--The variety of reducible forms
--given a form F how can we determine whether F is reducible, i.e. it can be written as F=GH?
--consider the case of F(x,y,z) of degree 3 and deg G=1, deg H=2
--the variety of cubics which splits as a deg 1 times a deg 2 form lives in P^9 and it has dimension 7
Powers::=Z/(2003)[x[0..9]];
--we need to perform elimination in
Tot::=Z/(2003)[a[0..2],b[0..5],x[0..9]];
--to construct the parameterization we need the ring
Use S::=Z/(2003)[y,z,t,a[0..2],b[0..5],x[0..9]];
--the general degree 1 form
L:=a[0]y+a[1]z+a[2]t;
--the general degree 2 form
A:=b[0]..b[5];
B:=Gens(Ideal(y,z,t)^2);
Q:=ScalarProduct(A,B);
--the variety of reducible form is the image of the map P^2xP^5 --> P^9 mapping [F,G] |--> [FG]
P:=Q*L;
--now we need the coefficients of P as a polynomial in y,z and t
Use Z/(2003)[y,z,t];
Cubes:=Gens(Ideal(y,z,t)^3);
--we move back to S
Use S;
Cubes:=BringIn(Cubes);
--we make a kist using the coefficients of P
Function:=[];
Foreach T In Cubes Do Append(Function,DerivationAction(T,P)); Endforeach;
--we make ready to perform elimination
Use Tot,
Function:=BringIn(Function);
J:=Ideal([x[I-1]-Function[I] | I In 1..10]);
--now the big elimination, may be too big
K:=Elim(a[0]..b[5],J);
--Ex 1b what is the connection between the variety of reducible quadratic forms and the quadratic Veronese variety?
--Ex 2b can the *generic* cubic form F(x,y,z) be written as F=LMN for some linear forms? how can we check whether a given specific cubic form can be written as F=LMN?
--Ex 3b how can we check whether the *generic* cubic form F(x,y,z) can be written as F=LQ+MQ' for some linear forms L and M? This answer says that "the generic plane cubic contains four points whichc are the complete intersection of to conics". What does this mean? Why is this true?