a*t^3 + b*t^4 + c*t^5 + b*t^6 + a*t^7; Conj. 2*a - c = deg Ga (in units of 1/16). Conj. b = 2*a Conj deg (E1-E3) = 4*a + 2*c [1,3,4,4,4,16], d:=16; A:=[1,3,4,4,4]; B:=[[3,1,1,1]]; D := P(d,A)-PA(A)-Pb(B); // (2*t^7 + 4*t^6 + 4*t^4 + 2*t^3)/&*[1-t^i : i in [1,1,4,4]]; // 2,4,0, deg Ga = 16/(4*4*4) = 4/(4*4) // E1 = Oh(-1), E3 := Oh(-3), E1-E3 = Oh(2), deg = 8 [1,3,4,4,8,20], [1,3,4,4,9,21], d:=21; A:=[1,3,4,4,9]; B:=[[9,1,4,4],[3,1,1,1],[3,1,1,1]]; D := P(d,A)-PA(A)-Pb(B); // (2*t^7 + 4*t^6 + 3*t^5 + 4*t^4 + 2*t^3)/&*[1-t^i : i in [1,1,4,4]]; // 2,4,3, deg Ga = 1/(4*4) // E1 = Oh(-1,-9,+21) = Oh(11), E3 = Oh(-3), E1-E3 = Oh(14), deg = 14 [1,3,4,4,12,24], d:=24; A:=[1,3,4,4,12]; B:=[[3,1,1,1],[3,1,1,1]]; D := P(d,A)-PA(A)-Pb(B); // (t^7 + 2*t^6 + 2*t^4 + t^3)/&*[1-t^i : i in [1,1,4,4]]; // 1,2,0, deg = 2/(4*4) // E1 = Oh(-1), E3 = Oh(-3), E1-E3 = Oh(2), deg = 4 [1,3,4,5,12,25], d:=25; A:=[1,3,4,5,12]; B:= //line of 3 with emb 12 [1,3,4,8,8,24], d:=24; A:=[1,3,4,8,8]; B:= // 2 x emb 1/8 [1,3,4,8,12,28], // line of 3 and emb 8, 12 [1,3,4,8,13,29], // emb 8 [1,3,4,8,15,31], [1,3,4,8,16,32], d:=32; A:=[1,3,4,8,16]; B:=[[3,1,1,1]]; // also emb 1/8 [1,3,4,12,16,36], d:=36; A:=[1,3,4,12,16]; B:=[[3,1,1,1]]; // also emb 1/16 [1,3,4,12,17,37], [1,3,4,12,19,39], [1,3,4,12,20,40], [1,3,4,16,24,48], // also emb 1/8 [3,4,4,4,5,20], d:=20; A:=[3,4,4,4,5]; B:=[[3,1,1,1]]; D := P(d,A)-PA(A)-Pb(B); // (-5*t^5)/&*[1-t^i : i in [1,1,4,4]]; // 0,0,-5, deg = 5 // E1 = Oh(-5), E3 = Oh(-3), E1-E3 = Oh(-2), deg = -10 [3,4,4,5,8,24], d:=24; A:=[3,4,4,5,8]; B:=[[5,3,3,4]]; D := P(d,A)-PA(A)-Pb(B); // (-3*t^5)/&*[1-t^i : i in [1,1,4,4]]; // 0,0,-3, deg Ga = 3/4^2 // E1 = Oh(-5), E3 = Oh(-3), E1-E3 = Oh(-2), deg = -6 [3,4,4,5,12,28], [3,4,4,6,7,24], d:=24; A:=[3,4,4,6,7]; B:=[[7,4,4,6],[3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1]]; D := P(d,A)-PA(A)-Pb(B); // (-4*t^7 - 6*t^6 - 8*t^5 - 6*t^4 - 4*t^3)/&*[1-t^i : i in [1,1,4,4]]; // -4,-6,-8 // NB -- this is a Freudian slip. There is no line of 4, and the formula // correctly predicts deg Ga = 0. Instead there is a curve of 1/2 of deg 1/4 // with emb 6 x 1/4(1,1,2) [3,4,4,9,16,36], // emb 1/16 [3,4,4,9,20,40], // line of 3 with emb 9 [3,4,5,8,12,32], // emb 1/12 [3,4,5,8,15,35], // emb 1/8 [3,4,5,8,20,40], // emb 1/8 [3,4,5,12,21,45], // emb 1/12 [3,4,5,12,24,48], // emb 1/12 [3,4,5,16,23,51], // emb 1/16 [3,4,5,20,28,60], // emb 1/28 [3,4,5,24,36,72], // emb 1/12 [4,4,5,5,7,25], d:=25; A:=[4,4,5,5,7]; B:=[[7,4,5,5],[5,2,4,4],[5,2,4,4],[5,2,4,4],[5,2,4,4],[5,2,4,4]]; D := P(d,A)-PA(A)-Pb(B); // (3*t^7 + 6*t^6 + 5*t^5 + 6*t^4 + 3*t^3)/&*[1-t^i : i in [1,1,4,4]]; // 3,6,5, deg = 1/4^2 // E1 = Oh(-5,-5,+25) = Oh(15), E3 = Oh(-7), E1-E3 = Oh(22), deg = 22 [4,4,5,7,8,28], // emb 1/8 [4,4,5,7,15,35], d:=35; A:=[4,4,5,7,15]; B:=[[15,4,4,7],[5,2,4,4],[5,2,4,4]]; D := P(d,A)-PA(A)-Pb(B); // (-2*t^7 - 4*t^6 - 5*t^5 - 4*t^4 - 2*t^3)/&*[1-t^i : i in [1,1,4,4]]; // -2,-4,-5 // E1 = Oh(-5), E3 = Oh(-7-15+35) = Oh(13), E1-E3 = Oh(-18), deg = -18 [4,4,5,7,20,40], d:=40; A:=[4,4,5,7,20]; B:=[[7,4,4,6],[5,2,4,4],[5,2,4,4]]; D := P(d,A)-PA(A)-Pb(B); // (t^7 + 2*t^6 + 2*t^4 + t^3)/&*[1-t^i : i in [1,1,4,4]]; // 1,2,0, deg = 2 // E1 = Oh(-5), E3 = Oh(-7), E1-E3 = Oh(2), deg = 4 [4,4,5,11,20,44], // emb 1/40 [4,4,7,13,24,52], // emb 1/24 [4,4,7,13,28,56], d:=56; A:=[4,4,7,13,28]; B:=[[7,4,4,6],[7,4,4,6],[13,2,4,7]]; D := P(d,A)-PA(A)-Pb(B); // (-t^7 - 2*t^6 - 4*t^5 - 2*t^4 - t^3)/&*[1-t^i : i in [1,1,4,4]]; // -1,-2,-4 // E1 = Oh(-13), E3 = Oh(-7) [4,4,7,15,30,60], d:=60; A:=[4,4,7,15,30]; B:=[[15,4,4,7],[15,4,4,7],[7,4,15,30]]; D := P(d,A)-PA(A)-Pb(B); // (-8*t^7 - 15*t^6 - 16*t^5 - 15*t^4 - 8*t^3)/&*[1-t^i : i in [1,1,4,4]]; // Again no line of 1/4, curve of deg 0. It's a line of 2 of deg 1/8 with // 15 x emb 1/4(1,1,2) [4,4,9,15,17,49], // line of 3 [4,4,9,15,28,60], // emb 1/28 [4,4,9,30,43,90], d:=90; A:=[4,4,9,30,43]; B:=[[43,4,9,30],[3,1,1,1]]; D := P(d,A)-PA(A)-Pb(B); // (4*t^7 + 9*t^6 + 7*t^5 + 9*t^4 + 4*t^3)/&*[1-t^i : i in [1,1,4,4]]; // 4,9,7 No line of 1/4. Why 2*a-c = 1 but deg = 0? [4,4,11,17,19,55], d:=55; A:=[4,4,11,17,19]; B:=[[19,4,4,11],[17,4,11,19]]; D := P(d,A)-PA(A)-Pb(B); // (-5*t^7 - 10*t^6 - 11*t^5 - 10*t^4 - 5*t^3)/&*[1-t^i : i in [1,1,4,4]]; // -5,-10,-11 // E1 = Oh(-17), E3 = Oh(-19,-11,+55) = Oh(25) [4,4,11,17,36,72], d:=72; A:=[4,4,11,17,36]; B:=[[17,2,4,11],[11,3,4,4]]; D := P(d,A)-PA(A)-Pb(B); // (-t^7 - 2*t^6 - 4*t^5 - 2*t^4 - t^3)/&*[1-t^i : i in [1,1,4,4]]; // -7,-2,-4 // E1 = Oh(-17), E3 = Oh(-11)