Steiner Roman surface (not a cone); pi = 0 (plane section: quartic with 3 nodes)
   F = -w*x*y*z + x**2*y**2 + x**2*z**2 + y**2*z**2
   d = 4, lines = 3, F in I(l)^2 for all l: True, irreducible over Q: True, smooth rational point: (-1, -1, -1, -3)
   alpha(I(L)) = 2 <= d-2 = 2;  dim I(L)_(d-2) = 3 >= d-1+pi = 3: True  (equality)

cone over a plane quintic with 6 nodes; pi = 0
   F = -25011*x**3*y**2 + 72113*x**3*y*z - 51876*x**3*z**2 - 6804*x**2*y**3 - 6602*x**2*y**2*z + 24008*x**2*y*z**2 - 1044*x**2*z**3 - 2896*x*y**3*z - 1897*x*y**2*z**2 - x*y*z**3 - 20*y**3*z**2 + 30*y**2*z**3
   d = 5, lines = 6, F in I(l)^2 for all l: True, irreducible over Q: True, smooth rational point: (0, -3, -2, -3)
   alpha(I(L)) = 3 <= d-2 = 3;  dim I(L)_(d-2) = 4 >= d-1+pi = 4: True  (equality)

ruled quartic of bidegree (2,2) with two skew double lines; pi = 1 (quartic with 2 nodes)
   F = w**2*x*y - 3*w**2*y**2 + w*x**2*z + 5*w*x*y*z + 2*x**2*z**2 + y**2*z**2
   d = 4, lines = 2, F in I(l)^2 for all l: True, irreducible over Q: True, smooth rational point: (-3, -1, 0, -3)
   alpha(I(L)) = 2 <= d-2 = 2;  dim I(L)_(d-2) = 4 >= d-1+pi = 4: True  (equality)

quartic x^2y^2 = z^3 w with two cuspidal double lines; pi = 1 (quartic with 2 cusps)
   F = -w*z**3 + x**2*y**2
   d = 4, lines = 2, F in I(l)^2 for all l: True, irreducible over Q: True, smooth rational point: (-3, -3, -3, -3)
   alpha(I(L)) = 1 <= d-2 = 2;  dim I(L)_(d-2) = 5 >= d-1+pi = 4: True

Enriques sextic, double along the 6 edges of the tetrahedron; pi = 4 (sextic with 6 nodes)
   F = w**3*x*y*z + w**2*x**2*y**2 + w**2*x**2*z**2 + w**2*y**2*z**2 + w*x**3*y*z - 8*w*x**2*y**2*z + w*x*y**3*z + w*x*y*z**3 + x**2*y**2*z**2
   d = 6, lines = 6, F in I(l)^2 for all l: True, irreducible over Q: True, smooth rational point: (-3, -3, -3, -3)
   alpha(I(L)) = 3 <= d-2 = 4;  dim I(L)_(d-2) = 13 >= d-1+pi = 9: True

sextic of bidegree (3,3) with two skew triple lines; pi = 4 (sextic with 2 triple points)
   F = -2*w**3*x*y**2 + w**3*y**3 + w**2*x**3*z + w**2*x**2*y*z + 5*w*x*y**2*z**2 + x**3*z**3 + x**2*y*z**3 - 3*y**3*z**3
   d = 6, lines = 2, F in I(l)^2 for all l: True, irreducible over Q: True, smooth rational point: (-3, 0, 0, -3)
   alpha(I(L)) = 2 <= d-2 = 4;  dim I(L)_(d-2) = 25 >= d-1+pi = 9: True

RESULT: all examples are absolutely irreducible, singular along L, and consistent with Lemma A
nice -n 10 $S/venv/bin/python lemmaA_examples.py  1.00s user 0.04s system 99% cpu 1.038 total
