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I am reading the section on Monoidal transformations from Hartshorne. I was wondering about the following:

Suppose if $X$ is non-singular projective surface. $C$ is a curve on $X$. If we blow up a point $p$ lying on $C$ to get the non-singular surface $\tilde{X}$. Let $f:\tilde{X}\rightarrow X$ be the corresponding morphism. Suppose $E\subset\tilde{X}$ denote the exceptional curve. Let $D\subset\tilde{X}$ be the strict transform of $C$.

I have a few clarifications:

  1. If $p$ is a point of multiplicity 1, then $f^*O(C)=O(D+E)$. Is this right?

2.If $V$ is a vector bundle on $C$, then we can consider $V'=f|_C^*V$, a vector bundle on the divisor $D+E$. What is $V'|_E$? I think it is trivial?

3.If we take a vector bundle $V$ on $X$, then what is $f^*V|_E$? It seems to me that this should be trivial also? Since essentially we are considering the pull back of $V|_{p}$ to $E$. But then if I take a $V=O_X(C)$, then by 1. what I want to consider is $O(D+E)|_E=O(E)|_E$ which is not trivial.

Where am I going wrong?

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In 1 and 2 you are correct, and in the first part of 3 also. The only wrong thing is that $O(D+E)\vert_E \ne O(E)\vert_E$ since $O(D)\vert_E \ne O_E$. Indeed, $D$ intersects $E$ and the intersection equals the multiplicity of $P$ on $C$. So, if it is 1, then $D\cdot E = 1$, so $O(D)\vert_E \cong O_E(1)$ and it cancels with $O_E(E) \cong O_E(-1)$.

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