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Hi guys!!I'm new in this forum. I have a simple question for you. Let $k$ an algebraically closed field. Consider $\mathbb{P}^1\times\mathbb{P}^1$ and $T_{\mathbb{P}^1\times\mathbb{P}^1}$ the tangent sheaf. How can I prove that $\dim_k \Gamma(\mathbb{P}^1\times\mathbb{P}^1,T_{\mathbb{P}^1\times\mathbb{P}^1})=6$???

Good...I'm so stupid...It's enough observe that $\Omega_{A\otimes_R B/R}\cong A\otimes\Omega_{B/R}\oplus\Omega_{A/R}\otimes B$. Then dualize this and use the Euler sequence for $\mathbb{P}^1$

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    $\begingroup$ Well, that's because it's the direct sum of the pullbacks of the tangent bundle on $\mathbb{P}^1$, so you just need that it has 3 dimensional global sections. $\endgroup$
    – Charles Siegel
    Commented May 23, 2010 at 14:45
  • $\begingroup$ Here's a nice lower bound: there is an action of the three dimensional group $PGL_2$ on each factor by automorphisms, and you can get tangent fields from the Lie algebra. $\endgroup$
    – S. Carnahan ♦
    Commented May 23, 2010 at 15:00
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    $\begingroup$ +1 for self-reflection and enthusiastic punctuation. I think you're being a little hard on yourself. $\endgroup$
    – S. Carnahan ♦
    Commented May 23, 2010 at 22:53

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