 5 '12 at 21:45
up vote 4 down vote accepted

Hyperquadrics of dimension at least three.

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