efficient way to compute anti-canonical bundle

[Chris Dare]

Happy holidays all —

I have a fairly simple problem that I was hoping Macaulay2 would be able to compute: I want to compute the anti-canonical bundle H = -KX  of a (2, 3)-complete intersection in P^5. My setup is as follows:

i1 : R = QQ[x_0..x_5]

o1 = R

o1 : PolynomialRing

i2 : f = random(2, R)

o2 : ...

o2 : R

i3 : g = random(3, R)

o3 : ...

o3 : R

i4 : I = ideal(f, g)

o4 : ...

o4 : Ideal of Q5

i5 : loadPackage "TorAlgebra"

o5 = TorAlgebra

o5 : Package

i6 : isCI(I)

o6 = true

i7 : X = Proj(Q5/I)

o7 = X

o7 : ProjectiveVariety

After ensuring this is indeed non-singular (using the idea Xiaowen kindly gave in my previous post), this (basically every time) yields a smooth prime Fano threefold of degree 6. In particular, Pic(X) should be generated by the anticanonical bundle H, so I would like to explicitly compute this H to do some homological computations (e.g. what is RHom(O, O(H))?  ). I first compute

i8 : omega = cotangentSheaf(3, X)

o8 : ...

                                                                                                             455
o8 : coherent sheaf on X, quotient of OO    (-6)
                                          X

This seems like a pretty huge rank to actually compute things over, so it doesnt come as much a surprise that when i try:

i9 : H = dual omega

my Mac runs for hours without producing any result. Is there a more efficient approach than what I'm doing?

Thanks in advance for any help.

— Best,

Chris Dare

[Devlin Mallory]

Hello Chris,

The “Divisor” package does a great job of this:

loadPackage(“Divisor”)

KX=canonicalDivisor(R/I,IsGraded=>true)

H=sheaf OO(-KX)

You can then compute the various cohomologies you want to.

Best,

Devlin 

PS: In your case, you know already from the adjunction formula that the anticanonical bundle will be the hyperplane section, so you can also do the following:

use R/I

H = dual sheaf module ideal(x_0)

(You can use any linear form in place of x_0.)  

In general, for a non-complete-intersection, where adjunction doesn’t tell you what the anti canonical is immediately, the Divisor package approach above will work as long as the computation doesn’t time out. 

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