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Let $X$ be a smooth projective surface over complex numbers. Let $p$ be a point in $X$. Suppose we blow up $p$ to get the surface $Y$. Consider a curve $C$ on $X$ where $p$ has multiplicity 1.

Then if $p:Y\rightarrow X$, then $p^*C= C'+ E$ where $C'$ is the strict transform of $C$ and $E$ is the exceptional divisor. Is $C'\cdot C'= C\cdot C$?

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    No, $C' \cdot C' = C \cdot C -1$. Chapter of Shafarevich's book cover this well. – 2017-02-21
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    Sorry, I meant to write Chapter IV. – 2017-02-21

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