The line through B₃ and Y₃

We give properties of lines through B₃ and Y₃. We show that every point on this line is a solution to the quadratic lineY₃B₃Charges_quad and a double point of the cubic lineY₃B₃_doublePoint.

References

The main reference for the material in this file is: Allanach, Madigan and Tooby-Smith

def MSSMACC.lineY₃B₃Charges (a b : ℚ) : MSSMACC.LinSols

The line through $Y_3$ and $B_3$ as LinSols.

Equations

• MSSMACC.lineY₃B₃Charges a b = a • MSSMACC.Y₃.toLinSols + b • MSSMACC.B₃.toLinSols

theorem MSSMACC.lineY₃B₃Charges_quad (a b : ℚ) : MSSMACCs.accQuad (lineY₃B₃Charges a b).val = 0

theorem MSSMACC.lineY₃B₃Charges_cubic (a b : ℚ) : MSSMACCs.accCube (lineY₃B₃Charges a b).val = 0

def MSSMACC.lineY₃B₃ (a b : ℚ) : MSSMACC.Sols

The line through $Y_3$ and $B_3$ as Sols.

Equations

• MSSMACC.lineY₃B₃ a b = MSSMACC.AnomalyFreeMk' (MSSMACC.lineY₃B₃Charges a b) ⋯ ⋯

theorem MSSMACC.doublePoint_Y₃_B₃ (R : MSSMACC.LinSols) : ((MSSMACCs.cubeTriLin Y₃.val) B₃.val) R.val = 0

theorem MSSMACC.lineY₃B₃_doublePoint (R : MSSMACC.LinSols) (a b : ℚ) : ((MSSMACCs.cubeTriLin (lineY₃B₃ a b).val) (lineY₃B₃ a b).val) R.val = 0