rreducparautommodg6_51xCN51.r The generic automorphism phi of C x g_{5,1} as computed by calculautom6_51xC.r\ ed : They fall into 4 kinds: The first kind (which contains the identity component) is: The parameters are subject to the supplementary conditions : b(2,2)*( - b(3,1)*b(1,3) + b(3,3)*b(1,1))neq 0 phi:= mat((b(1,1),b(1,2),b(1,3),b(1,4),0,0), ((b(3,1)*b(2,2)*b(1,4) - b(3,1)*b(2,4)*b(1,2) + b(3,2)*b(2,4)*b(1,1) - b(3,4)*b(2,2)*b(1,1))/( - b(3,1)*b(1,3) + b(3,3)*b(1,1)),b(2,2),( b(3,2)*b(2,4)*b(1,3) + b(3,3)*b(2,2)*b(1,4) - b(3,3)*b(2,4)*b(1,2) - b(3,4)*b(2,2)*b(1,3))/( - b(3,1)*b(1,3) + b(3,3)*b(1,1)),b(2,4),0,0), (b(3,1),b(3,2),b(3,3),b(3,4),0,0), 2 ((b(3,1) *b(1,3)*b(1,2) - b(3,2)*b(3,1)*b(1,3)*b(1,1) 2 - b(3,3)*b(3,1)*b(1,2)*b(1,1) + b(3,3)*b(3,2)*b(1,1) + b(4,2)*b(3,1)*b(2,2)*b(1,4) - b(4,2)*b(3,1)*b(2,4)*b(1,2) + b(4,2)*b(3,2)*b(2,4)*b(1,1) - b(4,2)*b(3,4)*b(2,2)*b(1,1))/(b(2,2) 2 *( - b(3,1)*b(1,3) + b(3,3)*b(1,1))),b(4,2),( - b(3,2)*b(3,1)*b(1,3) + b(3,3)*b(3,1)*b(1,3)*b(1,2) + b(3,3)*b(3,2)*b(1,3)*b(1,1) 2 - b(3,3) *b(1,2)*b(1,1) + b(4,2)*b(3,2)*b(2,4)*b(1,3) + b(4,2)*b(3,3)*b(2,2)*b(1,4) - b(4,2)*b(3,3)*b(2,4)*b(1,2) - b(4,2)*b(3,4)*b(2,2)*b(1,3))/(b(2,2) *( - b(3,1)*b(1,3) + b(3,3)*b(1,1))), - b(3,1)*b(1,3) + b(3,3)*b(1,1) + b(4,2)*b(2,4) --------------------------------------------------,0,0), b(2,2) (b(5,1),b(5,2),b(5,3),b(5,4), - b(3,1)*b(1,3) - b(3,2)*b(1,4) + b(3,3)*b(1,1) + b(3,4)*b(1,2),b(5,6)), (b(6,1),b(6,2),b(6,3),b(6,4),0,b(6,6))) 3 det(phi):=( - b(3,1)*b(1,3) - b(3,2)*b(1,4) + b(3,3)*b(1,1) + b(3,4)*b(1,2)) *b(6,6) condition11:=( - b(3,1)*b(1,3) + b(3,3)*b(1,1))*b(2,2) neq 0 generic derivation : delta:= mat((xi(1,1),xi(1,2),xi(1,3),xi(1,4),0,0), (xi(2,1),xi(2,2),xi(1,4),xi(2,4),0,0), (xi(3,1),xi(3,2),xi(3,3), - xi(2,1),0,0), (xi(3,2),xi(4,2), - xi(1,2), - ( - xi(3,3) - xi(1,1) + xi(2,2)),0,0), (xi(5,1),xi(5,2),xi(5,3),xi(5,4),xi(1,1) + xi(3,3),xi(5,6)), (xi(6,1),xi(6,2),xi(6,3),xi(6,4),0,xi(6,6))) [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx1 := [ ] [0 0 0 0 0 0] [ ] [0 0 1 0 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx2 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 1 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx3 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [0 -1 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(3,3):= - xi(1,1) xi(6,6):=0 And the matrix A:= [xi(1,1) xi(1,2) xi(1,3) xi(1,4) ] [ ] [xi(2,1) xi(2,2) xi(1,4) xi(2,4) ] [ ] [xi(3,1) xi(3,2) - xi(1,1) - xi(2,1)] [ ] [xi(3,2) xi(4,2) - xi(1,2) - xi(2,2)] is nilpotent. [xi(1,1) xi(1,2) xi(1,3) xi(1,4) ] [ ] [xi(2,1) xi(2,2) xi(1,4) xi(2,4) ] m := [ ] [xi(3,1) xi(3,2) - xi(1,1) - xi(2,1)] [ ] [xi(3,2) xi(4,2) - xi(1,2) - xi(2,2)] we may suppose that A is an element of sp(4,C)+, that is: xi(1,1):=0 xi(2,1):=0 xi(2,2):=0 xi(3,1):=0 xi(3,2):=0 xi(4,2):=0 M**1:= [0 xi(1,2) xi(1,3) xi(1,4)] [ ] [0 0 xi(1,4) xi(2,4)] [ ] [0 0 0 0 ] [ ] [0 0 - xi(1,2) 0 ] M**2:= [0 0 0 xi(2,4)*xi(1,2)] [ ] [0 0 - xi(2,4)*xi(1,2) 0 ] [ ] [0 0 0 0 ] [ ] [0 0 0 0 ] M**3:= [ 2 ] [0 0 - xi(2,4)*xi(1,2) 0] [ ] [0 0 0 0] [ ] [0 0 0 0] [ ] [0 0 0 0] M**4:= [0 0 0 0] [ ] [0 0 0 0] [ ] [0 0 0 0] [ ] [0 0 0 0] Trace(M**1)/2:=0$ Trace(M**2)/2:=0$ Trace(M**3)/2:=0$ Trace(M**4)/2:=0$ By subtracting adjoints one then may suppose:$ xi(5,1):=0,xi(5,2):=0,xi(5,3):=0,xi(5,4):=0$ delta:= mat((0,xi(1,2),xi(1,3),xi(1,4),0,0),(0,0,xi(1,4),xi(2,4),0,0),(0,0,0,0,0,0),(0,0 , - xi(1,2),0,0,0),(0,0,0,0,0,xi(5,6)),(xi(6,1),xi(6,2),xi(6,3),xi(6,4),0,0))$ We denote this delta by the shortform$ shortformdelta:={xi(1,2), xi(1,3), xi(1,4), ss, xi(2,4), ss, xi(5,6), ss, xi(6,1), xi(6,2), xi(6,3), xi(6,4)}$ paramindexeslist:={{1,2},{1,3},{1,4},{2,4},{5,6},{6,1},{6,2},{6,3},{6,4}}$ ************************************************************************$ *******Suppose xi(1,2) = 0.$ *******Suppose xi(1,3) = 0.$ *******Suppose xi(2,4) = 0.$ ************************************************************************$ xi(1,2):=0$ xi(1,3):=0$ xi(2,4):=0$ *****************************************************************************$ Suppose xi(1,4) neq 0.$ *****************************************************************************$ Then may suppose xi(1,4)=1.$ xi(1,4):=1$ delta:= [ 0 0 0 1 0 0 ] [ ] [ 0 0 1 0 0 0 ] [ ] [ 0 0 0 0 0 0 ] [ ] [ 0 0 0 0 0 0 ] [ ] [ 0 0 0 0 0 xi(5,6)] [ ] [xi(6,1) xi(6,2) xi(6,3) xi(6,4) 0 0 ] Now consider the automorphism defined by:$ x'(1)=x(1),x'(2)=x(4),x'(3)=x(3),x'(4)=-x(2),x'(5)=x(5)$ It is of the 2d kind.$ more precisely, it is the automorphism psi below:$ for i:=1:6 do for j:=1:6 do <>$ **** Case 1.2 : suppose b(2,2) = 0.$ psi:= [1 0 0 0 0 0] [ ] [0 0 0 -1 0 0] [ ] [0 0 1 0 0 0] [ ] [0 1 0 0 0 0] [ ] [0 0 0 0 1 0] [ ] [0 0 0 0 0 1] With the second kind automorphism one gets$ shortformdeltaprimemodadg:={-1, 0, 0, ss, 0, ss, xi(5,6), ss, xi(6,1), - xi(6,4), xi(6,3), xi(6,2)}$ deltaprimemodg(1,2):=-1$ deltaprimemodg(1,3):=0$ deltaprimemodg(1,4):=0$ deltaprimemodg(2,4):=0$ deltaprimemodg(5,6):=xi(5,6)$ deltaprimemodg(6,1):=xi(6,1)$ deltaprimemodg(6,2):= - xi(6,4)$ deltaprimemodg(6,3):=xi(6,3)$ deltaprimemodg(6,4):=xi(6,2)$ det(AUTOM):=1$ DELTAPRIMEMODADG:= [ 0 -1 0 0 0 0 ] [ ] [ 0 0 0 0 0 0 ] [ ] [ 0 0 0 0 0 0 ] [ ] [ 0 0 1 0 0 0 ] [ ] [ 0 0 0 0 0 xi(5,6)] [ ] [xi(6,1) - xi(6,4) xi(6,3) xi(6,2) 0 0 ] Hence we are back to some case where$ xi(1,2):=-1, xi(1,3):=0, xi(1,4):=0, xi(2,4):=0.$