This file essentially starts with the setup from the file “a5-blow-up.nb” and replaces the coefficients of the (Subscript[A, 5])-family from the Theorem the coefficients of a general divisor that vanishes at (R,(1:1)).  This eventually yields a quadric and quintic polynomial that may be entered into MAGMA and leads one to the find the coefficients of the (Subscript[A, 5])-family.

Setup

Equations for (Subscript[E, τ]) and the sections (Subscript[ψ, i]):

alpha1=0;
a=16τ^6;
b=1-6τ^2-3τ^4;
c[x_]:=a(x-alpha1)+b(x-alpha1)^2+(x-alpha1)^3;
(*The variable t will stand for 1/(x-alpha1), and will be substituted for this later*)
psi[0]=(x-alpha1)*Z^6+a*t;
psi[1]=(x-alpha1)^2*Z^6+a^2*t^2;
psi[2]=(x-alpha1)*Z^5+a*t*Z;
psi[3]=y*Z^5-a*y*t^2*Z;
psi[4]=Z^4+Z^2;
psi[5]=(x-alpha1)*Z^4+a*t*Z^2;
psi[6]=Z^3;

Equations for special sections (Subscript[λ, i]):

λ[0]=Table[psi[i],{i,0,6}].{1,0,1,0,0,0,0}
λ[1]=Table[psi[i],{i,0,6}].{1,0,0,0,0,-1,0}
λ[2]=Table[psi[i],{i,0,6}].{0,0,0,0,1,0,2}
λ[3]=Table[psi[i],{i,0,6}].{-8τ^2(1+τ^2),1,0,0,4τ^4(1+4τ^2+τ^4),0,(-2)*4τ^4(1+4τ^2+τ^4)}

This is the equation of the family with an (Subscript[A, 5])-singularity (denoted by μ(τ) in the article) at the point R, whose local coordinates are R=(x,Z)=(4τ^3,1). Since we only care about the family locally at the point, we clear denominators.

fam=Simplify[(x-alpha1)^2*({a0,a1,-4τ^3a0+8τ^5(2-τ+2τ^2)*a3,a3}.{λ[0],λ[1],λ[2],λ[3]}/.t->1/(x-alpha1))];

Equation of the family in local coordinates near the point R:

local=fam/.{x->u+4τ^3,Z->v+1};

Given a polynomial f(u,v), the following will give the coefficient of (u^i)(v^j) and print its k-jet at (0,0):

coef[i_,j_,f_]:=Factor[(1/i!)(1/j!)D[f,{u,i},{v,j}]/.{u->0,v->0}];
jet[k_,f_]:=
(
For[i=0,i≤k,i++,
For[j=0,j≤i,j++,
Print[coef[i-j,j,f], " * ", u^(i-j) v^j,"+"];
];
];
Print["terms of higher order"];
)

The following shows that the a generic member of the family has a double point at R and that the Hessian has rank 1:

jet[2,local]

Factor[Det[Factor[D[local,{{u,v},2}]/.{u->0,v->0}]]]

Hence the curve has a degenerate singularity at (R,(1:1)) exactly when the parameters are given by (a zero of) the following:

quadric=(a0^2)+4 a0 a1+4 (a1^2)-2 a0 a3 τ-8 a0 a3 (τ^2)-32 a1 a3 (τ^2)-28 a0 a3 (τ^3)-32 a1 a3 (τ^3)+8 (a3^2) (τ^3)-8 a0 a3 (τ^4)-32 a1 a3 (τ^4)+16 (a3^2) (τ^4)-2 a0 a3 (τ^5)+248 (a3^2) (τ^5)+32 (a3^2) (τ^6)+248 (a3^2) (τ^7)+16 (a3^2) (τ^8)+8 (a3^2) (τ^9);

First blow up

We now blow up (u,v)-plane at the origin and examine the pullback of the family. Recall that is defined by {(u,v,((Subscript[s, 0]):(Subscript[s, 1]))) ∈ (A^1)×(P^1): (Subscript[s, 1])u=(Subscript[s, 0])v}, and so the two coordinate patches may be given by (Subscript[s, 1])≠0 (where (Subscript[s, 1])=1 and u=(Subscript[s, 0])v) and (Subscript[s, 0])≠0 (where (Subscript[s, 0])=1 and v=(Subscript[s, 1])u):

First we calculate the equation of the proper transform of the family, and then compute where it intersects the exceptional divisor:

(local/.{u->s0 v})/v^2//Factor;
Factor[%/.v->0]

((local/.{v->s1 u})/u^2)//Factor;
Factor[%/.u->0]

From these equations we see that, for generic (Subscript[a, 0,])(Subscript[a, 1]),(Subscript[a, 3]), the proper transform intersects the exceptional divisor at two points (u,v,((Subscript[s, 0]):(Subscript[s, 1])))=(0,0,((Subscript[s, 0]):1)), where (Subscript[s, 0]) is a root of:

intersect=a0 (s0^2)-4 a3 (s0^2) (τ^2)+20 a0 s0 (τ^3)+8 a1 s0 (τ^3)+8 a3 (s0^2) (τ^3)-4 a3 (s0^2) (τ^4)-96 a3 s0 (τ^5)+96 a0 (τ^6)+64 a1 (τ^6)+96 a3 s0 (τ^6)+8 a3 (τ^7)-96 a3 s0 (τ^7)-544 a3 (τ^8)+304 a3 (τ^9)-544 a3 (τ^10)+8 a3 (τ^11);
Apart[intersect,s0]

But this should not be the case if (Subscript[a, 0,])(Subscript[a, 1]),(Subscript[a, 3]) belongs to the quadric curve described previously.  And indeed, if one computes the discriminant of the quadratic equation in (Subscript[s, 0]) above, one obtains:

Factor[Discriminant[intersect,s0]]

So assuming that this discriminant vanishes, one finds that the proper transform intersects the exceptional divisor at (0,0,((Subscript[s, 0]):1)), where (Subscript[s, 0]) is given by:

Factor[Solve[intersect==0,s0]]

Factor[-(2 (5 a0 (τ^3)+2 a1 (τ^3)-24 a3 (τ^5)+24 a3 (τ^6)-24 a3 (τ^7)+√((τ^6) *0)))/(a0-4 a3 (τ^2)+8 a3 (τ^3)-4 a3 (τ^4))]

We will now localize the proper transform at this point, abusing notation by using (u,v) for the local coordinates again:

proper1=Factor[Factor[(local/.{u->s0 v})/v^2]/.{s0->u-((2 (τ^3) (5 a0+2 a1-24 a3 (τ^2)+24 a3 (τ^3)-24 a3 (τ^4)))/(a0-4 a3 (τ^2)+8 a3 (τ^3)-4 a3 (τ^4)))}];
jet[2,proper1]

Comparing constant and linear terms with the equation of the quadric, we see that the proper transform of the family still has one singular point.  Furthermore, it will be a degenerate singular point if the hessian has zero determinant:

Factor[Det[D[proper1,{{u,v},2}]/.{u->0,v->0}]]

The long quintic factor now gives us another worthwhile constraint to consider:

quintic=-101 (a0^5)-536 (a0^4) a1-984 (a0^3) (a1^2)-736 (a0^2) (a1^3)-208 a0 (a1^4)+208 (a0^4) a3 τ+288 (a0^3) a1 a3 τ+128 (a0^2) (a1^2) a3 τ+2244 (a0^4) a3 (τ^2)+11296 (a0^3) a1 a3 (τ^2)+15648 (a0^2) (a1^2) a3 (τ^2)+7552 a0 (a1^3) a3 (τ^2)+832 (a1^4) a3 (τ^2)-16 (a0^3) (a3^2) (τ^2)+1568 (a0^4) a3 (τ^3)+704 (a0^3) a1 a3 (τ^3)-4480 (a0^2) (a1^2) a3 (τ^3)-4608 a0 (a1^3) a3 (τ^3)-1536 (a1^4) a3 (τ^3)-3776 (a0^3) (a3^2) (τ^3)-3968 (a0^2) a1 (a3^2) (τ^3)-1024 a0 (a1^2) (a3^2) (τ^3)+2244 (a0^4) a3 (τ^4)+11296 (a0^3) a1 a3 (τ^4)+15648 (a0^2) (a1^2) a3 (τ^4)+7552 a0 (a1^3) a3 (τ^4)+832 (a1^4) a3 (τ^4)-17344 (a0^3) (a3^2) (τ^4)-85760 (a0^2) a1 (a3^2) (τ^4)-80896 a0 (a1^2) (a3^2) (τ^4)-18432 (a1^3) (a3^2) (τ^4)+192 (a0^2) (a3^3) (τ^4)+208 (a0^4) a3 (τ^5)+288 (a0^3) a1 a3 (τ^5)+128 (a0^2) (a1^2) a3 (τ^5)-46912 (a0^3) (a3^2) (τ^5)-14464 (a0^2) a1 (a3^2) (τ^5)+53248 a0 (a1^2) (a3^2) (τ^5)+30720 (a1^3) (a3^2) (τ^5)+25472 (a0^2) (a3^3) (τ^5)+17920 a0 a1 (a3^3) (τ^5)+2048 (a1^2) (a3^3) (τ^5)-8672 (a0^3) (a3^2) (τ^6)-123392 (a0^2) a1 (a3^2) (τ^6)-140800 a0 (a1^2) (a3^2) (τ^6)-30720 (a1^3) (a3^2) (τ^6)+54464 (a0^2) (a3^3) (τ^6)+279552 a0 a1 (a3^3) (τ^6)+136192 (a1^2) (a3^3) (τ^6)-768 a0 (a3^4) (τ^6)-46912 (a0^3) (a3^2) (τ^7)-14464 (a0^2) a1 (a3^2) (τ^7)+53248 a0 (a1^2) (a3^2) (τ^7)+30720 (a1^3) (a3^2) (τ^7)+462848 (a0^2) (a3^3) (τ^7)+90112 a0 a1 (a3^3) (τ^7)-169984 (a1^2) (a3^3) (τ^7)-75776 a0 (a3^4) (τ^7)-26624 a1 (a3^4) (τ^7)-17344 (a0^3) (a3^2) (τ^8)-85760 (a0^2) a1 (a3^2) (τ^8)-80896 a0 (a1^2) (a3^2) (τ^8)-18432 (a1^3) (a3^2) (τ^8)-362368 (a0^2) (a3^3) (τ^8)+283648 a0 a1 (a3^3) (τ^8)+250880 (a1^2) (a3^3) (τ^8)-46592 a0 (a3^4) (τ^8)-331776 a1 (a3^4) (τ^8)+1024 (a3^5) (τ^8)-3776 (a0^3) (a3^2) (τ^9)-3968 (a0^2) a1 (a3^2) (τ^9)-1024 a0 (a1^2) (a3^2) (τ^9)+993536 (a0^2) (a3^3) (τ^9)+316416 a0 a1 (a3^3) (τ^9)-327680 (a1^2) (a3^3) (τ^9)-1873920 a0 (a3^4) (τ^9)-157696 a1 (a3^4) (τ^9)+83968 (a3^5) (τ^9)-16 (a0^3) (a3^2) (τ^10)-362368 (a0^2) (a3^3) (τ^10)+283648 a0 a1 (a3^3) (τ^10)+250880 (a1^2) (a3^3) (τ^10)+3068672 a0 (a3^4) (τ^10)+360448 a1 (a3^4) (τ^10)-46080 (a3^5) (τ^10)+462848 (a0^2) (a3^3) (τ^11)+90112 a0 a1 (a3^3) (τ^11)-169984 (a1^2) (a3^3) (τ^11)-6897664 a0 (a3^4) (τ^11)-1126400 a1 (a3^4) (τ^11)+2691072 (a3^5) (τ^11)+54464 (a0^2) (a3^3) (τ^12)+279552 a0 a1 (a3^3) (τ^12)+136192 (a1^2) (a3^3) (τ^12)+6454272 a0 (a3^4) (τ^12)+1679360 a1 (a3^4) (τ^12)-6667264 (a3^5) (τ^12)+25472 (a0^2) (a3^3) (τ^13)+17920 a0 a1 (a3^3) (τ^13)+2048 (a1^2) (a3^3) (τ^13)-6897664 a0 (a3^4) (τ^13)-1126400 a1 (a3^4) (τ^13)+14858240 (a3^5) (τ^13)+192 (a0^2) (a3^3) (τ^14)+3068672 a0 (a3^4) (τ^14)+360448 a1 (a3^4) (τ^14)-21844992 (a3^5) (τ^14)-1873920 a0 (a3^4) (τ^15)-157696 a1 (a3^4) (τ^15)+24502272 (a3^5) (τ^15)-46592 a0 (a3^4) (τ^16)-331776 a1 (a3^4) (τ^16)-21844992 (a3^5) (τ^16)-75776 a0 (a3^4) (τ^17)-26624 a1 (a3^4) (τ^17)+14858240 (a3^5) (τ^17)-768 a0 (a3^4) (τ^18)-6667264 (a3^5) (τ^18)+2691072 (a3^5) (τ^19)-46080 (a3^5) (τ^20)+83968 (a3^5) (τ^21)+1024 (a3^5) (τ^22);

These two polynomials were put into MAGMA, with the substitution (Subscript[a, 3])=1, and a radical decomposition of the ideal they generate was computed quickly (in about 15 seconds on a laptop).  The most interesting piece of this decomposition is given by the following component:

Factor[Solve[({a0+a1*t^3+12*a1*t^2+37*a1*t+8*a1-24*t^6-224*t^5-252*t^4-192*t^3-4*t^2,a1*t^4+12*a1*t^3+38*a1*t^2+12*a1*t+a1-24*t^7-224*t^6-272*t^5-224*t^4-24*t^3}/.{t->τ})=={0,0},{a0,a1}]]

We had assumed that (Subscript[a, 3])=1, and the original assumption that the curve passes through (R,(1:1)) determined the value of (Subscript[a, 2]) from these:

Factor[(-4τ^3a0+8τ^5(2-τ+2τ^2)*a3)/.{a0->((4 (((-1+τ))^2) (τ^2) (1+14 τ+34 (τ^2)+14 (τ^3)+(τ^4)))/(((1+6 τ+(τ^2)))^2)),a1->((8 (τ^3) (3+28 τ+34 (τ^2)+28 (τ^3)+3 (τ^4)))/(((1+6 τ+(τ^2)))^2)),a3->1}]

Hence if we clear denominators, we arrive at the coefficients of our family:

(((1+6 τ+(τ^2)))^2)*{((4 (((-1+τ))^2) (τ^2) (1+14 τ+34 (τ^2)+14 (τ^3)+(τ^4)))/(((1+6 τ+(τ^2)))^2)),((8 (τ^3) (3+28 τ+34 (τ^2)+28 (τ^3)+3 (τ^4)))/(((1+6 τ+(τ^2)))^2)),-((8 (τ^6) (1-52 τ-90 (τ^2)-52 (τ^3)+(τ^4)))/(((1+6 τ+(τ^2)))^2)),1}//MatrixForm