-- -*- M2-comint -*- {* hash: -652190796 *} i1 : kk = ZZ/101; i2 : S=kk[x_0..x_4]; i3 : i=monomialCurveIdeal(S,{2,3,5,6}) 2 3 2 2 2 2 o3 = ideal (x x - x x , x - x x , x x - x x , x - x x , x x - x x , x x 2 3 1 4 2 0 4 1 2 0 3 3 2 4 1 3 0 4 0 3 ------------------------------------------------------------------------ 2 2 3 2 - x x , x x - x x x , x - x x ) 1 4 1 3 0 2 4 1 0 4 o3 : Ideal of S i4 : time V1 = reesIdeal i; -- used 0.076989 seconds o4 : Ideal of S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 i5 : time V2 = reesIdeal(i,i_0); -- used 0.183972 seconds o5 : Ideal of S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 i6 : numgens V1 o6 = 15 i7 : numgens V2 o7 = 15 i8 : M1 = gens gb V1; 1 84 o8 : Matrix (S[w , w , w , w , w , w , w , w ]) <--- (S[w , w , w , w , w , w , w , w ]) 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 i9 : M2 = gens gb V2; 1 84 o9 : Matrix (S[w , w , w , w , w , w , w , w ]) <--- (S[w , w , w , w , w , w , w , w ]) 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 i10 : use ring M2 o10 = S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 o10 : PolynomialRing i11 : M1 = substitute(M1, ring M2); 1 84 o11 : Matrix (S[w , w , w , w , w , w , w , w ]) <--- (S[w , w , w , w , w , w , w , w ]) 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 i12 : M1 == M2 o12 = true i13 : numgens source M2 o13 = 84 i14 : S=kk[a,b,c] o14 = S o14 : PolynomialRing i15 : m=matrix{{a,0},{b,a},{0,b}} o15 = | a 0 | | b a | | 0 b | 3 2 o15 : Matrix S <--- S i16 : i=minors(2,m) 2 2 o16 = ideal (a , a*b, b ) o16 : Ideal of S i17 : time reesIdeal i -- used 0.027996 seconds 2 o17 = ideal (b*w - a*w , b*w - a*w , w - w w ) 1 2 0 1 1 0 2 o17 : Ideal of S[w , w , w ] 0 1 2 i18 : res i 1 3 2 o18 = S <-- S <-- S <-- 0 0 1 2 3 o18 : ChainComplex i19 : m=random(S^3,S^{4:-1}) o19 = | -10a-10b+17c -40a-5b+31c -29a+28b-18c -31a-20b+48c | | -20a+2b-c -37a+11b+40c 20a+25b-40c 26a+14b+22c | | -47a+36b+12c -8a-15b-2c -3a+4b+9c -2a-49b-48c | 3 4 o19 : Matrix S <--- S i20 : i=minors(3,m) 3 2 2 3 2 2 2 o20 = ideal (- 42a + 4a b - 43a*b + 26b - 11a c - 21a*b*c + 42b c - 3a*c ----------------------------------------------------------------------- 2 3 3 2 2 3 2 2 + 10b*c - 25c , - 48a - 5a b + 6a*b + 13b + 10a c - 22a*b*c + 9b c ----------------------------------------------------------------------- 2 2 3 3 2 2 3 2 + 17a*c + 41b*c + 37c , - 34a - 37a b + 31a*b + 36b - 14a c + ----------------------------------------------------------------------- 2 2 3 3 2 2 3 2 33a*b*c - 23b c - b*c + 19c , - 26a + 23a b + 47a*b - 35b + 18a c + ----------------------------------------------------------------------- 2 2 2 3 23a*b*c + 41b c + 30a*c + 7b*c + 27c ) o20 : Ideal of S i21 : time I=reesIdeal (i,i_0); -- used 0.011998 seconds o21 : Ideal of S[w , w , w , w ] 0 1 2 3 i22 : transpose gens I o22 = {-1, -4} | w_0c+39w_1a+26w_1b+45w_1c-45w_2a-31w_2b+10w_2c+49w_3a-28w_ {-1, -4} | w_0b+14w_1a-25w_1b+27w_1c-32w_2a-44w_2b-46w_2c+16w_3a+27w_ {-1, -4} | w_0a-18w_1a+41w_1b-20w_1c+50w_2a-39w_2b-21w_2c-26w_3a-22w_ {-3, -9} | w_0^3+2w_0^2w_1+39w_0w_1^2+17w_1^3+16w_0^2w_2-40w_0w_1w_2- ----------------------------------------------------------------------- 3b+22w_3c 3b+44w_3c 3b+16w_3c 18w_1^2w_2-2w_0w_2^2+13w_1w_2^2-21w_2^3+23w_0^2w_3-6w_0w_1w_3+47w_1^2w_ ----------------------------------------------------------------------- 3+38w_0w_2w_3+9w_1w_2w_3+11w_2^2w_3+19w_0w_3^2+29w_1w_3^2-14w_2w_3^2+ ----------------------------------------------------------------------- | | | 44w_3^3 | 4 1 o22 : Matrix (S[w , w , w , w ]) <--- (S[w , w , w , w ]) 0 1 2 3 0 1 2 3 i23 : i=minors(2,m); o23 : Ideal of S i24 : time I=reesIdeal (i,i_0); -- used 0.075989 seconds o24 : Ideal of S[w , w , w , w , w , w , w , w , w , w , w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 i25 : R = ZZ/32003[x,y,z] o25 = R o25 : PolynomialRing i26 : I = ideal(x,y) o26 = ideal (x, y) o26 : Ideal of R i27 : cusp = ideal(x^2*z-y^3) 3 2 o27 = ideal(- y + x z) o27 : Ideal of R i28 : RI = reesIdeal(I) o28 = ideal(y*w - x*w ) 0 1 o28 : Ideal of R[w , w ] 0 1 i29 : S = ring RI o29 = S o29 : PolynomialRing i30 : totalTransform = substitute(cusp, S) + RI 3 2 o30 = ideal (- y + x z, y*w - x*w ) 0 1 o30 : Ideal of S i31 : D = decompose totalTransform -- the components are the proper transform of the cuspidal curve and the exceptional curve 3 2 2 2 2 o31 = {ideal (y*w - x*w , y - x z, x*z*w - y w , z*w - y*w ), ideal (y, 0 1 0 1 0 1 ----------------------------------------------------------------------- x)} o31 : List i32 : totalTransform = first flattenRing totalTransform 3 2 o32 = ideal (- y + x z, w y - w x) 0 1 ZZ o32 : Ideal of -----[w , w , x, y, z] 32003 0 1 i33 : L = primaryDecomposition totalTransform 3 2 2 2 2 2 o33 = {ideal (w y - w x, y - x z, w x*z - w y , w z - w y), ideal (y , x*y, 0 1 0 1 0 1 ----------------------------------------------------------------------- 2 x , w y - w x)} 0 1 o33 : List i34 : apply(L, i -> (degree i)/(degree radical i)) o34 = {1, 2} o34 : List i35 : use ring L_0 ZZ o35 = -----[w , w , x, y, z] 32003 0 1 o35 : PolynomialRing i36 : singular = ideal(singularLocus(L_0)); ZZ o36 : Ideal of -----[w , w , x, y, z] 32003 0 1 i37 : SL = saturate(singular, ideal(x,y,z)); ZZ o37 : Ideal of -----[w , w , x, y, z] 32003 0 1 i38 : saturate(SL, ideal(w_0,w_1)) -- we get 1 so it is smooth. o38 = ideal 1 ZZ o38 : Ideal of -----[w , w , x, y, z] 32003 0 1 i39 :