# Simple normal crossing divisors

I found the following definition.

A Weil divisor $D = \sum_{i}D_i \subset X$ on a smooth variety $X$ is simple normal crossing if for every point $p \in X$ a local equation of $D$ is $x_1\cdot...\cdot x_r$ for independent local parameters $x_i$ in $O_{p,X}$. A log resolution of the pair $(X,D)$ is a birational morphism $f:Y\rightarrow X$ such that $Y$ is smooth and $f^{-1}D \cup Exc(f)$ is simple normal crossing.

From this definition it seems to me that if $D = L\cup R\cup C\subset\mathbb{P}^2$ is the union of three lines passing through the same point, then $D$ is simple normal crossing.

On the other hand I found an example showing that if $C\subset\mathbb{P}^2$ is a cusp, in order to get a log resolution we have to blow-up three times. However after two blow-ups we end up with the strict transform $\tilde{C}$ of $C$ and two exceptional divisors $E_1, E_2$ intersecting transversally in a point $p = \tilde{C}\cap E_1\cap E_2$. Why is not the divisor $\tilde{C}\cup E_1\cup E_2$ simple normal crossing ?

• That does not look like the standard definition of simple normal crossing: the components of $D$ should also be smooth. What you have is just normal crossing.
– user5117
May 23, 2014 at 13:33

A Weil divisor $$D = \sum_{i}D_i \subset X$$ on a smooth variety $$X$$ of dimension $$n$$ is simple normal crossing if any component $$D_i$$ is smooth and for every point $$p \in X$$ a local equation of $$D$$ is $$x_1\cdot...\cdot x_r$$ for independent local parameters $$x_i$$ in $$O_{p,X}$$ with $$r\leq n$$.
What goes wrong in the case of three line passing through the same point in $$\mathbb{P}^2$$ is that you need three local parameters in a neighborhood of $$p$$ and $$3 > 2$$. The same for the cusp.
More generally, let $$D = \sum H_i\subset\mathbb{P}^n$$ be a reduced divisor, where the $$H_i$$'s are hyperplanes. Consider the points $$h_i\in\mathbb{P}^{n*}$$ dual to the the $$H_i$$'s. Then $$D$$ is simple normal crossing if and only if the $$h_i$$'s are in linear general position in $$\mathbb{P}^{n*}$$. You see that three lines through the same point in $$\mathbb{P}^2$$ correspond to three points on the same line in $$\mathbb{P}^{2*}$$. Therefore this case does not work. More generally, for a curve $$C$$ on a surface, simple normal crossing means any irreducible component is smooth and $$C$$ has at most nodes as singularities.