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)

```


A question on a Macaulay2 computation
Commutative algebra is NOT the same as algebraic geometry, especially projective algebraic geometry.


The variety in $\mathbb{P}^9$ defined by $I$ and the variety in $\mathbb{P}^9$ defined by $I\_0$ are the same variety.


If you were to work in $\mathbb{A}^{10}$, then $I$ and $I\_0$ would define different affine schemes; the scheme defined by $I$ has some extra fatness at the origin, but since the origin is not part of projective space, you don't see this difference in $\mathbb{P}^9$.


EDIT: The relevant bit of Hartshorne is Exercise II.5.10


