normalBundleLineInQ -- compute the normal bundle of a line in Q

Synopsis

Usage:

normalBundleLineInQ(subsLine,relPfaf)
Inputs:
- subsLine, a matrix, row matrix containing the variables of the normalized set-up with the a-variables being replaced by the assignments from the chosen line
- relPfaf, a matrix, row matrix containing the quadratic (Pfaffian) relations
Outputs:
- normalBundle, a coherent sheaf,

Description

If the chosen line $l$ does not intersect the singular locus of the complete intersection $Q$, the the normal sheaf ${\cal N}_{l|Q}$ is locally free, hence isomorphic to a line bundle on $\mathbb{P}^1$. Note that if $h^1({\cal N}_{l|Q}) = 0$, then the Fano scheme of lines in $Q$ is smooth at the corresponding point.

i1 : kk = ZZ/nextPrime(32001);

i2 : s = "1111";

i3 : (relLin,relPfaf,d1',d2,Ms) = setupGodeaux(kk,s);

i4 : (randLine,subsLine) = randomLineTorsZ5(d1',relPfaf);

i5 : normalBundle = normalBundleLineInQ(subsLine,relPfaf)

                       2                         4
o5 = OO                 (1) ++ OO
       Proj(kk[x ..x ])          Proj(kk[x ..x ])
                0   1                     0   1

o5 : coherent sheaf on Proj(kk[x ..x ])
                                0   1

i6 : HH^1(normalBundle)

o6 = 0

o6 : kk-module

Caveat

We do not check whether the line intersects the singular locus or not.

Ways to use `normalBundleLineInQ` :

"normalBundleLineInQ(Matrix,Matrix)"

For the programmer