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We want to check that for each \(s\), the scheme \( P_{s} \cap OG(5,10)\) has the desired \( \mathbb{G}_m\) and \(Z_2\) actions.
We break the proof into cases.
Case 1: Suppose that \(c_1\) and \(c_3\) are nonzero. Then after scaling we may assume \(c_1 = c_3 = 1\)(.
We choose seven variables to parametrize \(P_s\), and use the nine equations to write the remaining variables in terms of the parameters.
We rewrite Mukai's equations for the orthogonal Grassmannian in the variables \(y_i\), and then check that the ideal defined by these quadrics has the same automorphisms as the balanced ribbon.
i1 : R=frac(QQ[c_2,c_4,t,Degrees=>{0,0,0}]);
i2 : S=R[y_0..y_6];
i3 : -- Variables used to parametrize P_s
x_1245=y_0;
i4 : x_1345=y_1;
i5 : x_2345=y_2;
i6 : x_14=y_3;
i7 : x_15=y_4;
i8 : x_25=2*y_5;
i9 : x_35=y_6;
i10 : -- Nine equations defining the remaining variables
x_0=0;
i11 : x_12=-c_4*x_1345;
i12 : x_13=-c_2*x_2345;
i13 : x_23=x_14;
i14 : x_24=-2*c_2*x_15;
i15 : x_34=-1/2*c_4*x_25;
i16 : x_45=0;
i17 : x_1234=0;
i18 : x_1235=0;
i19 : L = {x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235};
i20 : print toString(L) << endl
{2*c_4*y_5^2-2*c_2*y_4*y_6, -c_4*y_1^2+c_2*y_0*y_2, c_4*y_4*y_5+y_3*y_6, -c_4*y_1*y_2-y_0*y_3, 2*c_2*y_4^2+2*y_3*y_5, -c_2*y_2^2-y_1*y_3, -y_3*y_4-2*c_2*y_2*y_5+c_4*y_1*y_6, y_2*y_3+2*c_2*y_1*y_4-c_4*y_0*y_5, -y_3^2+2*c_2^2*y_2*y_4-c_4^2*y_1*y_5, y_2*y_4-2*y_1*y_5+y_0*y_6}
i21 : I=ideal L;
o21 : Ideal of S
i22 : F1=map(S,S,diagonalMatrix({t^-3,t^-2,t^-1,t^0,t^1,t^2,t^3}));
o22 : RingMap S <--- S
i23 : F1(I)==I
o23 = true
i24 : F2=map(S,S,reverse gens S);
o24 : RingMap S <--- S
i25 : F2(I)==I
o25 = trueCase 2: Suppose that \(c_1=0\) and \(c_3\) is nonzero. Since \(c_1=0\), we must have \(c_2\) nonzero. Then after scaling we may assume \(c_2 = c_3 = 1\).
We get \(x_{2345}=0\) and \(x_{15}\)=0 as equations, so we use \(x_{13}\) and \(x_{24}\) as parameters.
i26 : R=frac(QQ[c_4,t,Degrees=>{0,0}]);
i27 : S=R[y_0..y_6];
i28 : -- Variables used to parametrize P_s
x_1245=y_0;
i29 : x_1345=y_1;
i30 : x_13=y_2;
i31 : x_14=y_3;
i32 : x_24=2*y_4;
i33 : x_25=2*y_5;
i34 : x_35=y_6;
i35 : -- Nine equations defining the remaining variables
x_0=0;
i36 : x_12=-c_4*x_1345;
i37 : x_2345=0;
i38 : x_23=x_14;
i39 : x_15 = 0;
i40 : x_34=-1/2*c_4*x_25;
i41 : x_45=0;
i42 : x_1234=0;
i43 : x_1235=0;
i44 : L = {x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235};
i45 : I=ideal L;
o45 : Ideal of S
i46 : F1=map(S,S,diagonalMatrix({t^-3,t^-2,t^-1,t^0,t^1,t^2,t^3}));
o46 : RingMap S <--- S
i47 : F1(I)==I
o47 = true
i48 : F2=map(S,S,reverse gens S);
o48 : RingMap S <--- S
i49 : F2(I)==I
o49 = trueCase 3: Suppose that \(c_1\) is nonzero and \(c_3=0\). Since \(c_3=0\), we must have \(c_4\) nonzero. Then after scaling we may assume \(c_1 = c_4 = 1\).
We get \(x_{1345}=0\) and \(x_{25}\)=0 as equations, so we use \(x_{12}\) and \(x_{34}\) as parameters.
i50 : R=frac(QQ[c_2,t,Degrees=>{0,0}]);
i51 : S=R[y_0..y_6];
i52 : -- Variables used to parametrize P_s
x_1245=y_0;
i53 : x_12 = y_1;
i54 : x_2345=y_2;
i55 : x_14=y_3;
i56 : x_15=y_4;
i57 : x_34 = y_5;
i58 : x_35=y_6;
i59 : -- Nine equations defining the remaining variables
x_0=0;
i60 : x_1345=0;
i61 : x_13=-c_2*x_2345;
i62 : x_23=x_14;
i63 : x_24=-2*c_2*x_15;
i64 : x_25=0;
i65 : x_45=0;
i66 : x_1234=0;
i67 : x_1235=0;
i68 : L = {x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235};
i69 : I=ideal L;
o69 : Ideal of S
i70 : F1=map(S,S,diagonalMatrix({t^-3,t^-2,t^-1,t^0,t^1,t^2,t^3}));
o70 : RingMap S <--- S
i71 : F1(I)==I
o71 = true
i72 : F2=map(S,S,reverse gens S);
o72 : RingMap S <--- S
i73 : F2(I)==I
o73 = trueCase 4: Suppose that \(c_1=c_3 = 0.\) Then we must have \(c_2\) and \(c_4\) nonzero. After scaling we may assume \(c_2 = c_4 = 1\).
i74 : R=frac(QQ[t,Degrees=>{0}]);
i75 : S=R[y_0..y_6];
i76 : -- Variables used to parametrize P_s
x_1245=y_0;
i77 : x_12 = y_1;
i78 : x_13=y_2;
i79 : x_14=y_3;
i80 : x_24=2*y_4;
i81 : x_34 = y_5;
i82 : x_35=y_6;
i83 : -- Nine equations defining the remaining variables
x_0=0;
i84 : x_1345 = 0;
i85 : x_2345=0;
i86 : x_23=x_14;
i87 : x_15 = 0;
i88 : x_25 = 0;
i89 : x_45=0;
i90 : x_1234=0;
i91 : x_1235=0;
i92 : L = {x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235};
i93 : I=ideal L;
o93 : Ideal of S
i94 : F1=map(S,S,diagonalMatrix({t^-3,t^-2,t^-1,t^0,t^1,t^2,t^3}));
o94 : RingMap S <--- S
i95 : F1(I)==I
o95 = true
i96 : F2=map(S,S,reverse gens S);
o96 : RingMap S <--- S
i97 : F2(I)==I
o97 = true