A.4.4 T1 and T2

, resp. , of an ideal usually denote the modules of infinitesimal deformations, resp. of obstructions. In SINGULAR there are procedures T_1 and T_2 in sing.lib such that T_1(j) and T_2(j) compute a standard basis of a presentation of these modules. If are finite dimensional K-vector spaces (e.g., for isolated singularities), a basis can be computed by applying kbase(T_1(j));, resp. kbase(T_2(j));, the dimensions by applying vdim. For a complete intersection j the procedure Tjurina also computes , but faster ( in this case). For a non complete intersection, it is faster to use the procedure T_12 instead of T_1 and T_2. Type help T_1; (or help T_2; or help T_12;) to obtain more detailed information about these procedures.

We give three examples, the first being a hypersurface, the second a complete intersection, the third not a complete intersection:

• load sing.lib
• check whether the ideal j is a complete intersection. It is, if number of variables = dimension + minimal number of generators
• compute the Tjurina number
• compute a vector space basis (kbase) of

• compute the Hilbert function of

• create a polynomial encoding the Hilbert series
• compute the dimension of

LIB "sing.lib"; ring R=32003,(x,y,z),ds; // --------------------------------------- // hypersurface case (from series T[p,q,r]): int p,q,r = 3,3,4; poly f = x^p+y^q+z^r+xyz; tjurina(f); ==> 8 // Tjurina number = 8 kbase(Tjurina(f)); ==> // Tjurina number = 8 ==> _[1]=z3 ==> _[2]=z2 ==> _[3]=yz ==> _[4]=xz ==> _[5]=z ==> _[6]=y ==> _[7]=x ==> _[8]=1 // --------------------------------------- // complete intersection case (from series P[k,l]): int k,l =3,2; ideal j=xy,x^k+y^l+z2; dim(std(j)); // Krull dimension ==> 1 size(minbase(j)); // minimal number of generators ==> 2 tjurina(j); // Tjurina number ==> 6 module T=Tjurina(j); ==> // Tjurina number = 6 kbase(T); // a sparse output of the k-basis of T_1 ==> _[1]=z*gen(1) ==> _[2]=gen(1) ==> _[3]=y*gen(2) ==> _[4]=x2*gen(2) ==> _[5]=x*gen(2) ==> _[6]=gen(2) print(kbase(T)); // columns of matrix are a k-basis of T_1 ==> z,1,0,0, 0,0, ==> 0,0,y,x2,x,1 // --------------------------------------- // general case (cone over rational normal curve of degree 4): ring r1=0,(x,y,z,u,v),ds; matrix m[2][4]=x,y,z,u,y,z,u,v; ideal i=minor(m,2); // 2x2 minors of matrix m module M=T_1(i); // a presentation matrix of T_1 ==> // dim T_1 = 4 vdim(M); // Tjurina number ==> 4 hilb(M); // display of both Hilbert series ==> (-4t5+20t4-40t3+40t2-20t+4) / (1-t)^5 ==> (4) / (1-t)^0 ==> // dimension (local) = 0 ==> // multiplicity = 4 bigintvec v1=hilb(M,1); // first Hilbert series as intvec bigintvec v2=hilb(M,2); // second Hilbert series as intvec v1; ==> 4,-20,40,-40,20,-4,0 v2; ==> 4,0 v1[3]; // 3rd coefficient of the 1st Hilbert series ==> 40 module N=T_2(i); ==> // dim T_2 = 3

// In some cases it might be useful to have a polynomial in some ring // encoding the Hilbert series. This polynomial can then be // differentiated, evaluated etc. It can be done as follows: ring H = 0,t,ls; poly h1; int ii; for (ii=1; ii<=size(v1); ii=ii+1) { h1=h1+v1[ii]*t^(ii-1); } h1; // 1st Hilbert series ==> 4-20t+40t2-40t3+20t4-4t5 diff(h1,t); // differentiate h1 ==> -20+80t-120t2+80t3-20t4 subst(h1,t,1); // substitute t by 1 ==> 0 // The procedures T_1, T_2, T_12 may be called with two arguments and then // they return a list with more information (type help T_1; etc.) // e.g., T_12(i,<any>); returns a list with 9 nonempty objects where // _[1] = std basis of T_1-module, _[2] = std basis of T_2-module, // _[3]= vdim of T_1, _[4]= vdim of T_2 setring r1; // make r1 again the basering list L = T_12(i,1); ==> // dim T_1 = 4 ==> // dim T_2 = 3 kbase(L[1]); // kbase of T_1 ==> _[1]=1*gen(2) ==> _[2]=1*gen(3) ==> _[3]=1*gen(6) ==> _[4]=1*gen(7) kbase(L[2]); // kbase of T_2 ==> _[1]=1*gen(6) ==> _[2]=1*gen(8) ==> _[3]=1*gen(9) L[3]; // vdim of T_1 ==> 4 L[4]; // vdim of T_2 ==> 3