I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables.

Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\mathbb{P}^9$, and that $I$ is not radical.

When I ask for the primary decomposition of $I$ Macaulay2 gives me two ideals $I_0,I_1$. He says that $I_0$ is radical and defines an irreducible variety of dimension $5$ and degree $10$, while $I_1$ is not radical and defines a variety of dimension $-1$. Indeed, when I ask for the radical of $I_1$ Macaulay2 gives me the ideal generated by all the homogeneous coordinates of $\mathbb{P}^9$ thus corresponding to the empty set. This last fact is really confusing me.

What is then the meaning of $I_1$ in the primary decomposition of $I$?

Thank you very much in advance.

Here are the ideals I am considering

```
P9 = QQ[a00,a01,a02,a03,a11,a12,a13,a22,a23,a33]
F0 = a00*a12-a01*a02+a01*a13-a03*a11
F1 = a00*a23+a01*a33-a02*a03-a03*a13
F2 = a01*a22-a02*a12+a11*a23-a12*a13
F3 = a02*a23-a03*a22+a12*a33-a13*a23
F4 = a00*a22-a02^2-a11*a33+a13^2
J = ideal(F0,F1,F2,F3,F4)
M = matrix{{a00,a01,a02,a03},{a01,a11,a12,a13},{a02,a12,a22,a23},{a03,a13,a23,a33}}
I = J + minors(3,M)
L = primaryDecomposition(I)
X = variety L_0
Y = variety L_1
dim(X)
dim(Y)
```