A.4.9 Resolution of singularities

Resolution of singularities and applications thereof are provided by the libraries resolve.lib and reszeta.lib; graphical output may be generated automatically by using external programs surf and dot respectively to which a specialized interface is provided by the library resgraph.lib. In this example, the basic functionality of the resolution of singularities package is illustrated by the computation of the intersection matrix and genera of the exceptional curves on a surface obtained from resolving the A6 surface singularity. A separate tutorial, which introduces the complete functionality of the package and explains the rather complicated data structures appearing in intermediate results, can be found at https://www.singular.uni-kl.de/tutor_resol.pdf.

LIB"resolve.lib"; // load the resolution algorithm LIB"reszeta.lib"; // load its application algorithms ring R=0,(x,y,z),dp; // define the ring Q[x,y,z] ideal I=x7+y2-z2; // an A6 surface singularity list L=resolve(I); // compute the resolution list iD=intersectionDiv(L); // compute intersection properties iD; // show the output ==> [1]: ==> -2,0,1,0,0,0, ==> 0,-2,0,1,0,0, ==> 1,0,-2,0,1,0, ==> 0,1,0,-2,0,1, ==> 0,0,1,0,-2,1, ==> 0,0,0,1,1,-2 ==> [2]: ==> 0,0,0,0,0,0 ==> [3]: ==> [1]: ==> [1]: ==> 2,1,1 ==> [2]: ==> 4,1,1 ==> [2]: ==> [1]: ==> 2,1,2 ==> [2]: ==> 4,1,2 ==> [3]: ==> [1]: ==> 4,2,1 ==> [2]: ==> 6,2,1 ==> [4]: ==> [1]: ==> 4,2,2 ==> [2]: ==> 6,2,2 ==> [5]: ==> [1]: ==> 6,3,1 ==> [2]: ==> 7,3,1 ==> [6]: ==> [1]: ==> 6,3,2 ==> [2]: ==> 7,3,2 ==> [4]: ==> 1,1,1,1,1,1 // The output is a list whose first entry contains the intersection matrix // of the exceptional divisors. The second entry is the list of genera // of these divisors. The third and fourth entry contain the information // how to find the corresponding divisors in the respective charts.