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A topological group structure lifts to a covering space. Therefore, other than $\mathbb RP^1$ the only real projective space with a group structure is $\mathbb RP^3\simeq SO(3)$.


There is a typo in the book. In fact, $K_X = -5E$. One way to see it is with your argument: $F_4$ is a divisor of class $H = 2E$, so if $K_X = -rE$ then by adjunction $$ K_{F_4} = (K_X + F_4)\vert_{F_4} = (-rE + 2E)\vert_{F_4} = -(r-2)E\vert_{F_4}, $$ which gives $r - 2 = 3$, so $r = 5$. Another way to see this is by identifying $X$ with the weighted ...


Showing that the diagonal is closed in $\mathbb{P}^1 \times \mathbb{P}^1$ is a step in showing that $\mathbb{P}^1$ itself is a variety, not that the product is a variety. Once you've shown that $\mathbb{P}^1$ is a variety, then show that the product of any two varieties is again a variety. If your question is why $\mathbb{P}^1$ is a variety to begin with, ...


As far as I know, the naming is coincidental.* I think projective spaces as understood in projective geometry get the "projective" notion from the idea of light and images. That is, one can understand 2-d perspective as light rays being collapsed into points ("projected onto") a canvas from different angles. But I think the term for modules arises from ...

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