A.4.7 Branches of space curve singularities

In this example, the number of branches of a given quasihomogeneous isolated space curve singularity will be computed as an example of the pitfalls appearing in the use of primary decomposition. When dealing with singularities, two situations are possible in which the primary decomposition algorithm might not lead to a complete decomposition: first of all, one of the computed components could be globally irreducible, but analytically reducible (this is impossible for quasihomogeneous singularities) and, as a second possibility, a component might be irreducible over the rational numbers, but reducible over the complex numbers.

ring r=0,(x,y,z),ds; ideal i=x^4-y*z^2,x*y-z^3,y^2-x^3*z; // the space curve singularity qhweight(i); ==> 1,2,1 // The given space curve singularity is quasihomogeneous. Hence we can pass // to the polynomial ring. ring rr=0,(x,y,z),dp; ideal i=imap(r,i); resolution ires=mres(i,0); ires; ==> 1 3 2 ==> rr <-- rr <-- rr ==> ==> 0 1 2 ==> // From the structure of the resolution, we see that the Cohen-Macaulay // type of the given singularity is 2 // // Let us now look for the branches using the primdec library. LIB "primdec.lib"; primdecSY(i); ==> [1]: ==> [1]: ==> _[1]=z3-xy ==> _[2]=x3+x2z+xz2+xy+yz ==> _[3]=x2z2+x2y+xyz+yz2+y2 ==> [2]: ==> _[1]=z3-xy ==> _[2]=x3+x2z+xz2+xy+yz ==> _[3]=x2z2+x2y+xyz+yz2+y2 ==> [2]: ==> [1]: ==> _[1]=x-z ==> _[2]=z2-y ==> [2]: ==> _[1]=x-z ==> _[2]=z2-y def li=_[1]; ideal i2=li[2]; // call the first ideal i1 // The curve seems to have 2 branches by what we computed using the // algorithm of Shimoyama-Yokoyama. // Now the same computation by the Gianni-Trager-Zacharias algorithm: primdecGTZ(i); ==> [1]: ==> [1]: ==> _[1]=-z2+y ==> _[2]=x-z ==> [2]: ==> _[1]=-z2+y ==> _[2]=x-z ==> [2]: ==> [1]: ==> _[1]=z8+yz6+y2z4+y3z2+y4 ==> _[2]=xz5+z6+yz4+y2z2+y3 ==> _[3]=-z3+xy ==> _[4]=x2z2+xz3+xyz+yz2+y2 ==> _[5]=x3+x2z+xz2+xy+yz ==> [2]: ==> _[1]=z8+yz6+y2z4+y3z2+y4 ==> _[2]=xz5+z6+yz4+y2z2+y3 ==> _[3]=-z3+xy ==> _[4]=x2z2+xz3+xyz+yz2+y2 ==> _[5]=x3+x2z+xz2+xy+yz // Having computed the primary decomposition in 2 different ways and // having obtained the same number of branches, we might expect that the // number of branches is really 2, but we can check this by formulae // for the invariants of space curve singularities: // // mu = tau - t + 1 (for quasihomogeneous curve singularities) // where mu denotes the Milnor number, tau the Tjurina number and // t the Cohen-Macaulay type // // mu = 2 delta - r + 1 // where delta denotes the delta-Invariant and r the number of branches // // tau can be computed by using the corresponding procedure T1 from // sing.lib. setring r; LIB "sing.lib"; T_1(i); ==> // dim T_1 = 13 ==> _[1]=gen(6)+2z*gen(5) ==> _[2]=gen(4)+3x2*gen(2) ==> _[3]=gen(3)+gen(1) ==> _[4]=x*gen(5)-y*gen(2)-z*gen(1) ==> _[5]=x*gen(1)-z2*gen(2) ==> _[6]=y*gen(5)+3x2z*gen(2) ==> _[7]=y*gen(2)-z*gen(1) ==> _[8]=2y*gen(1)-z2*gen(5) ==> _[9]=z2*gen(5) ==> _[10]=z2*gen(1) ==> _[11]=x3*gen(2) ==> _[12]=x2z2*gen(2) ==> _[13]=xz3*gen(2) ==> _[14]=z4*gen(2) setring rr; // Hence tau is 13 and therefore mu is 12. But then it is impossible that // the singularity has two branches, since mu is even and delta is an // integer! // So obviously, we did not decompose completely. Because the second branch // is smooth, only the first ideal can be the one which can be decomposed // further. // Let us now consider the normalization of this first ideal i1. LIB "normal.lib"; normal(i2); ==> ==> // 'normal' created a list, say nor, of two elements. ==> // To see the list type ==> nor; ==> ==> // * nor[1] is a list of 1 ring(s). ==> // To access the i-th ring nor[1][i], give it a name, say Ri, and type ==> def R1 = nor[1][1]; setring R1; norid; normap; ==> // For the other rings type first (if R is the name of your base ring) ==> setring R; ==> // and then continue as for R1. ==> // Ri/norid is the affine algebra of the normalization of R/P_i where ==> // P_i is the i-th component of a decomposition of the input ideal id ==> // and normap the normalization map from R to Ri/norid. ==> ==> // * nor[2] is a list of 1 ideal(s). Let ci be the last generator ==> // of the ideal nor[2][i]. Then the integral closure of R/P_i is ==> // generated as R-submodule of the total ring of fractions by ==> // 1/ci * nor[2][i]. ==> [1]: ==> [1]: ==> // characteristic : 0 ==> // number of vars : 6 ==> // block 1 : ordering dp ==> // : names T(1) T(2) T(3) ==> // block 2 : ordering dp ==> // : names x y z ==> // block 3 : ordering C ==> [2]: ==> [1]: ==> _[1]=y ==> _[2]=xz ==> _[3]=x2 ==> _[4]=z2 def rno=_[1][1]; setring rno; norid; ==> norid[1]=-T(2)*z+x ==> norid[2]=T(1)*x-z ==> norid[3]=T(2)*x-T(3)*z ==> norid[4]=T(1)*z+T(2)*z+T(3)*x+T(3)*z+z ==> norid[5]=-T(2)*y+z^2 ==> norid[6]=T(1)*z^2-y ==> norid[7]=T(2)*z^2-T(3)*y ==> norid[8]=T(1)*y+T(2)*y+T(3)*z^2+T(3)*y+y ==> norid[9]=T(1)^2+T(1)+T(2)+T(3)+1 ==> norid[10]=T(1)*T(2)-1 ==> norid[11]=T(2)^2-T(3) ==> norid[12]=T(1)*T(3)-T(2) ==> norid[13]=T(2)*T(3)+T(1)+T(2)+T(3)+1 ==> norid[14]=T(3)^2-T(1) ==> norid[15]=z^3-x*y ==> norid[16]=x^3+x^2*z+x*z^2+x*y+y*z ==> norid[17]=x^2*z^2+x^2*y+x*y*z+y*z^2+y^2 // The ideal is generated by a polynomial in one variable of degree 4 which // factors completely into 4 polynomials of type T(2)+a. // From this, we know that the ring of the normalization is the direct sum of // 4 polynomial rings in one variable. // Hence our original curve has these 4 branches plus a smooth one // which we already determined by primary decomposition. // Our final result is therefore: 5 branches.