Example of 2-dimensional hypersurface singularity whose exceptional locus of minimal resolution is not normal crossing - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T08:23:09Z http://mathoverflow.net/feeds/question/83183 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83183/example-of-2-dimensional-hypersurface-singularity-whose-exceptional-locus-of-mini Example of 2-dimensional hypersurface singularity whose exceptional locus of minimal resolution is not normal crossing tarosano 2011-12-11T16:57:27Z 2011-12-11T18:36:23Z <p>Let $U= (f=0) \subset \mathbb{C}^3$ be an isolated hypersurface singularity of dimension $2$. Let $\mu: \tilde{U} \rightarrow U$ be its minimal resolution. </p> <p><strong>Question</strong> Is there an example of $U$ such that the exceptional locus $E$ of $\mu$ is not normal crossing? </p> http://mathoverflow.net/questions/83183/example-of-2-dimensional-hypersurface-singularity-whose-exceptional-locus-of-mini/83196#83196 Answer by rita for Example of 2-dimensional hypersurface singularity whose exceptional locus of minimal resolution is not normal crossing rita 2011-12-11T18:36:23Z 2011-12-11T18:36:23Z <p>Yes. </p> <p>Take $f=z^2+(x^3+y^3)(y^3+x^4)$. To compute the resolution, consider the projection $U\to {\mathbb C}^2$ onto the $x,y$ coordinates: it is a double cover branched on the curve $B:={(x^3+y^3)(y^3+x^4)=0}$. Blow up the origin in ${\mathbb C}^2$, and let $U'$ be the surface obtained by base change and normalization: $U'$ is a double cover branched on the strict transform $B'$ of $B$ and it is smooth since $B'$ is smooth. The exceptional locus of the resolution $U'\to U$ is the inverse image $Z\subset U'$ of the exceptional curve of the blow up. It is a standard computation to show that $Z$ is irreducible with $Z^2=-2$, $p_a(Z)=2$ and $Z$ has a simple cusp as singularity. So the resolution is minimal and $Z$ is not normal crossings. </p>