My Macaulay2 calculation for the 2-secant variety of the twisted cubic.

The 2-secant variety of the twisted cubic is the image of the morphism 

 sigma: T times T times Delta -> A^3

where T is the twisted cubic (sitting in A^3), Delta is {(t1,t2)| t1+t2=1}
and sigma sends (w,z,t1,t2) to t1w+t2z.

Let Gamma_{sigma} be the graph of sigma.

We know I(Gamma_{sigma}).

We calculate the intersection of I(Gamma_{sigma}) and k[x1,x2,x3].

This intersection turns out to be zero. We conclude that the closure of the 
2-secant variety of the twisted cubic is all of A^3.

Here are my computer calculations:

i20 : QQ[t1,t2,w1,w2,w3,z1,z2,z3,x1,x2,x3,MonomialOrder => {GRevLex=>8, GRevLex=>3}]

o20 = QQ [t1, t2, w1, w2, w3, z1, z2, z3, x1, x2, x3]

i22 : gens  gb ideal{z3-z1*z2,z2-z1^2,w3-w1*w2,w2-w1^2,t1+t2-1,x1-t1*z1-t2*w1,x2-t1*z2-t2*w2,x3-t1*z3-t2*w3}

The output is fairly messy. It is a row vector of 46 ugly polynomials. I want to see that none of these polynomials is in QQ[x1,x2,x3]. It would suffice to see that none of the leading terms is in QQ[x1,x2,x3]. Alas, the output is so messy that it is hard to see where one polynomial stops and the next one starts.

I can look at them one at at time using o22_0, ..., o22_45, for example,

i25 : o22_0

o25 = | z2x1^3x3+z2x2^3-3z2x1x2x3+z2x3^2-z3x1^3x2+z3x1x2^2+z3x1^2x3-z3x2x3-x2^4+2x1x2^2x3-x1^2x3^2 |

but that is boring!

I could transpose o22 and turn this matrix into a column vector

i35 : transpose o22

o35 = {-5} | z2x1^3x3+z2x2^3-3z2x1x2x3+z2x3^2-z3x1^3x2+z3x1x2^2+z3x1^2x3-z3x2x3-x2^4+2x1x2^2x3-x1^2x3^2                                      
      {-3} | z1x2^2-z1x1x3-z2x1x2+z2x3+z3x1^2-z3x2  

etc. But even this output is hideous. It does not really fit on one screen from left to right.

I then used the command leadTerm to pick out the leading terms


i42 : leadTerm o22

o42 = | z2x1^3x3 z1x2^2 z1x1x2 z1x1^2x3 w3x1^6 w2x2^2 w2x1^3 w1x2^2 w1x1x2 w1x1^2 3t2x1^2x2^2 t1 z3^2x1^6 z2z3x2^2 z2z3x1^3 z1z3x1^2 w3z3
      ---------------------------------------------------------------------------------------------------------------------------------------
      w2z3 w1z3 2t2z3x2^2 t2z3x1x2 2t2z3x1^2 z2^2 z1z2 w3z2 w2z2 w1z2 3t2z2x3 3t2z2x2 3t2z2x1 z1^2 w3z1 w2z1 w1z1 3t2z1x3 3t2z1x2 2t2z1x1
      ---------------------------------------------------------------------------------------------------------------------------------------
      t2w3 w2^2 w1w2 t2w2 w1^2 t2w1 t2z3^2 3t2z2z3 3t2z1z3 |

                                                                  
o42 : Matrix (QQ [t1, t2, w1, w2, w3, z1, z2, z3, x1, x2, x3])^1  <--- (QQ [t1, t2, w1, w2, w3, z1, z2, z3, x1, x2, x3])^46

Now I can see that the intersection of  I(Gamma_{sigma}) and QQ[x1,x2,x3] is zero.