(Normal crossings)
Let \(k\) be an algebraically closed field. A variety \(Y\) of dimension \(n\) has a normal crossing singularity at a closed point \(p\) if there is an isomorphism of complete local rings \[ \hat{\OO}_{Y,p} \cong k[[x_0, \ldots, x_n]] / (x_0 x_1 \cdots x_r) \] for some \(0 \leq r \leq n\). That is, the completion of \(Y\) at \(p\) is isomorphic to the completion at the origin of the union of the \(r+1\) coordinate hyperplanes \(V(x_0),\ldots,V(x_r)\) in \(\AA^{n+1}\). \(Y\) has normal crossings if it has a normal crossing singularity or a smooth point at every closed point; the case \(r = 0\) is a smooth point.
(Simple normal crossing divisor)
Let \(X\) be a smooth variety and \(D = \sum_i D_i\) a reduced effective divisor. \(D\) is a normal crossing divisor if at each closed point \(p \in D\) there are local parameters \(x_1, \ldots, x_n\) of \(\hat{\OO}_{X,p}\) with \(D\) defined by \(x_1 \cdots x_r = 0\). \(D\) is a simple normal crossing divisor if moreover every component \(D_i\) is smooth, and at each point the components through \(p\) are cut out by distinct members of a single system of local parameters in \(\OO_{X,p}\) itself.
In characteristic not \(2\), the node \(y^2 = x^2(x+1)\) in \(\AA^2\) has a normal crossing singularity at the origin: \(x\sqrt{x+1}\) is a power series, and \(y^2 - x^2(x+1) = (y - x\sqrt{x+1})(y + x\sqrt{x+1})\) is a product of two parameters. As a divisor on \(\AA^2\) it is normal crossing but not simple normal crossing, since its one component is singular. The cusp \(y^2 = x^3\) is not a normal crossing singularity, since its tangent cone \(y^2 = 0\) is not a pair of distinct lines.
Dependencies
D-0SYCYSmooth and singular points; Jacobian and regularity criteriauses
Backlinks
D-VARLOGRESLog resolutionsuses