Let $C$ be a twisted cubic in $\mathbb P^3$. I’d like to compute the splitting type of normal bundle $N_{C/\mathbb P^3}$? I understood that $T_{\mathbb P^3}|_C=\mathcal O(4)^{\oplus 3}.$ So we have an short exact sequence $$ 0 \to \mathcal O(2) \to \mathcal O(4)^{\oplus 3} \to N_{C/\mathbb P^3} \to 0. $$ So $N_{C/\mathbb P^3} =\mathcal O(4) […]

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don’t really have time to learn all basics of sheaves theory and the connection to vector bundles. I just need a definition […]

On page 362 of Ravi Vakil’s notes, the author says “It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves – as $\mathcal O_X$-modules – to be locally free; we could define quasi-coherent sheaves to be those $\mathcal O_X$ […]

I’ve been working on the Hartshorne exercise in the title for quite a while, which goes like this: let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes, $\mathscr{L}$ a very ample invertible sheaf on $X$ relative to $Y$, and $\mathscr{M}$ a very ample invertible sheaf on $Y$ relative […]

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