// This file takes the polynomials obtained midway through the Mathematica file "general-blow-up.nb" and, with the substitution a3=1 and tau=t, computes the radical decomposition of their intersection.

r<a0,a1,t> := PolynomialRing(Rationals(),3);
quadric := a0^2 + 4*a0*a1 + 4*a1^2 - 2*a0*1*t - 8*a0*1*t^2 - 32*a1*1*t^2 - 
   28*a0*1*t^3 - 32*a1*1*t^3 + 8*1^2*t^3 - 8*a0*1*t^4 - 32*a1*1*t^4 + 
   16*1^2*t^4 - 2*a0*1*t^5 + 248*1^2*t^5 + 32*1^2*t^6 + 248*1^2*t^7 + 
   16*1^2*t^8 + 8*1^2*t^9;
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*1*t + 288*a0^3*a1*1*t + 128*a0^2*a1^2*1*t + 2244*a0^4*1*t^2 + 
     11296*a0^3*a1*1*t^2 + 15648*a0^2*a1^2*1*t^2 + 7552*a0*a1^3*1*t^2 + 
     832*a1^4*1*t^2 - 16*a0^3*1^2*t^2 + 1568*a0^4*1*t^3 + 704*a0^3*a1*1*t^3 - 
     4480*a0^2*a1^2*1*t^3 - 4608*a0*a1^3*1*t^3 - 1536*a1^4*1*t^3 - 
     3776*a0^3*1^2*t^3 - 3968*a0^2*a1*1^2*t^3 - 1024*a0*a1^2*1^2*t^3 + 
     2244*a0^4*1*t^4 + 11296*a0^3*a1*1*t^4 + 15648*a0^2*a1^2*1*t^4 + 
     7552*a0*a1^3*1*t^4 + 832*a1^4*1*t^4 - 17344*a0^3*1^2*t^4 - 
     85760*a0^2*a1*1^2*t^4 - 80896*a0*a1^2*1^2*t^4 - 18432*a1^3*1^2*t^4 + 
     192*a0^2*1^3*t^4 + 208*a0^4*1*t^5 + 288*a0^3*a1*1*t^5 + 
     128*a0^2*a1^2*1*t^5 - 46912*a0^3*1^2*t^5 - 14464*a0^2*a1*1^2*t^5 + 
     53248*a0*a1^2*1^2*t^5 + 30720*a1^3*1^2*t^5 + 25472*a0^2*1^3*t^5 + 
     17920*a0*a1*1^3*t^5 + 2048*a1^2*1^3*t^5 - 8672*a0^3*1^2*t^6 - 
     123392*a0^2*a1*1^2*t^6 - 140800*a0*a1^2*1^2*t^6 - 30720*a1^3*1^2*t^6 + 
     54464*a0^2*1^3*t^6 + 279552*a0*a1*1^3*t^6 + 136192*a1^2*1^3*t^6 - 
     768*a0*1^4*t^6 - 46912*a0^3*1^2*t^7 - 14464*a0^2*a1*1^2*t^7 + 
     53248*a0*a1^2*1^2*t^7 + 30720*a1^3*1^2*t^7 + 462848*a0^2*1^3*t^7 + 
     90112*a0*a1*1^3*t^7 - 169984*a1^2*1^3*t^7 - 75776*a0*1^4*t^7 - 
     26624*a1*1^4*t^7 - 17344*a0^3*1^2*t^8 - 85760*a0^2*a1*1^2*t^8 - 
     80896*a0*a1^2*1^2*t^8 - 18432*a1^3*1^2*t^8 - 362368*a0^2*1^3*t^8 + 
     283648*a0*a1*1^3*t^8 + 250880*a1^2*1^3*t^8 - 46592*a0*1^4*t^8 - 
     331776*a1*1^4*t^8 + 1024*1^5*t^8 - 3776*a0^3*1^2*t^9 - 
     3968*a0^2*a1*1^2*t^9 - 1024*a0*a1^2*1^2*t^9 + 993536*a0^2*1^3*t^9 + 
     316416*a0*a1*1^3*t^9 - 327680*a1^2*1^3*t^9 - 1873920*a0*1^4*t^9 - 
     157696*a1*1^4*t^9 + 83968*1^5*t^9 - 16*a0^3*1^2*t^10 - 
     362368*a0^2*1^3*t^10 + 283648*a0*a1*1^3*t^10 + 250880*a1^2*1^3*t^10 + 
     3068672*a0*1^4*t^10 + 360448*a1*1^4*t^10 - 46080*1^5*t^10 + 
     462848*a0^2*1^3*t^11 + 90112*a0*a1*1^3*t^11 - 169984*a1^2*1^3*t^11 - 
     6897664*a0*1^4*t^11 - 1126400*a1*1^4*t^11 + 2691072*1^5*t^11 + 
     54464*a0^2*1^3*t^12 + 279552*a0*a1*1^3*t^12 + 136192*a1^2*1^3*t^12 + 
     6454272*a0*1^4*t^12 + 1679360*a1*1^4*t^12 - 6667264*1^5*t^12 + 
     25472*a0^2*1^3*t^13 + 17920*a0*a1*1^3*t^13 + 2048*a1^2*1^3*t^13 - 
     6897664*a0*1^4*t^13 - 1126400*a1*1^4*t^13 + 14858240*1^5*t^13 + 
     192*a0^2*1^3*t^14 + 3068672*a0*1^4*t^14 + 360448*a1*1^4*t^14 - 
     21844992*1^5*t^14 - 1873920*a0*1^4*t^15 - 157696*a1*1^4*t^15 + 
     24502272*1^5*t^15 - 46592*a0*1^4*t^16 - 331776*a1*1^4*t^16 - 
     21844992*1^5*t^16 - 75776*a0*1^4*t^17 - 26624*a1*1^4*t^17 + 
     14858240*1^5*t^17 - 768*a0*1^4*t^18 - 6667264*1^5*t^18 + 
     2691072*1^5*t^19 - 46080*1^5*t^20 + 83968*1^5*t^21 + 1024*1^5*t^22;
inter := ideal<r|quadric,quintic>;
time RadicalDecomposition(inter);



Here is the output:

[
    Ideal of Polynomial ring of rank 3 over Rational Field
    Order: Lexicographical
    Variables: a0, a1, t
    Inhomogeneous, Dimension 1, Radical, Prime
    Groebner basis:
    [
        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
    ],
    Ideal of Polynomial ring of rank 3 over Rational Field
    Order: Lexicographical
    Variables: a0, a1, t
    Inhomogeneous, Dimension 1, Radical, Prime
    Groebner basis:
    [
        a0 + 2*a1 - 24,
        t - 1
    ],
    Ideal of Polynomial ring of rank 3 over Rational Field
    Order: Lexicographical
    Variables: a0, a1, t
    Homogeneous, Dimension 1, Radical, Prime
    Groebner basis:
    [
        a0 + 2*a1,
        t
    ],
    Ideal of Polynomial ring of rank 3 over Rational Field
    Order: Lexicographical
    Variables: a0, a1, t
    Inhomogeneous, Dimension 1, Radical, Prime
    Groebner basis:
    [
        a0 - 4*t^4 + 8*t^3 - 4*t^2,
        a1 - 2*t^4 - 8*t^3 - 2*t^2
    ],
    Ideal of Polynomial ring of rank 3 over Rational Field
    Order: Lexicographical
    Variables: a0, a1, t
    Inhomogeneous, Dimension 1, Radical, Prime
    Groebner basis:
    [
        a0 - 4*t^4 - 4*t^2,
        a1 - 4*t^4 - 4*t^2
    ]
]
Time: 14.570


Notes:
* The first ideal in the decomposition is the one giving the A_5-family.
* The second and third ideals will not interest us, since they correspond to values of tau where the E_tau is not smooth.
* The fourth ideal corresponds to a family that has an A_3-singularity at (R,(1:1)).  It also appears to have an A_1 singularity at another point.
* The fifth ideal corresponds to a (nonreduced) double curve in the linear system |Lambda|