Finding six conics tangent to one conic and four lines in the plane

Question

Indeed the number of conics tangent to five plane curves of degrees $d_1,d_2,d_3,d_4,d_5$ in ${\Bbb P}^2_{\Bbb C}$ is

$$d_{{1}}d_{{2}}d_{{3}}d_{{4}}d_{{5}} \left( d_{{1}}d_{{2}}d_{{3}}d_{{4} }+d_{{1}}d_{{2}}d_{{3}}+3\,d_{{1}}+3\,d_{{2}}+3\,d_{{3}}+3\,d_{{4}}+3 \,d_{{5}}-3\,d_{{1}}d_{{2}}+d_{{1}}d_{{4}}d_{{5}}+d_{{3}}d_{{4}}d_{{5} }+d_{{2}}d_{{3}}d_{{4}}+d_{{3}}d_{{4}}d_{{1}}+d_{{3}}d_{{1}}d_{{5}}+d_ {{2}}d_{{4}}d_{{1}}+d_{{2}}d_{{1}}d_{{5}}+d_{{2}}d_{{4}}d_{{5}}+d_{{2} }d_{{3}}d_{{5}}+d_{{2}}d_{{3}}d_{{4}}d_{{5}}+d_{{1}}d_{{3}}d_{{4}}d_{{ 5}}+d_{{2}}d_{{3}}d_{{1}}d_{{5}}+d_{{2}}d_{{1}}d_{{4}}d_{{5}}-3\,d_{{3 }}d_{{5}}-3\,d_{{2}}d_{{4}}-3\,d_{{3}}d_{{4}}-3\,d_{{2}}d_{{5}}-3\,d_{ {4}}d_{{5}}-3\,d_{{2}}d_{{3}}-3\,d_{{1}}d_{{5}}-3\,d_{{4}}d_{{1}}-3\,d _{{3}}d_{{1}}+d_{{1}}d_{{2}}d_{{3}}d_{{4}}d_{{5}} \right)$$

as can be shown e.g. by computing in the space of complete conics like in the following schubert for maple code

proj(5, H, all); # currentvariety_ is PH, DIM is 5
proj(2, h, all); # currentvariety_ is Ph, DIM is 2
morphism(i, Ph, PH, [H = 2*h]); # i
unprotect(Bi);
blowup(i); # currentvariety_ is Bi, DIM is 5
factor(integral(Bi, product('d || j*(d || j+1)*H-d || j*Ei', j = 1 .. 5)));

This formula gives the famous $3264$ for $d_1=d_2=d_3=d_4=d_5=2$ and $1$ for $d_1=d_2=d_3=d_4=d_5=1.$ But for $d_1=2, d_2=d_3=d_4=d_5=1$ it gives $6$. A first question is how to find the six or if they all can be real.

I can see that if we have one like in the above figure, there will be another by considering the dual situation of going through four points and the dual conic.

Is there an elementary way, and does this give six real solutions?

Answer 1

I'm still looking for elementary solutions, but inspired by 3264 conics in a second given the conic and four lines

$\begin{align}(5/100)x^2+(2/100)xy +(6/100)y^2+x/6+y/2+1=0\\ x/2+y/3+1=0\\ x/6-y+1=0\\ 6x+y+1=0\\ x/6+y+1=0\end{align}$

the solution conics are $u_1x^2+u_2xy+u_3y^2+u_4x+u_5y+1=0$

R=QQ[u_1..u_5][x,y]
U=u_1*x^2+u_2*x*y+u_3*y^2+u_4*x+u_5*y+1
A=(5/100)*x^2+(2/100)*x*y +(6/100)*y^2+x/6+y/2+1
L1=x/2+y/3+1
L2=x/6-y+1
L3=6*x+y+1
L4=x/6+y+1
D1=diff(x,L1)*diff(y,U)-diff(y,L1)*diff(x,U)
D2=diff(x,L2)*diff(y,U)-diff(y,L2)*diff(x,U)
D3=diff(x,L3)*diff(y,U)-diff(y,L3)*diff(x,U)
D4=diff(x,L4)*diff(y,U)-diff(y,L4)*diff(x,U)
D5=diff(x,A)*diff(y,U)-diff(y,A)*diff(x,U)
J1=ideal(U,L1,D1)
J2=ideal(U,L2,D2)
J3=ideal(U,L3,D3)
J4=ideal(U,L4,D4)
J5=ideal(U,A,D5)
K=ideal((gens gb J1)_0_0,(gens gb J2)_0_0,(gens gb J3)_0_0,(gens gb J4)_0_0,(gens gb J5)_0_0)
primaryDecomposition K

There are some double line solutions, but among the eight solutions to one of the primary components

solve([779328*u_3-426300*u_4-202353*u_5+193778,15744*u_2-1044*u_4-13019*u_5+6438,377856*u_1-193980*u_4+25839*u_5+39026,144*u_4^2-72*u_4*u_5+297*u_5^2-936*u_4-150*u_5+296,87597407842465168927913032800*u_4*u_5^2+97837260030578925685153377000*u_5^3-487698925557077528651918553600*u_4*u_5+485810392929259766965799846400*u_5^2-1613407383119823950704340858892*u_4-172970680683284616608594041533*u_5+537939273006881393060106495050],[u_1,u_2,u_3,u_4,u_5]);

there corresponds five real conics which are tangent to the given conic and four lines, plus one real that isn't tangent to the given conic (but to the four lines) and I haven't checked the two complex conics.

[[u_1 = 0.8750688958295058,u_2 = -1.395065003979835,u_3 = 0.351181862035697,
  u_4 = 1.728426395939086,u_5 = -1.331160601122578],
 [u_1 = 0.02802059633576817,u_2 = -0.0710532899674756,
  u_3 = 0.0189357084050595,u_4 = 0.3069139966273187,u_5 = 0.3839710210550147],
 [u_1 = 0.02802059633576817,u_2 = 0.1036310752486395,
  u_3 = 0.08843621910321699,u_4 = 0.3347557072754427,
  u_5 = 0.5929859224499876],
 [u_1 = 0.02772525394661624,u_2 = -0.06384418709580388,
  u_3 = 0.021504375349097,u_4 = 0.3074936850968285,u_5 = 0.3926412614980289],
 [u_1 = 0.02772525394661624,u_2 = 0.09628990509059535,
  u_3 = 0.08521591666161953,u_4 = 0.3330164012360351,
  u_5 = 0.5842475386779185],
 [u_1 = 0.02647068604303149,u_2 = 0.05107972592341037,
  u_3 = 0.06595614571503451,u_4 = 0.3233927188226181,u_5 = 0.530346232179226],
 [u_1 = 0.02635190019910318-1.461471234038826E-4*%i,
  u_2 = 0.04509188119602943-0.01037090865736362*%i,
  u_3 = 0.06345333307394292-0.004274417712843975*%i,
  u_4 = 0.3222094373896304-0.001934618733077926*%i,
  u_5 = 0.5231999327840873-0.01238650003411568*%i],
 [u_1 = 1.461471234038826E-4*%i+0.02635190019910318,
  u_2 = 0.01037090865736362*%i+0.04509188119602943,
  u_3 = 0.004274417712843358*%i+0.06345333307394241,
  u_4 = 0.001934618733076755*%i+0.3222094373896294,
  u_5 = 0.01238650003410853*%i+0.5231999327840805]]$