The exact presentation gives a direct product decomposition
O_sing ≅ Z_3 × R_pair,
R_pair = Z_3[eta]/(eta^2 - 3 eta).Therefore every ideal of O_sing splits uniquely as
I = 3^a Z_3 × I_pair,where a >= 0 and I_pair is an ideal of the pair factor.
So the full singular ideal theory is exactly the product of:
Z_3 ⊃ 3Z_3 ⊃ 3^2Z_3 ⊃ ... on the V7 branch,For the finite quotients
O_k = O_sing / 3^k O_sing,the number of ideals is therefore
#Ideals(O_k) = (k+1) * ((k+1) + 4(k-1)) = (k+1)(5k-3).| quotient | ideals on V7 branch |
ideals on pair factor | total ideals |
|---|---|---|---|
O/3^1O |
2 |
2 |
4 |
O/3^2O |
3 |
7 |
21 |
O/3^3O |
4 |
12 |
48 |
O/3^4O |
5 |
17 |
85 |
O/3^5O |
6 |
22 |
132 |
O/3^6O |
7 |
27 |
189 |
So the full singular node is a Cartesian product of a chain and a projective-line bouquet. The split V7 branch contributes only linear depth; all branching happens inside the glued V1/V5 pair factor.