Posts filed under ‘Macaulay’

Experimental algebraic geometry I : the grassmannian

I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.

Macaulay provides a command-line interface using the readline library (like many other command-line programs) : here is what input/output looks like

i1 : V = Grassmannian(1,3)
o1 = ideal(p   p    - p   p    + p   p   )
            1,2 0,3    0,2 1,3    0,1 2,3
o1 : Ideal of ZZ[p   , p   , p   , p   , p   , p   ]
                  0,1   0,2   1,2   0,3   1,3   2,3

When I type a command at i1, I get an output o1 with a value \mathrm{ideal}(p_{1,2} p_{0,3} - p_{0,2} p_{1,3} + p_{0,1} p_{2,3}) and a type: this output is an ideal of the ring \mathbb{Z}[p_{0,1}, p_{0,2}, p_{1,2}, p_{0,3}, p_{1,3}, p_{2,3}]. Many features of the Grassmannian as an algebraic variety are available: first define

i1 : V = Grassmannian(1,3,CoefficientRing => QQ);
o1 : Ideal of QQ[p   , p   , p   , p   , p   , p   ]
                  0,1   0,2   1,2   0,3   1,3   2,3
i2 : X = Proj(ring V / V)
o2 = X
o2 : ProjectiveVariety

the projective variety X over \mathbb Q defined by the homogeneous ideal V : here ring V denotes the ambient ring of V. We see that V is a non-singular quadric in 5-dimensional projective space, and check several well-known facts (more…)

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7 February 2009 at 11:17 pm Leave a comment


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