Fibers of a map

[Sara Muli]

Dear all, can you help me?

I need to compute fibers of a given map

f=map(ImS,P3,{L1,L2,L3,L4})

P9=ZZ/32003[x_0..x_9]

P3=ZZ/32003[x_0,x_1,x_2,x_3]

ImS=P9/Is

L1,L2,L3,L4 are linear forms.

Thank you in advance!

[Daniel R. Grayson]

Do you mean the fibers of f or the fibers of Spec f?  If the latter, perhaps you can use the

function here: https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.16/share/doc/Macaulay2/Macaulay2Doc/html/_preimage.html

[Sara Muli]

Dear Profesor

I meant fibers of f.

Thank you very much!

[Daniel R. Grayson]

The fibers of a ring homomorphism are not usually considered, but they are the cosets of the kernel of f, and the kernel of f can be computed with the function "kernel".

[KSchwede]

What do you mean by *compute* fibers of f?  Do you want a specific fiber?  A random fiber over the base?  That has infinitely many geometric points, do you just want fibers over rational points?  (In which case, for a smaller prime, you could write them all).  To compute a particular fiber, you can simply take an ideal on your base defining a point, and extend it to your total space, then mod out by the result.  In your case, I assume that you will take an ideal in P^3 defining a point and apply f to it.  In other words, f(Ideal) is an Ideal.  I'd be tempted to work on a chart of the base, as I tend to confuse myself about how having things homogeneous can mess things up.  Of course, every point on the base has an affine neighborhood...