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Let $x_3\rightarrow x_2 \rightarrow x_1$ be a sequence of infinitely near points on $\mathbb P^2$ :

  • $x_1 $ is a point on the complex projective plane $X_1=\mathbb P^2$;

  • $x_2$ is a point on the exceptional divisor $E_2$ of the blow-up $X_2= {\rm Bl}_{x_1}(\mathbb P^2)\rightarrow \mathbb P^2$ of $X_1$ at $x_1$;

  • $x_3$ is a point on the exceptional divisor $E_3$ of the blow-up $X_3= {\rm Bl}_{x_2}(X_2)\rightarrow X_2$ of $X_2$ at $x_2$.

Question: what does it means for $x_1,x_2,x_3$ to be collinear?

For instance, this notion appears page 401 of Dolgachev book "Classical Algebraic Geometry".

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1 Answer 1

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The point $x_2$ determines a line $L$ in $X_1$ passing through $x_1$. So, the total transform of this line in $X_2$ is $L_2+E_2$ and $\{x_2\}=L_2\cap E_2$. When we blow up $x_2$, the inverse image of $L_2$ is $L_3+E_3$ and $L_3\cap E_3$ is a single point. If this point is $x_3$, one says $x_1,x_2,x_3$ are collinear.

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