Blowup of $\mathbb{P}^1\times\mathbb{P}^1$ and a sheaf computation

Asked Jan 11 '15 at 22:17

Active Jan 12 '15 at 02:27

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Let $Y_1=Y_2=\mathbb{P}^1$, $Y=Y_1\times Y_2$, $p_i:Y\rightarrow Y_i$, $i=1,2$ be a canonnical projections and $\pi:X\rightarrow Y$ be a blowup of $Y$ in a finite set of points. How to compute $(p_2\pi)_*(p_1\pi)^*(\mathcal{O}_{Y_1}(1))$?

I tried to use a projection formula in the following way $$(p_2\pi)_*(p_1\pi)^*(\mathcal{O}_{Y_1}(1))\cong p_{2*}\pi_*\pi^*(p_1^*(\mathcal{O}_{Y_1}(1)))\cong p_{2*}p_1^*(\mathcal{O}_{Y_1}(1))\cong\mathcal{O}_{Y_2}(1),$$ but the answer seems not to be correct.

edited Jan 12 '15 at 02:27

asked Jan 11 '15 at 22:17

cmplx_variable

(I deleted my incorrect answer so somebody else can answer.) – DCT Jan 12 '15 at 08:26
I have another silly comment. Are you sure that $(p_2){}p_1^{}\mathscr{O}{Y_1}(1)=\mathscr{O}{Y_2}(1)$? I think it is instead $\mathscr{O}{Y_2}^2$ (check on open affines)... – DCT Jan 12 '15 at 21:46
Yes, you are right. – cmplx_variable Jan 14 '15 at 22:12

Blowup of $\mathbb{P}^1\times\mathbb{P}^1$ and a sheaf computation

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