If the chosen line l does not intersect the singular locus of the complete intersection Q, the the normal sheaf Nl|Q is locally free, hence isomorphic to a line bundle on ℙ1. Note that if h1(Nl|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 ]), free
0 1
|
i6 : HH^1(normalBundle) o6 = 0 o6 : kk-module |
We do not check whether the line intersects the singular locus or not.