D6link

D6-link from the Burkhardt quartic X4 to a smooth quadric


The page displays in equations the D6-link from the Burkhardt quartic X4 to a smooth quadric introduced in the paper.
The source code of all equations and matrices involved can be found here.





Consider the Burkhardt quartic X4 in P4[y1:y2:y3:y4:y5] given by the equation 
		y14 - y1*y23 - y1*y33 - y1*y43 + 3*y2*y3*y4*y5 - y1*y53.
Fix the D6-action on X4 generated by matrices (action from the right) 
				 		     m1=[1/3,              -2/3*z3,       1/3*(2*z3 + 2),   -2/3,              -2/3,]
							[-1/3,             -1/3*z3,       1/3*(z3 + 1),     -1/3,              2/3,]
							[1/3*(z3+1),       2/3,           -1/3*z3,          1/3*(z3 + 1),      1/3*(z3 + 1),]
							[1/3*(z3+1),       -1/3,          2/3*z3,           1/3*(z3 + 1),      1/3*(z3 + 1),]
							[1/3*(z3+1),       -1/3,          -1/3*z3,          1/3*(-2*z3 - 2),   1/3*(z3 + 1)]

							z3=the (first) primitive third root of unity
and 
					             m2=[1           0           0           0           0]
							[0           0           0           0           1]
							[0           0           0          z32          0]
							[0           0           z3          0           0]
							[0           1           0           0           0]
The linear system 
| 4*(−KX4) − D8 |
	

where D8 is the formal sum of eight planes given by equations

	plane1: y1=y5=0,
	plane2: y1=y2=0,
	plane3: (-z3+1)*y1+(-2*z3-1)*y5=z3*y1-z32*y2-y3-y4-z32*y5=0,
	plane4: z3*y1-y2-y3-z3*y4-y5=z3*y1-z32*y2-y3-y4-z32*y5=0,
	plane5: z3*y1-y2-z3*y3-y4-y5=z3*y1-z32*y2-y3-y4-z32*y5=0,
	plane6: z32*y1-z32*y2-y3-y4-y5=(z3+2)*y1+(-z3-2)*y2=0,
	plane7: y1-y5=y2+z3*2y3+z3*y4=0
	plane8: z32*y1-y2-z3*y3-z32*y4-z32*y5=z3*y1-z3*y2-y3-y4-y5=0


has projective dimension 4 and a basis of five polynomials

f1=y13*y2-y1*y23+(z3+1)*y12*y32-z3*y1*y2*y32+z3*y1*y33+(-z3-1)*y13*y4+z3*y12*y2*y4+y1*y22*y4+(-z3+1)*y12*y3*y4-y1*y2*y3*y4+(z3+1)*y1*y32*y4-z3*y12*y42-y1*y2*y42+y1*y3*y42+1/3*(-z3+1)*y13*y5-y12*y2*y5+(z3+1)*y1*y22*y5+1/3*(-2*z3-1)*y23*y5+1/3*(-z3+4)*y12*y3*y5+1/3*(-z3-5)*y1*y2*y3*y5+1/3*(5*z3+4)*y22*y3*y5+1/3*(2*z3+4)*y1*y32*y5+1/3*(-2*z3-4)*y2*y32*y5+1/3*(-z3-2)*y12*y4*y5+1/3*(-z3-2)*y1*y2*y4*y5+1/3*(2*z3+1)*y22*y4*y5+1/3*(-2*z3-1)*y1*y3*y4*y5+1/3*(-4*z3+1)*y2*y3*y4*y5+1/3*(-z3-2)*y1*y42*y5+1/3*(-2*z3-1)*y2*y42*y5+1/3*(-z3-2)*y12*y52+1/3*(5*z3+4)*y1*y2*y52+1/3*(-z3-2)*y22*y52+1/3*(-2*z3-1)*y1*y3*y52+1/3*(2*z3+1)*y2*y3*y52+1/3*(z3-1)*y1*y4*y52+1/3*(-z3-2)*y2*y4*y52+1/3*(2*z3+1)*y1*y53+1/3*(z3-1)*y2*y53,

f2=y12*y22-y1*y23+z3*y12*y32-y1*y33-z3*y13*y4-y12*y2*y4+(z3+1)*y1*y22*y4+(z3+1)*y12*y3*y4-y1*y32*y4+(z3+2)*y12*y42+(-z3-1)*y1*y2*y42+z3*y1*y3*y42+z3*y1*y43+1/3*(z3+2)*y13*y5+(-z3-1)*y12*y2*y5+z3*y1*y22*y5+1/3*(-z3+1)*y23*y5+1/3*(4*z3+2)*y12*y3*y5+1/3*(-5*z3-4)*y1*y2*y3*y5+1/3*(z3-1)*y22*y3*y5+1/3*(z3-1)*y1*y32*y5+1/3*(-z3+1)*y2*y32*y5+1/3*(z3+2)*y12*y4*y5+1/3*(-5*z3-1)*y1*y2*y4*y5+1/3*(z3-1)*y22*y4*y5+1/3*(2*z3-2)*y1*y3*y4*y5+1/3*(-5*z3+8)*y2*y3*y4*y5+1/3*(z3+2)*y1*y42*y5+1/3*(-z3+1)*y2*y42*y5+1/3*(-2*z3-1)*y12*y52+1/3*(z3-1)*y1*y2*y52+1/3*(-2*z3-1)*y22*y52+1/3*(2*z3+1)*y1*y3*y52+1/3*(-2*z3-1)*y2*y3*y52+1/3*(-z3+1)*y1*y4*y52+1/3*(-2*z3-1)*y2*y4*y52+1/3*(z3-1)*y1*y53+1/3*(-z3-2)*y2*y53,

f3=y13*y3+(z3+1)*y12*y32+1/3*(-z3+1)*y13*y4+1/3*(2*z3+1)*y12*y2*y4+1/3*(-z3+1)*y1*y22*y4+1/3*(2*z3+4)*y12*y3*y4+1/3*(z3-1)*y1*y2*y3*y4+1/3*(2*z3+1)*y1*y32*y4+1/3*(-z3+1)*y12*y42+1/3*(z3-1)*y1*y2*y42+1/3*(z3+2)*y1*y3*y42+1/3*(-z3+1)*y1*y43+1/3*(z3+2)*y12*y2*y5+1/3*(z3+2)*y1*y22*y5+1/3*(-2*z3-1)*y23*y5+1/3*(z3-1)*y1*y2*y3*y5+1/3*(2*z3+1)*y22*y3*y5+1/3*(-2*z3-1)*y2*y32*y5+1/3*(-z3-2)*y12*y4*y5+1/3*(-z3-2)*y1*y2*y4*y5+1/3*(2*z3+1)*y22*y4*y5+1/3*(-2*z3-1)*y1*y3*y4*y5+1/3*(-z3-2)*y2*y3*y4*y5+1/3*(-2*z3-1)*y1*y42*y5+1/3*(-2*z3-1)*y2*y42*y5+1/3*(z3-1)*y1*y2*y52+1/3*(-z3-2)*y22*y52+1/3*(-z3-2)*y2*y3*y52+1/3*(-z3-2)*y1*y4*y52+1/3*(-z3-2)*y2*y4*y52+1/3*(z3-1)*y2*y53,

f4=y12*y2*y3+z3*y12*y32+(z3+1)*y1*y2*y32-y1*y33+1/3*(-z3+1)*y13*y4+1/3*(-z3+1)*y12*y2*y4+1/3*(2*z3+1)*y1*y22*y4+1/3*(5*z3+4)*y12*y3*y4+1/3*(z3+2)*y1*y2*y3*y4+1/3*(2*z3-2)*y1*y32*y4+1/3*(2*z3+4)*y12*y42+1/3*(-2*z3-1)*y1*y2*y42+1/3*(4*z3+2)*y1*y3*y42+1/3*(2*z3+1)*y1*y43+(z3+1)*y13*y5+1/3*(-2*z3-1)*y12*y2*y5+1/3*(z3+2)*y1*y22*y5+1/3*(-2*z3-1)*y23*y5+(2*z3+1)*y12*y3*y5+1/3*(-5*z3-4)*y1*y2*y3*y5+1/3*(2*z3-2)*y22*y3*y5+z3*y1*y32*y5+1/3*(-5*z3-1)*y2*y32*y5+1/3*(-z3+1)*y12*y4*y5+1/3*(-4*z3-5)*y1*y2*y4*y5+1/3*(2*z3+1)*y22*y4*y5+1/3*(z3+2)*y1*y3*y4*y5+1/3*(-7*z3+1)*y2*y3*y4*y5+1/3*(-2*z3+2)*y1*y42*y5+1/3*(-2*z3-1)*y2*y42*y5-z3*y12*y52+1/3*(z3-4)*y1*y2*y52+1/3*(-z3-2)*y22*y52+y1*y3*y52+1/3*(-z3-5)*y2*y3*y52+1/3*(-4*z3-2)*y1*y4*y52+1/3*(-z3-2)*y2*y4*y52-y1*y53+1/3*(z3-1)*y2*y53,

f5=y1*y22*y3+y12*y32+(z3+1)*y1*y33+1/3*(2*z3+1)*y13*y4+1/3*(-z3+1)*y12*y2*y4+1/3*(-z3+1)*y1*y22*y4+1/3*(-z3+1)*y12*y3*y4+1/3*(z3-1)*y1*y2*y3*y4+1/3*(2*z3+4)*y1*y32*y4+1/3*(-z3-2)*y12*y42+1/3*(z3-1)*y1*y2*y42+1/3*(-2*z3+2)*y1*y3*y42+1/3*(-z3+1)*y1*y43+y13*y5+1/3*(z3+2)*y12*y2*y5+1/3*(-2*z3-1)*y1*y22*y5+1/3*(z3-1)*y23*y5+y12*y3*y5+1/3*(z3-1)*y1*y2*y3*y5+1/3*(2*z3+1)*y22*y3*y5+y1*y32*y5+1/3*(-2*z3-4)*y2*y32*y5+1/3*(-z3-2)*y12*y4*y5+1/3*(5*z3-2)*y1*y2*y4*y5+1/3*(-z3+1)*y22*y4*y5+1/3*(-5*z3-1)*y1*y3*y4*y5+1/3*(-4*z3-8)*y2*y3*y4*y5+1/3*(-5*z3-4)*y1*y42*y5+1/3*(z3-1)*y2*y42*y5+(-z3-1)*y12*y52+1/3*(4*z3+2)*y1*y2*y52+1/3*(2*z3+1)*y22*y52+(-2*z3-1)*y1*y3*y52+1/3*(5*z3+1)*y2*y3*y52+1/3*(-z3-5)*y1*y4*y52+1/3*(2*z3+1)*y2*y4*y52+z3*y1*y53+1/3*(z3+2)*y2*y53



The map


	
X4 ---> P4,  
	

[y1, y2, y3, y4, y5] ---> [f1, f2, f3, f4, f5]




is birational onto its image. The image is a quadric cone QC in P4 given by 


	
y12-y1*y2+y22+1/3*(2*z3-2)*y1*y3+1/3*(-4*z3-2)*y2*y3+1/3*(5*z3+4)*y1*y4+1/3*(-z3-5)*y2*y4+1/3*(-z3-2)*y3*y4+1/3*(2*z3+1)*y42+1/3*(-z3+1)*y1*y5+1/3*(2*z3+1)*y2*y5+1/3*(z3-1)*y3*y5+y4*y5+1/3*(-2*z3-1)*y52
	



Its inverse rational map 


	

QC ---> X4,

                                   
[y1, y2, y3, y4, y5] ---> [h1, h2, h3, h4, h5]

	


 
is given by 


	
h1=(-2*z3-11)*y1*y32+(-3*z3+5)*y2*y32+6*y33+(-5*z3+6)*y1*y3*y4+(2*z3+2)*y2*y3*y4+(-4*z3-9)*y32*y4+(3*z3+2)*y1*y42+2*z3*y2*y42+y3*y42+(z3+1)*y43+(3*z3-2)*y1*y3*y5-5*z3*y2*y3*y5+(-7*z3-1)*y32*y5+(-7*z3-1)*y1*y4*y5+(3*z3+1)*y2*y4*y5+(13*z3-4)*y3*y4*y5+(-5*z3+4)*y42*y5+(-z3-3)*y1*y52+(z3+1)*y2*y52+(6*z3+8)*y3*y52+(-5*z3-6)*y4*y52+z3*y53,

h2=(5*z3+6)*y1*y32+(-2*z3-11)*y2*y32+(3*z3-8)*y1*y3*y4+(-5*z3+6)*y2*y3*y4+(3*z3+5)*y32*y4+(-5*z3-2)*y1*y42+(3*z3+2)*y2*y42+(-4*z3-7)*y3*y42-z3*y43+(2*z3+2)*y1*y3*y5+(3*z3-2)*y2*y3*y5+z3*y32*y5+4*z3*y1*y4*y5+(-7*z3-1)*y2*y4*y5+(-8*z3-4)*y3*y4*y5+(11*z3+3)*y42*y5+2*y1*y52+(-z3-3)*y2*y52+(z3-5)*y3*y52+10*y4*y52+(-3*z3-2)*y53,

h3=(2*z3-3)*y1*y32+(z3+1)*y2*y32+6*y33+(-9*z3-2)*y1*y3*y4+(z3+3)*y2*y3*y4+(3*z3-10)*y32*y4+(4*z3+4)*y1*y42+(-3*z3-4)*y2*y42+(-7*z3+2)*y3*y42+(5*z3+3)*y43+(-z3-10)*y1*y3*y5+(-3*z3+7)*y2*y3*y5+(z3+6)*y32*y5+(-5*z3+3)*y1*y4*y5+(5*z3-1)*y2*y4*y5+(-2*z3-10)*y3*y4*y5+(-4*z3+3)*y42*y5-y1*y52+(-z3+3)*y2*y52+(-2*z3+1)*y3*y52+(3*z3-5)*y4*y52+(3*z3+4)*y53,

h4=(z3-5)*y1*y32+(3*z3+8)*y2*y32+6*y33+z3*y1*y3*y4+(-4*z3-10)*y2*y3*y4+(-4*z3-9)*y32*y4+(-3*z3-1)*y1*y42+(-z3+3)*y2*y42+y3*y42+(z3+1)*y43+(9*z3+10)*y1*y3*y5+(-2*z3-3)*y2*y3*y5+(-4*z3-4)*y32*y5+(-z3-7)*y1*y4*y5+(-3*z3-2)*y2*y4*y5+(10*z3-1)*y3*y4*y5+(-2*z3+1)*y42*y5+(2*z3+3)*y1*y52+(-2*z3-5)*y2*y52+(3*z3+2)*y3*y52+(z3+6)*y4*y52+(-5*z3-3)*y53,

h5=(2*z3+3)*y1*y32+(-5*z3-5)*y2*y32-6*y33+(3*z3-2)*y1*y3*y4-2*z3*y2*y3*y4+5*y32*y4+(-2*z3-2)*y1*y42-y2*y42+(2*z3-1)*y3*y42-z3*y43+(2*z3-1)*y1*y3*y5-2*y2*y3*y5+(z3+3)*y32*y5+z3*y1*y4*y5+(-z3-1)*y2*y4*y5+(-2*z3-1)*y3*y4*y5+2*z3*y42*y5+(3*z3+2)*y1*y52-z3*y2*y52+(-5*z3-2)*y3*y52+(3*z3+1)*y4*y52+y53




Equivalently, the inverse is given by blow-up of six lines and two conics in QC (Figure 1), given by 

line1: L1=y4-y5=y3-y5=y1+z3*y2=0,
line2: L2=2y3-y4-y5=2y2-y4+y5=2y1-z32*y4+z32*y5=0,
line3: L3=y4-z3*y5=2y3+z32*y5=y1+z32*y2-(z3+2)*y5=0,
line4: L4=y4+z32*y5=y3=y1+z32*y2+z3*y5=0,
line5: L5=y3+z3*y4+y5=y2-y4-z32*y5=y1+2*z3*y4+(z3+2)*y5=0,
line6: L6=y4-z3*y5=y3-z3*y5=y1+z3*y2=0,
conic1: R=2y3-z32*y4-z3*y5=2y1+2*z3*y2-z32*y4+z3*y5=0 (in QC),
conic2: R'=y3-y4=y1+z32*y2+z3*y4-z3*y5=0 (in QC).





	

      
      
Figure 1

    




The corresponding C6-action generated by m1 is linear on QC, given by the matrix

	
						[-z3-1,          0,               1/3*(z3-1),        1/3*(z3+2),      1/3*(-2*z3-1),]
						[1,              0,               1/3*(z3+2),        1/3*(4*z3+2),    1/3*(4*z3+5), ]
						[1/3,            1/3*(z3+1),      1/3*(-2*z3+2),     1/3*(-2*z3-1),   1/3*(z3+2),   ]
						[1/3*(2*z3+2),   1/3*(-z3-3),     1/3*(2*z3-2),      1/3*(2*z3-2),    1/3*(-z3+1),  ]
						[1/3*(z3+3),     1/3*(3*z3-1),    z3+1,              2*z3+1,          1             ]

	

And the C6-action on QC fixes a point



	[z3-2 : -3*z3-8 : 3*z3+1 : 3*z3+1 : 7]




 So the projection from a fixed point gives a linearization of the original C6-action on X4.


The corresponding C2 involution generated by m2 is birational on QC, given by the map


	

ι: QC ---> QC

[y1, y2, y3, y4, y5] ---> [t1, t2, t3, t4, t5]
	


 
where

t1=y1*y3+1/19*(18*z3+7)*y1*y4+1/19*(-6*z3+4)*y1*y5+1/19*(-5*z3-3)*y2*y3+1/19*(-5*z3-3)*y2*y4+1/19*(4*z3-9)*y2*y5+1/19*(2*z3-14)*y32+1/19*(2*z3+5)*y3*y4+1/19*(11*z3+18)*y3*y5+1/19*(3*z3-2)*y42+1/19*(-9*z3+6)*y4*y5+1/19*(-9*z3-13)*y52,

t2=1/19*(3*z3+17)*y1*y3+1/19*(6*z3-4)*y1*y4+1/19*(-3*z3+2)*y1*y5+1/19*(-z3-12)*y2*y3+1/19*(-10*z3-6)*y2*y4+1/19*(-z3-12)*y2*y5+1/19*(4*z3-28)*y32+1/19*(-5*z3+16)*y3*y4+z3*y3*y5+1/19*(12*z3+11)*y42+1/19*(-18*z3+12)*y4*y5+1/19*(-12*z3-11)*y52,

t3=1/19*(12*z3+11)*y1*y3+1/19*(-2*z3-5)*y1*y4+1/19*(5*z3+3)*y1*y5+1/19*(-z3-12)*y2*y3+1/19*(-3*z3+2)*y2*y4+1/19*(-2*z3-5)*y2*y5+1/19*(8*z3+1)*y32+1/19*(-2*z3-5)*y3*y4+1/19*(2*z3+5)*y3*y5+1/19*(-3*z3+2)*y42+1/19*(-5*z3-3)*y52,

t4=1/19*(20*z3+12)*y1*y3+1/19*(-9*z3-13)*y1*y4+1/19*(4*z3+10)*y1*y5+1/19*(-2*z3-5)*y2*y3+1/19*(5*z3+3)*y2*y4+1/19*(-9*z3-13)*y2*y5+1/19*(8*z3+1)*y32+1/19*(-9*z3-13)*y3*y4+1/19*(12*z3-8)*y3*y5+1/19*(-4*z3+9)*y42+1/19*(6*z3+15)*y4*y5+1/19*(-13*z3-4)*y52,

t5=(z3+1)*y1*y3+1/19*(-z3-12)*y1*y4+1/19*(-3*z3+2)*y1*y5+1/19*(-9*z3-13)*y2*y3+1/19*(4*z3+10)*y2*y4+1/19*(-z3-12)*y2*y5+1/19*(-10*z3+13)*y32+1/19*(23*z3-9)*y3*y4+1/19*(10*z3+6)*y3*y5+1/19*(-17*z3-14)*y42+1/19*(6*z3+15)*y4*y5+1/19*(-12*z3-11)*y52;



The line in QC


	
L: y1+1/3*(z3+2)*y4+1/3*(-z3+1)*y5=y2+1/3*(5*z3+4)*y4+1/3*(z3+5)*y5=y3-2*z3*y4+(-z3-1)*y5=0
	



passes through the C6-fixed point mentioned above and the vertex of QC. The projection of the line


	

pr: QC ---> P2

	



and its composition with the involution pr∘ι define a product map


	
pr × pr∘ι : QC ---> P2[x0:x1:x2] × P2[z0:z1:z2] 
	



This is a birational map onto its image, a divisor of degree (1,1) given by 


	

x0*z0-1/2*x0*z1+1/3*(z3-1)*x0*z2-1/2*x1*z0-1/2*x1*z1+1/6*(5*z3+4)*x1*z2+1/3*(z3-1)*x2*z0+1/6*(5*z3+4)*x2*z1+1/6*(-2*z3+5)*x2*z2
	
	



The D6 acts biregularly on P2 × P2, with C6 acting via two matrices on each factor

							[-z3-1,          0,             0,       ]     [1,       0,              0,          ]
							[z3+2,           3*z3+1,        3,       ]  ×  [-2       -2*z3-1,        -3,         ]
							[ 1/3*(-z3+2),   1/3*(2*z3+6),  -2*z3 - 1]     [5/3*z3,  1/3*(-2*z3-6),  3*z3+1      ]


and C2 acting as switching two factors of P2. Moreover, the D6 action leaves invariant the subscheme


	

3*x1-(3*z3+2)*x2=3*z1-(3*z3+2)*z2=0




 
In its comeplement, the divisor of degree (1,1) in P2 × P2 is an affine quadric. Therefore it is D6-equivariant birational to a smooth quadric in P4, given by the birational map 


	

P2 × P2 ---> P4[y1:y2:y3:y4:y5]


[x0*(3*z1-(3*z3+2)*z2) : x1*(3*z1-(3*z3+2)*z2) : z0*(3*x1-(3*z3+2)*x2) : z1*(3*x1-(3*z3+2)*x2) : (3*x1-(3*z3+2)*x2)*(3*z1-(3*z3+2)*z2)]
	
	



The image is a smooth quadric Q given by 




y1*y3+1/14*(10*z6-9)*y2*y3+1/14*(10*z6-9)*y1*y4+1/98*(-43*z6+10)*y2*y4+1/21*(-5*z6+1)*y1*y5+1/294*(29*z6+32)*y2*y5+1/21*(-5*z6+1)*y3*y5+1/294*(29*z6+32)*y4*y5+1/294*(-5*z6-41)*y52
	
z6 = the (first) primitive 6-th root of unity





The D6-action on Q is given by matrices



[-z6,                0,                0,                 0,             0,]
[1/7*(-z6+10),       z6,               0,                 0,             0,]
[0,                  0,                z6-1,              0,             0,]
[0,                  0,                1/7*(-9*z6-1),     -z6+1,         0,]
[1/21*(8*z6-3),      1/3*(2*z6-2),     1/21*(-5*z6+15),   2/3*z6,        -1]


[0, 0, 1, 0, 0,]
[0, 0, 0, 1, 0,]
[1, 0, 0, 0, 0,]
[0, 1, 0, 0, 0,]
[0, 0, 0, 0, 1 ]





A change of variable shows the D6-action on X4 we started with is equivariant birational to a smooth quadric

		y1*y4 - y2*y3 + y52


with the D6-action generated by

	[-z6+1,  0,      0,      0,     0,]
	[0,      z6-1,   0,      0,     0,]
	[0,      0,      -z6,    0,     0,]
	[0,      0,      0,      z6,    0,]
	[0,      0,      0,      0,     -1]


	[0, 0, 0, 1, 0,]
	[0, 0, 1, 0, 0,]
	[0, 1, 0, 0, 0,]
	[1, 0, 0, 0, 0,]
	[0, 0, 0, 0, 1 ]