Oliver Labs
RICAM (Linz, Austria)

www.OliverLabs.net
mail@OliverLabs.net

www.ricam.oeaw.ac.at
Oliver.Labs@oeaw.ac.at


This Singular code computes the objects used in the article
'A Septic with 99 Real Nodes'. 
So, it might be helpful if one wants to work
with these objects, e.g. with the conics through 6 singularities. 

The script also computes ideals describing the singularities. 
This can e.g. be used to check the reality assertion of the theorem.

Finally, we check using these explicit coordinates of the singularities
that the surface actually has 99 nodes and no other singularities.

This script does not follow the path of the paper, but it assumes from the beginning
that we already know the condition on alpha.


This version: November 2005, RICAM (Linz, Austria).
Original version: August 2004, University of Mainz (Mainz, Germany).
-----


// ** redefining sign **

The parameters:
t = (-7*alpha)
a_1 = (-12/7*alpha^2-384/49*alpha-8/7)
a_2 = (-32/7*alpha^2+24/49*alpha-4)
a_3 = (-4*alpha^2+24/49*alpha-4)
a_4 = (-8/7*alpha^2+8/49*alpha-8/7)
a_5 = (49*alpha^2-7*alpha+50)


---------------------------------------


the 3 generic singularities on Sy1:
I_gen_Sy1[1]=x+(-alpha^2-1)*z+(-alpha^2-1)
I_gen_Sy1[2]=52432351*z^3+(-53783576*alpha^2+11313484*alpha-54063467)*z^2+(-53438812*alpha^2+7576128*alpha-53981931)*z+(-7645764*alpha^2+1089340*alpha-7799625)
// dimension (affine) = 0
// degree (affine)  = 3
[1]:
   [1]:
      _[1]=x+(-alpha^2-1)*z+(-alpha^2-1)
   [2]:
      _[1]=52432351*z^3+(-53783576*alpha^2+11313484*alpha-54063467)*z^2+(-53438812*alpha^2+7576128*alpha-53981931)*z+(-7645764*alpha^2+1089340*alpha-7799625)
   [3]:
      1
   [4]:
      3
   [5]:
      1
   [6]:
      1
   [7]:
      0
the first coordinate:
_[1]=x^3+(-211360520/52432351*alpha^2+7683368/52432351*alpha-212976732/52432351)*x^2+(213592176/52432351*alpha^2-30206008/52432351*alpha+214696836/52432351)*x+(-61166112/52432351*alpha^2+8714720/52432351*alpha-62397000/52432351)

the conic C_0: 7*x^2+(56*alpha^2+84)*x*z+(84*alpha^2-8*alpha+56)*z^2+(56*alpha^2+56)*x+(140*alpha^2-16*alpha+140)*z+(56*alpha^2-8*alpha+56)

the two points of C_0 on the x=0 axes:
// dimension (proj.)  = 0
// degree (proj.)   = 2
[1]:
   [1]:
      _[1]=99*z+(49*alpha^2+7*alpha+50)
   [2]:
      _[1]=99*z+(49*alpha^2+7*alpha+50)
   [3]:
      1
   [4]:
      1
   [5]:
      1
   [6]:
      1
   [7]:
      1
[2]:
   [1]:
      _[1]=57*z+(98*alpha^2-28*alpha+106)
   [2]:
      _[1]=57*z+(98*alpha^2-28*alpha+106)
   [3]:
      1
   [4]:
      1
   [5]:
      1
   [6]:
      1
   [7]:
      1
the first coordinate:
_[1]=5643*z^2+(12495*alpha^2-2373*alpha+13344)*z+(5096*alpha^2-658*alpha+5398)

the two possible values of beta:
[1]:
   (-84*alpha^2-84)
[2]:
   (84*alpha^2+84)

the conic C_1:
C_1[1]=x^2+(-4*alpha^2-8)*z^2+(-4*alpha^2-4)*z

the 3 generic singularities on C_1:
I_gen_C1[1]=471796*z^2+(23422*alpha^2+24353*alpha+36652)*x+(180908*alpha^2+98280*alpha+211042)*z+(-66346*alpha^2+10472*alpha-67302)
I_gen_C1[2]=(-49*alpha^2)*x*z+(196*alpha^2-42*alpha-2)*z^2+(-42*alpha-4)*z-2
I_gen_C1[3]=(-343*alpha^2)*x^2+(1372*alpha^2-294*alpha-14)*x*z+(-56*alpha^2+192*alpha-168)*z^2+(-294*alpha-28)*x+(-28*alpha^2-204*alpha-28)*z+(28*alpha^2-4*alpha+84)

the 3 special singularities of Sy6
I_E[1]=274202*z^2+(-120099*alpha^2+34643*alpha-127351)*x+(234906*alpha^2-30170*alpha+254796)*z+(-9898*alpha^2+2072*alpha-4902)
I_E[2]=20843*x*z+(-7314*alpha^2-25200*alpha+67486)*z^2+(-15043*alpha^2+4550*alpha-13800)*x+(-16686*alpha^2+6902*alpha-14014)*z+(-42986*alpha^2+5600*alpha-42300)
I_E[3]=20843*x^2+(-7314*alpha^2-25200*alpha+67486)*x*z+(-12900*alpha^2-3500*alpha-14700)*x+(-201344*alpha^2+37996*alpha-197600)*z+(-141172*alpha^2+19796*alpha-142400)
// dimension (affine) = 0
// degree (affine)  = 3
[1]:
   [1]:
      _[1]=274202*z^2+(-120099*alpha^2+34643*alpha-127351)*x+(234906*alpha^2-30170*alpha+254796)*z+(-9898*alpha^2+2072*alpha-4902)
      _[2]=7*x*z+(2*alpha+14)*z^2+(-14*alpha^2+4*alpha-14)*z+(-14*alpha^2+2*alpha-14)
      _[3]=840693*x^2+(-1681386*alpha^2-240198*alpha-34972)*x*z+(-3362772*alpha^2+470404*alpha-6795488)*z^2+(-240198*alpha)*x+(-3292828*alpha^2+460412*alpha-3292828)*z+(69944*alpha^2-9992*alpha+69944)
   [2]:
      _[1]=274202*z^2+(-120099*alpha^2+34643*alpha-127351)*x+(234906*alpha^2-30170*alpha+254796)*z+(-9898*alpha^2+2072*alpha-4902)
      _[2]=7*x*z+(2*alpha+14)*z^2+(-14*alpha^2+4*alpha-14)*z+(-14*alpha^2+2*alpha-14)
      _[3]=840693*x^2+(-1681386*alpha^2-240198*alpha-34972)*x*z+(-3362772*alpha^2+470404*alpha-6795488)*z^2+(-240198*alpha)*x+(-3292828*alpha^2+460412*alpha-3292828)*z+(69944*alpha^2-9992*alpha+69944)
   [3]:
      0
   [4]:
      3
   [5]:
      0
   [6]:
      3
   [7]:
      1
the first coordinate:
_[1]=x^3+(-999698/137101*alpha^2+209272/137101*alpha-1043506/137101)*x^2+(1567804/137101*alpha^2-214172/137101*alpha+1607208/137101)*x+(1058400/137101*alpha^2-145544/137101*alpha+1078008/137101)

the three generic singularities on C_2:
I_gen_C2[1]=5869247*z^2+(-2403940*alpha^2-139944*alpha-3537604)*x+(-21145656*alpha^2+2708328*alpha-21301912)*z+(-10572828*alpha^2+1354164*alpha-10650956)
I_gen_C2[2]=8176539*x*z+(-28657762*alpha^2+4527752*alpha+4963024)*z^2+(8664082*alpha^2-1067612*alpha+9415658)*x+(24397856*alpha^2-3183520*alpha+25410160)*z+(12198928*alpha^2-1591760*alpha+12705080)
I_gen_C2[3]=8176539*x^2+(-28657762*alpha^2+4527752*alpha+4963024)*x*z+(-28028560*alpha^2+4450220*alpha-31057016)*x+(-42619108*alpha^2+9901808*alpha-43839236)*z+(-37662632*alpha^2+4950904*alpha-38272696)
// dimension (affine) = 0
// degree (affine)  = 3
[1]:
   [1]:
      _[1]=5869247*z^2+(-2403940*alpha^2-139944*alpha-3537604)*x+(-21145656*alpha^2+2708328*alpha-21301912)*z+(-10572828*alpha^2+1354164*alpha-10650956)
      _[2]=14*x*z+(-28*alpha^2+4*alpha+35)*z^2+(-56*alpha^2+8*alpha-56)*z+(-28*alpha^2+4*alpha-28)
      _[3]=25241370*x^2+(151448220*alpha^2-4272996*alpha+180629477)*x*z+(497787290*alpha^2+3532660*alpha+115904684)*z^2+(75724110*alpha^2-7211820*alpha+75724110)*x+(238333480*alpha^2-19384096*alpha+234212440)*z+(68684000*alpha^2-9692048*alpha+66623480)
   [2]:
      _[1]=5869247*z^2+(-2403940*alpha^2-139944*alpha-3537604)*x+(-21145656*alpha^2+2708328*alpha-21301912)*z+(-10572828*alpha^2+1354164*alpha-10650956)
      _[2]=14*x*z+(-28*alpha^2+4*alpha+35)*z^2+(-56*alpha^2+8*alpha-56)*z+(-28*alpha^2+4*alpha-28)
      _[3]=25241370*x^2+(151448220*alpha^2-4272996*alpha+180629477)*x*z+(497787290*alpha^2+3532660*alpha+115904684)*z^2+(75724110*alpha^2-7211820*alpha+75724110)*x+(238333480*alpha^2-19384096*alpha+234212440)*z+(68684000*alpha^2-9692048*alpha+66623480)
   [3]:
      0
   [4]:
      3
   [5]:
      0
   [6]:
      3
   [7]:
      1
the first coordinate:
_[1]=x^3+(22513344/5869247*alpha^2-10962896/5869247*alpha-7360/5869247)*x^2+(-99212848/5869247*alpha^2+8320144/5869247*alpha-98100656/5869247)*x+(-56281792/5869247*alpha^2+8085952/5869247*alpha-57066752/5869247)

the conic C_2:
1250778404579529*x^2+(-3335409078878744*alpha^2+1667704539439372)*x*z+(1667704539439372*alpha^2+476487011268392*alpha)*z^2+(-3335409078878744*alpha^2-3335409078878744)*x+(-1667704539439372*alpha^2+952974022536784*alpha-1667704539439372)*z+(-3335409078878744*alpha^2+476487011268392*alpha-3335409078878744)

the two other singularities, O_12, on Sy1:
O_12[1]=x+(-alpha^2-1)*z+(-alpha^2-1)
O_12[2]=33783*z^2+(-318108*alpha^2+46998*alpha-322866)*z+(-197960*alpha^2+27790*alpha-201025)
// dimension (affine) = 0
// degree (affine)  = 2
[1]:
   [1]:
      _[1]=x+(-alpha^2-1)*z+(-alpha^2-1)
   [2]:
      _[1]=33783*z^2+(-318108*alpha^2+46998*alpha-322866)*z+(-197960*alpha^2+27790*alpha-201025)
   [3]:
      1
   [4]:
      2
   [5]:
      1
   [6]:
      1
   [7]:
      0
the first coordinate:
_[1]=(33783*alpha^2)*x^2+(-6714*alpha^2+10332*alpha-6492)*x+(2452*alpha^2-392*alpha+3176)

the singularity on the rotation axes x=y=0, P_x:
P_x[1]=x
P_x[2]=z+1


---------------------------------------



the 15 singularities of the plane septic:

P_x:
// dimension (affine) = 0
// degree (affine)  = 1
[1]:
   [1]:
      _[1]=z+1
   [2]:
      _[1]=x
   [3]:
      1
   [4]:
      1
   [5]:
      1
   [6]:
      1
   [7]:
      1
the first coordinate:
_[1]=x

O_12:
// dimension (affine) = 0
// degree (affine)  = 2
[1]:
   [1]:
      _[1]=x+(-alpha^2-1)*z+(-alpha^2-1)
   [2]:
      _[1]=33783*z^2+(-318108*alpha^2+46998*alpha-322866)*z+(-197960*alpha^2+27790*alpha-201025)
   [3]:
      1
   [4]:
      2
   [5]:
      1
   [6]:
      1
   [7]:
      0
the first coordinate:
_[1]=(33783*alpha^2)*x^2+(-6714*alpha^2+10332*alpha-6492)*x+(2452*alpha^2-392*alpha+3176)

I_gen_Sy1:
// dimension (affine) = 0
// degree (affine)  = 3
[1]:
   [1]:
      _[1]=x+(-alpha^2-1)*z+(-alpha^2-1)
   [2]:
      _[1]=52432351*z^3+(-53783576*alpha^2+11313484*alpha-54063467)*z^2+(-53438812*alpha^2+7576128*alpha-53981931)*z+(-7645764*alpha^2+1089340*alpha-7799625)
   [3]:
      1
   [4]:
      3
   [5]:
      1
   [6]:
      1
   [7]:
      0
the first coordinate:
_[1]=x^3+(-211360520/52432351*alpha^2+7683368/52432351*alpha-212976732/52432351)*x^2+(213592176/52432351*alpha^2-30206008/52432351*alpha+214696836/52432351)*x+(-61166112/52432351*alpha^2+8714720/52432351*alpha-62397000/52432351)

I_gen_C1:
// dimension (affine) = 0
// degree (affine)  = 3
[1]:
   [1]:
      _[1]=471796*z^2+(23422*alpha^2+24353*alpha+36652)*x+(180908*alpha^2+98280*alpha+211042)*z+(-66346*alpha^2+10472*alpha-67302)
      _[2]=7*x*z+(-28*alpha^2-2*alpha-56)*z^2+(-14*alpha^2-4*alpha-14)*z+(14*alpha^2-2*alpha+14)
      _[3]=81977*x^2+(163954*alpha^2+364364*alpha-435911)*x*z+(-1632848*alpha^2-911314*alpha+744324)*z^2+(-163954*alpha^2+23422*alpha-163954)*x+(-634634*alpha^2+530156*alpha-315630)*z+(-641326*alpha^2+77702*alpha-627942)
   [2]:
      _[1]=471796*z^2+(23422*alpha^2+24353*alpha+36652)*x+(180908*alpha^2+98280*alpha+211042)*z+(-66346*alpha^2+10472*alpha-67302)
      _[2]=7*x*z+(-28*alpha^2-2*alpha-56)*z^2+(-14*alpha^2-4*alpha-14)*z+(14*alpha^2-2*alpha+14)
      _[3]=81977*x^2+(163954*alpha^2+364364*alpha-435911)*x*z+(-1632848*alpha^2-911314*alpha+744324)*z^2+(-163954*alpha^2+23422*alpha-163954)*x+(-634634*alpha^2+530156*alpha-315630)*z+(-641326*alpha^2+77702*alpha-627942)
   [3]:
      0
   [4]:
      3
   [5]:
      0
   [6]:
      3
   [7]:
      1
the first coordinate:
_[1]=x^3+(223146/117949*alpha^2+28196/117949*alpha+240125/117949)*x^2+(-6958/117949*alpha^2-3780/117949*alpha+956/117949)*x+(-132692/117949*alpha^2+20944/117949*alpha-134604/117949)

I_gen_C2:
// dimension (affine) = 0
// degree (affine)  = 3
[1]:
   [1]:
      _[1]=5869247*z^2+(-2403940*alpha^2-139944*alpha-3537604)*x+(-21145656*alpha^2+2708328*alpha-21301912)*z+(-10572828*alpha^2+1354164*alpha-10650956)
      _[2]=14*x*z+(-28*alpha^2+4*alpha+35)*z^2+(-56*alpha^2+8*alpha-56)*z+(-28*alpha^2+4*alpha-28)
      _[3]=25241370*x^2+(151448220*alpha^2-4272996*alpha+180629477)*x*z+(497787290*alpha^2+3532660*alpha+115904684)*z^2+(75724110*alpha^2-7211820*alpha+75724110)*x+(238333480*alpha^2-19384096*alpha+234212440)*z+(68684000*alpha^2-9692048*alpha+66623480)
   [2]:
      _[1]=5869247*z^2+(-2403940*alpha^2-139944*alpha-3537604)*x+(-21145656*alpha^2+2708328*alpha-21301912)*z+(-10572828*alpha^2+1354164*alpha-10650956)
      _[2]=14*x*z+(-28*alpha^2+4*alpha+35)*z^2+(-56*alpha^2+8*alpha-56)*z+(-28*alpha^2+4*alpha-28)
      _[3]=25241370*x^2+(151448220*alpha^2-4272996*alpha+180629477)*x*z+(497787290*alpha^2+3532660*alpha+115904684)*z^2+(75724110*alpha^2-7211820*alpha+75724110)*x+(238333480*alpha^2-19384096*alpha+234212440)*z+(68684000*alpha^2-9692048*alpha+66623480)
   [3]:
      0
   [4]:
      3
   [5]:
      0
   [6]:
      3
   [7]:
      1
the first coordinate:
_[1]=x^3+(22513344/5869247*alpha^2-10962896/5869247*alpha-7360/5869247)*x^2+(-99212848/5869247*alpha^2+8320144/5869247*alpha-98100656/5869247)*x+(-56281792/5869247*alpha^2+8085952/5869247*alpha-57066752/5869247)

I_E:
// dimension (affine) = 0
// degree (affine)  = 3
[1]:
   [1]:
      _[1]=274202*z^2+(-120099*alpha^2+34643*alpha-127351)*x+(234906*alpha^2-30170*alpha+254796)*z+(-9898*alpha^2+2072*alpha-4902)
      _[2]=7*x*z+(2*alpha+14)*z^2+(-14*alpha^2+4*alpha-14)*z+(-14*alpha^2+2*alpha-14)
      _[3]=840693*x^2+(-1681386*alpha^2-240198*alpha-34972)*x*z+(-3362772*alpha^2+470404*alpha-6795488)*z^2+(-240198*alpha)*x+(-3292828*alpha^2+460412*alpha-3292828)*z+(69944*alpha^2-9992*alpha+69944)
   [2]:
      _[1]=274202*z^2+(-120099*alpha^2+34643*alpha-127351)*x+(234906*alpha^2-30170*alpha+254796)*z+(-9898*alpha^2+2072*alpha-4902)
      _[2]=7*x*z+(2*alpha+14)*z^2+(-14*alpha^2+4*alpha-14)*z+(-14*alpha^2+2*alpha-14)
      _[3]=840693*x^2+(-1681386*alpha^2-240198*alpha-34972)*x*z+(-3362772*alpha^2+470404*alpha-6795488)*z^2+(-240198*alpha)*x+(-3292828*alpha^2+460412*alpha-3292828)*z+(69944*alpha^2-9992*alpha+69944)
   [3]:
      0
   [4]:
      3
   [5]:
      0
   [6]:
      3
   [7]:
      1
the first coordinate:
_[1]=x^3+(-999698/137101*alpha^2+209272/137101*alpha-1043506/137101)*x^2+(1567804/137101*alpha^2-214172/137101*alpha+1607208/137101)*x+(1058400/137101*alpha^2-145544/137101*alpha+1078008/137101)

Check that each of the ideals is reduced
and check also that these are all ordinary double points of Sy:

Okay!

Okay!

Okay!

Okay!

Okay!

Okay!


---------------------------------------


Check that these are really 15 distinct points:
15

Check that there are no other singularities on the proj. curve Sy:
// dimension (proj.)  = 0
// degree (proj.)   = 15

check that (1:i:0:0) does not lie on the surface,
i.e. the following is not zero: 64

Check that each of the ideals is reduced
and check also that these are all ordinary double points of S:

Okay!

Okay!

Okay!

Okay!

Okay!

Okay!

Time needed for the whole computation:
0h, 0m, 8s