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I could not understand a part of the book: Algebraic geometry: a first course by Joe Harris.

On page 64, line - 4, why the kernel of the map $\phi(\omega)$ will be the annihilator of the kernel of $\psi(\omega)$? Why the images of the transpose of the two maps are annihilator of each other? On page 65, line 6, why $\Theta_{\alpha, \beta}$ are quadratic?

Could someone explain Plucker embedding more easily and clearly? Thank you very much.

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