-- This M2 file accompanies the paper "A Tight Bound on the Projective -- Dimension of Four Quadrics" by Huneke, Mantero, McCullough and Seceleanu -- Here we provide details for the lemmas in the appendix. -- In every case, x,y,z represent independent linear forms, i.e. variables. -- We wish to show that each ideal remains unmixed or even (x,y,z)-primary. -- By the Buchsbaum-Eisenbud exactness criterion, we must show that the i-th -- ideal of minors of the expected rank r_i has height at least i for -- i = 1, 2, 3 and has height i+1 for i > 3. To do this we exhibit a regular -- sequence of the appropriate length or an explicit ideal of the appropriate -- height in each ideal of minors. -- (Note: "x % I == 0" returns true if x is in I and false otherwise. -- When this technique is too computationally expensive, we construct -- the appropriate minors by hand. F.dd_i^L_M returns the submatrix of -- F.dd_i with rows indexed by the entries of the list L and columns -- indexed by the entries of the list M.) -- Lemma A.1 restart S = QQ[A,B,C,x,y,z] L = ideal(x^2,y^2,z^2,x*y*z,C*x*y+B*x*z+A*y*z) F = res L F.dd x^2 % minors(1,F.dd_1) == 0 x^6 % minors(4,F.dd_2) == 0, y^6 % minors(4,F.dd_2) == 0 x^5 % minors(5,F.dd_3) == 0, y^5 % minors(5,F.dd_3) == 0, z^5 % minors(5,F.dd_3) == 0 ass J degree J -- Lemma A.2 restart S = QQ[a,b,c,d,e,f,x,y,z] J = (ideal(x,y,z))^2 + ideal(a*x+b*y+c*z,d*x+e*y+f*z) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 -- We construct the regular sequences in I_7(dd_2) and I_7(dd_3) directly determinant F.dd_2^{0,1,3,4,5,6,7}_{0,1,4,5,8,10,12} determinant F.dd_2_{2,3,4,5,9,11,13}^{0,1,2,3,5,6,7} determinant F.dd_3_{2,3,4,5,6,7,8}^{0,1,2,3,4,5,10} determinant F.dd_3_{0,1,2,3,6,7,8}^{2,3,6,7,9,11,13} determinant F.dd_3_{0,1,4,5,6,7,8}^{0,1,6,7,8,9,12} x^2 % minors(2,F.dd_4) == 0, y^2 % minors(2,F.dd_4) == 0, z^2 % minors(2,F.dd_4) == 0 a*e-b*d % minors(2,F.dd_4) == 0, a*f-c*d % minors(2,F.dd_4) == 0, b*f-c*e % minors(2,F.dd_4) == 0 ass J degree J -- Lemma A.3 restart S = QQ[a,b,c,x,y,z] J = (ideal(x,y,z))^2+ideal(b*x-a*y,c*x-a*z,c*y-b*z) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 -- check by hand determinant F.dd_2_{9,10,11,12,13,15,16,17}^{0,1,2,3,4,5,6,7} determinant F.dd_2_{1,2,4,6,8,11,14,16}^{0,1,3,4,5,6,7,8} determinant F.dd_3_{5,6,7,8,9,10,11,12,13,14}^{0,1,2,3,4,5,6,7,8,14} determinant F.dd_3_{1,2,3,4,8,9,11,12,13,14}^{0,1,2,5,9,10,11,12,13,16} determinant F.dd_3_{0,2,3,4,6,7,10,11,13,14}^{0,3,4,7,9,10,12,13,15,17} x^5 % minors(5,F.dd_4) == 0, y^5 % minors(5,F.dd_4) == 0, z^5 % minors(5,F.dd_4) == 0, a^5 % minors(5,F.dd_4) == 0, b^5 % minors(5,F.dd_4) == 0, c^5 % minors(5,F.dd_4) == 0 minors(1,F.dd_5) ass J degree J -- Lemma A.4 restart S = QQ[a,b,c,x,y,z] J = (ideal(x,y,z))^2+ideal(a*x+b*y+c*z) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 x^6 % minors(6,F.dd_2) == 0, y^6 % minors(6,F.dd_2) == 0 x^5 % minors(5,F.dd_3) == 0, y^5 % minors(5,F.dd_3) == 0, z^5 % minors(5,F.dd_3) == 0 minors(1,F.dd_4) ass J degree J -- Lemma A.5 restart S = QQ[a,x,y,z] J = ideal(x,y^2,y*z,z^3,a*y+z^2) F = res J F.dd x % minors(1,F.dd_1) == 0 x^3 % minors(3,F.dd_2) == 0, y^4 % minors(3,F.dd_2) == 0 x^2 % minors(2,F.dd_3) == 0, y^2 % minors(2,F.dd_3) == 0, z^2+a*y % minors(2,F.dd_3) == 0 ass J degree J -- Lemma A.6 restart S = QQ[a,b,c,d,x,y,z] J = ideal(x^2,x*y,x*z,y^2,y*z,a*x+b*y+z^2,c*x+d*y) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 x^6 % minors(6,F.dd_2) == 0, y^6 % minors(6,F.dd_2) == 0 x^5 % minors(5,F.dd_3) == 0, y^5 % minors(5,F.dd_3) == 0, (b*y*z^3 + z^5) % minors(5,F.dd_3) == 0 minors(1,F.dd_4) L = ideal(x^2,x*y,y^2,a*x+b*y+z^2,c*x*z-d*y*z) G = res L G.dd x^2 % minors(1,G.dd_1) == 0 x^5 % minors(4,G.dd_2) == 0, y^5 % minors(4,G.dd_2) == 0 x^4 % minors(4,G.dd_3) == 0, y^4 % minors(4,G.dd_3) == 0, z^6 % minors(4,G.dd_3) == 0 minors(1,G.dd_4) ass J degree J -- Lemma A.7 restart S = QQ[a,b,x,y,z] J = ideal(x,y^3,y^2*z,y*z^2,z^3,a*y+b*z) F = res J F.dd x % minors(1,F.dd_1) == 0 x^5 % minors(5,F.dd_2) == 0, y^8 % minors(5,F.dd_2) == 0 x^6 % minors(6,F.dd_3) == 0, y^7 % minors(6,F.dd_3) == 0, z^7 % minors(6,F.dd_3) == 0 x^2 % minors(2,F.dd_4) == 0, y^2 % minors(2,F.dd_4) == 0, z^2 % minors(2,F.dd_4) == 0, a^2 % minors(2,F.dd_2)==0, b^2 % minors(2,F.dd_2) == 0 ass J degree J -- Lemma A.8 restart S = QQ[a,b,c,d,x,y,z] J = ideal(x^2,x*y,x*z,y*z^2,z^3,a*x+b*y+c*z,d*x+y^2) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 x^6 % minors(6,F.dd_2) == 0, y^8 % minors(6,F.dd_2) == 0 x^6 % minors(6,F.dd_3) == 0, y^8 % minors(6,F.dd_3) == 0, z^8 % minors(6,F.dd_3) == 0 x^2 % minors(2,F.dd_4) == 0, y^2 % minors(2,F.dd_4) == 0, z^2 % minors(2,F.dd_4) == 0, c^2 % minors(2,F.dd_4)==0, (b^2*d-a*b*y) % minors(2,F.dd_4) == 0 ass J degree J -- Lemma A.9 restart S = QQ[a,b,c,d,x,y,z] J = ideal(x^2,x*y,x*z,y^3,z^3,a*x+b*y+c*z,d*x+y*z) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 x^6 % minors(6,F.dd_2) == 0, y^9 % minors(6,F.dd_2) == 0 x^6 % minors(6,F.dd_3) == 0, y^9 % minors(6,F.dd_3) == 0, z^9 % minors(6,F.dd_3) == 0 x^2 % minors(2,F.dd_4) == 0, y^2 % minors(2,F.dd_4) == 0, z^2 % minors(2,F.dd_4) == 0, b*c % minors(2,F.dd_4)==0, b^2*d % minors(2,F.dd_4) == 0, c^2*d % minors(2,F.dd_4) == 0 ass J degree J -- Lemma A.10 restart S = QQ[a,b,c,x,y,z] J = ideal(x^2,x*y,x*z,z^3,c*x+y^2,b*x-y*z,a*x+b*y+c*z) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 x^6 % minors(6,F.dd_2) == 0, y^6 % minors(6,F.dd_2) == 0 x^5 % minors(5,F.dd_3) == 0, y^5 % minors(5,F.dd_3) == 0, z^8 % minors(5,F.dd_3) == 0 minors(1,F.dd_4) ass J degree J -- Lemma A.11 restart S = QQ[a,b,c,d,e,f,x,y,z] J = (ideal(x,y,z))^3+ideal(a*x+b*y+c*z,d*x+e*y+f*z) F = res J F.dd x^3 % minors(1,F.dd_1) == 0 determinant F.dd_2^{0,1,3,4,5,6,7,8,9,10,11}_{0,1,5,8,9,12,16,18,22,24,26} determinant F.dd_2^{0,1,2,3,4,6,7,8,9,10,11}_{5,6,7,8,9,15,17,19,23,25,27} determinant F.dd_3^{2,3,4,6,7,10,11,13,14,15,17,19,20,21,23,25,27}_{0,2,3,4,5,7,8,10,12,13,16,17,18,19,22,23,24} determinant F.dd_3^{0,1,2,3,4,10,11,12,13,14,15,17,20,21,22,23,26}_{1,2,3,4,5,9,11,13,14,15,16,17,20,21,22,23,24} determinant F.dd_3^{0,1,2,3,4,5,6,7,8,9,10,11,13,14,16,18,24}_{6,7,8,9,11,13,14,15,16,17,18,19,20,21,22,23,24} determinant F.dd_4^{1,6,9,11,14,15,20,21}_{0,1,3,4,5,6,7,8} determinant F.dd_4^{0,6,7,8,9,11,18,19}_{0,2,3,4,5,6,7,8} determinant F.dd_4^{0,1,2,3,4,5,10,12}_{1,2,3,4,5,6,7,8} factor determinant F.dd_4^{2,7,9,13,16,17,22,23}_{0,1,2,4,5,6,7,8} factor determinant F.dd_4^{3,8,11,13,16,17,22,23}_{0,1,2,3,5,6,7,8} factor determinant F.dd_4^{5,11,15,17,13,16,22,23}_{0,1,2,5,3,4,7,8} factor determinant F.dd_4^{4,9,14,17,13,16,22,23}_{0,1,2,6,3,4,7,8} -- By symmetry, we have (a^4,b^4,d^4,e^4)(ae-bd)^2 + (a^4,c^4,d^4,f^4)(af-cd)^2+(b^4,c^4,e^4,f^4)(bf-cd)^2 -- in I_8(d_4). The associated primes of this ideal are (a,b,c,d,e,f) and -- (ae-bd, af-cd, bf-ce). Hence, under the assumptions of the lemma, -- ht(I_8(d_4)) is at least 5, as desired. minors(1,F.dd_5) ass J degree J -- Lemma A.12 restart S = QQ[a,b,x,y,z] J = (ideal(x,y,z))^3 + ideal(a*x+b*y,a*y+b*z,x*z-y^2) F = res J F.dd x^3 % minors(1,F.dd_1) == 0 determinant F.dd_2^{0,1,2,4,5,6,7,8,9}_{2,3,4,5,7,10,13,16,18} determinant F.dd_2^{0,1,2,3,4,5,6,7,8}_{7,9,12,13,14,15,17,18,19} determinant F.dd_3^{0,1,6,8,9,11,12,14,15,17,19}_{0,1,2,4,5,7,8,10,11,13,14} determinant F.dd_3^{2,3,6,7,8,9,12,13,14,15,18}_{0,1,3,5,6,7,9,11,12,13,14} determinant F.dd_3^{0,2,3,4,5,6,8,10,12,13,16}_{1,3,5,6,7,9,10,11,12,13,14} x^4 % minors(4,F.dd_4) == 0, y^4 % minors(4,F.dd_4) == 0, z^4 % minors(4,F.dd_4) == 0, a^4 % minors(4,F.dd_4) == 0, b^4 % minors(4,F.dd_4) == 0 ass J degree J -- Lemma A.13 restart S = QQ[a,b,x,y,z] J = ideal(x^2,x*y,x*z,y^2,y*z,z^3,a*x+b*y+z^2) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 x^5 % minors(5,F.dd_2) == 0, y^6 % minors(5,F.dd_2) == 0 x^3 % minors(3,F.dd_3) == 0, y^3 % minors(3,F.dd_3) == 0, z^4 % minors(3,F.dd_3) == 0 ass J degree J -- Lemma A.14 restart S = QQ[a,b,c,x,y,z] J = ideal(x^2,x*y,x*z,y^3,y^2*z,y*z^2,z^3,a*x+b*y+c*z) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 x^7 % minors(7,F.dd_2) == 0, y^10 % minors(7,F.dd_2) == 0 determinant F.dd_3^{3,4,6,8,9,11,13}_{0,1,2,4,5,7,8} determinant F.dd_3^{0,2,6,7,9,10,12}_{0,1,3,4,6,7,8} determinant F.dd_3^{1,4,5,0,6,7,10}_{0,3,4,5,6,7,8} x^2 % minors(2,F.dd_4) == 0, y^2 % minors(2,F.dd_4) == 0, z^2 % minors(2,F.dd_4) == 0, b^2 % minors(2,F.dd_4) == 0, c^2 % minors(2,F.dd_4) == 0 ass J degree J -- Lemma A.15 restart S = QQ[a,b,c,d,x,y,z] J = ideal(x^2,x*y,y^2,a*x+b*y+z^2,c*x+d*y) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 x^5 % minors(4,F.dd_2) == 0, y^5 % minors(4,F.dd_2) == 0 x^4 % minors(4,F.dd_3) == 0, y^4 % minors(4,F.dd_3) == 0, z^8 % minors(4,F.dd_3) == 0 minors(1,F.dd_4) ass J degree J -- Lemma A.16 restart S = QQ[a,b,c,d,e,f,x,y,z] J = ideal(x^2,x*y,y^2,a*x+b*y,c*x+d*y,a*d-b*c+e*x+f*y) F = res J F.dd x^2 % minors(1,F.dd_1) == 0 x^5 % minors(5,F.dd_2) == 0, y^5 % minors(5,F.dd_2) == 0 x^3 % minors(3,F.dd_3) == 0, y^3 % minors(3,F.dd_3) == 0, (a*(a*d-b*c)-b*e*y+a*f*y) % minors(3,F.dd_3) == 0, (d*(a*d-b*c)+d*e*x-c*f*x)% minors(3,F.dd_3) == 0, (c*(a*d-b*c)-d*e*y+c*f*y) % minors(3,F.dd_3) == 0, (b*(a*d-b*c)+b*e*x-a*f*x) % minors(3,F.dd_3) == 0 ass J degree J -- Lemma A.17 restart S = QQ[a,b,c,u,v,x,y,z] J = ideal(x,y,z)*ideal(u,v) + ideal(a*x+b*y+c*z) F = res J F.dd x*u % minors(1,F.dd_1) == 0 v^2*z^4 % minors(6,F.dd_2) == 0, u^2*x^4 % minors(6,F.dd_2) == 0 -- Note: Here we show that (x^4, y^4, z^4)(u,v) + (x^3, y^3, z^3)(ax+by+cz) -- is a height 3 subideal of I_5(dd_3) determinant F.dd_3^{0,2,3,4,5}_{0,2,3,4,5} determinant F.dd_3^{1,2,3,4,5}_{0,2,3,4,5} factor determinant F.dd_3^{2,3,4,5,6}_{0,2,3,4,5} determinant F.dd_3^{0,3,6,9,10}_{0,1,2,4,5} determinant F.dd_3^{1,3,6,9,10}_{0,1,2,4,5} factor determinant F.dd_3^{2,3,6,9,10}_{0,1,2,4,5} determinant F.dd_3^{0,2,6,7,8}_{0,1,3,4,5} determinant F.dd_3^{1,2,6,7,8}_{0,1,3,4,5} factor determinant F.dd_3^{2,3,6,7,8}_{0,1,3,4,5} minors(1,F.dd_4) ass J degree J --Lemma A.18 restart S = QQ[a,b,c,d,w,x,y,z] J = ideal(x^2,x*y,x*z,y^2,y*z,a*x+b*y+w*z,c*x+d*y) F = res J ass J x^2 % minors(1,F.dd_1) == 0 x^6 % minors(6,F.dd_2) == 0, y^6 % minors(6,F.dd_2) == 0 x^5 % minors(5,F.dd_3) == 0, y^5 % minors(5,F.dd_3) == 0, z^4*w + b*y*z^3 % minors(5,F.dd_3) == 0 minors(1,F.dd_4) ass J degree J