## Posts tagged ‘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 and a type: this output is an ideal of the ring . 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 defined by the homogeneous ideal V : here `ring V`

denotes the ambient ring of V. We see that is a non-singular quadric in 5-dimensional projective space, and check several well-known facts (more…)

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