The output of this script is a list of all configuration types for which the dimension of the tangent space determined by the given configuration of r points is smaller than for r generic points; the (affine) dimension of the tangent space for each deficient configuration type is also given. This script works only if 1 <= r <= 8. In this case, there are no deficient configuration types when d > 15, so d is restricted to being at most 15.
• S = k[P2] = S0 ⊕ S1 ⊕ … ⊕ Si &oplus …
• Φr,d : (S1)r → Sd where Φr,d(l1,…,lr) = l1d + … + lrd
• P(Im(Φ)) = Secr(Vd), where Vd = d-uple image of P2
• Im(dΦ(l1,…,lr)) = l1d-1S1 + … + lrd-1S1 ≅ (S/(I(p1)2∩…∩ I(pr)2))d
where pi is the point dual to the form li
Points (l1,…,lr) ∈ ((P2)∧)r for which
dim Im(dΦ(l1,…,lr)) fails to achieve its maximum value we will call deficient.
Definition: Z = m1p1+…+mrpr ⊂ P2 denotes the subscheme, called a fat point subscheme, defined by the ideal I(p1)m1∩…∩ I(pr)mr ⊆ S.
Definition: For Z = m1p1+…+mrpr, the Hilbert function H(Z, •) of Z is H(Z, d) = dim (S/(I(p1)m1∩…∩ I(pr)mr))d
Definition: (p1,…,pr) ∈ P2 and (p'1,…,p'r) ∈ P2 have the same configuration type if