My second attack on   the 2-secant variety of the twisted cubic.

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

 sigma: A^1 times A^1 times Delta -> A^3

(s,u,t1,t2) -> (t1s+t2u,t1s^2+t2u^2,t1s^3+t2u^3)

So, 2-secant variety of twisted cubic= V( I(im sigma))
and

I(im sigma)= (t1+t2-1,y1-s*t1-u*t2,y2-s^2*t1-u^2*t2,y3-s^3*t1-u^3*t2) intersected with k[y1,y2,y3]

We use Macaulay2 to calculate that this intersection is (0). Hence,  

2-secant variety of twisted cubic= V( I(im sigma))=V(0)=A^3.

[kustin@localhost m2-747-class]$ M2
Macaulay 2, version 1.1
with packages: Classic, Core, Elimination, IntegralClosure, LLLBases, Parsing, PrimaryDecomposition, SchurRings, TangentCone

i1 : QQ[s,u,t1,t2,y1,y2,y3,MonomialOrder => {GRevLex=>4, GRevLex=>3}]

o1 = QQ [s, u, t1, t2, y1, y2, y3]

o1 : PolynomialRing

i2 : gens gb ideal{t1+t2-1,y1-s*t1-u*t2,y2-s^2*t1-u^2*t2,y3-s^3*t1-u^3*t2}

o2 = | t1+t2-1 2uy1^5-5uy1^3y2+3uy1y2^2+uy1^2y3-uy2y3+3t2y1^2y2^2-4t2y1^3y3-4t2y2^3+6t2y1y2y3-t2y3^2-y1^4y2+3y1^3y3+2y2^3-5y1y2y3+y3^2
     ----------------------------------------------------------------------------------------------------------------------------------------
     sy1^2-sy2+uy1^2-uy2-y1y2+y3 3t2^2y1^2y2^2-4t2^2y1^3y3-4t2^2y2^3+6t2^2y1y2y3-t2^2y3^2-3t2y1^2y2^2+4t2y1^3y3+4t2y2^3-6t2y1y2y3+t2y3^2+y1^6
     ----------------------------------------------------------------------------------------------------------------------------------------
     -3y1^4y2+3y1^2y2^2-y2^3 2ut2y2^2-2ut2y1y3-2uy1^4+3uy1^2y2-uy2^2-3t2y1y2^2+4t2y1^2y3-t2y2y3+y1^3y2-2y1^2y3+y2y3
     ----------------------------------------------------------------------------------------------------------------------------------------
     ut2y1y2-ut2y3-uy1^3+uy1y2-2t2y2^2+2t2y1y3+y2^2-y1y3 2ut2y1^2-2ut2y2-uy1^2+uy2-t2y1y2+t2y3-y1^3+2y1y2-y3 st2-ut2-s+y1
     ----------------------------------------------------------------------------------------------------------------------------------------
     u2y1^2-u2y2-uy1y2+uy3+y2^2-y1y3 su-sy1-uy1+y2 u2t2-2ut2y1+t2y2+y1^2-y2 |

                                           1                                     11
o2 : Matrix (QQ [s, u, t1, t2, y1, y2, y3])  <--- (QQ [s, u, t1, t2, y1, y2, y3])

 
i4 : leadTerm o2

o4 = | t1 2uy1^5 sy1^2 3t2^2y1^2y2^2 2ut2y2^2 ut2y1y2 2ut2y1^2 st2 u2y1^2 su u2t2 |

                                           1                                     11
o4 : Matrix (QQ [s, u, t1, t2, y1, y2, y3])  <--- (QQ [s, u, t1, t2, y1, y2, y3])

i5 : 

Notice that none of the leading terms involve only y's.