Verify my solution: dim of quotient ring using Jacobian criterion (algebraic geometry)

Question

Compute the (Krull) dimension of $\mathbb C[x_1, \ldots , x_6]/I$ for $I = (x_5 x_5-x_4 x_6,x_4 x_5-x_3 x_6,x_3 x_5-x_2 x_6,x_2 x_5-x_1 x_6,x_4 x_4-x_2 x_6,x_3 x_4-x_1 x_6,x_2 x_4-x_1 x_5,x_3 x_3-x_1 x_5,x_2 x_3-x_1 x_4,x_2 x_2-x_1 x_3)$

In SageMath, I obtained that rank of Jacobian matrix (10 ideals treated as 10 rows of matrix) is 6. Does that mean that dim of that ring is 0 (6-6=0)?

Space of dimensions of (ideal one = 0,..., ideal ten = 0) is 2 dimensional (I checked that in Wolfram) + solution (0,0,0,0,0,0).

Does therefore vanishing locus of $I$ not have regular points (because 6-2=4 is not equal rank of Jacobian matrix)?

I dont know how to forge all facts.

Answer 1

I have not checked your example but your reasoning does look correct. I used M2, Macaulay2, to compute the following example.

Let $P$ be the kernel of the map $\phi \colon k[x,y,z]\to k[t]$, where $\phi(x) = t^3, \phi(y) = t^4, \phi (z) = t^5$ and $k = \mathbb{Q}$. Then $P$ is a height $2$ ideal whose jacobian matrix is a $3$ by $3$ matrix of rank $3$.