Question about irreducible components of affine varieties defined by homogeneous polynomials

Question

Suppose I have $F_1, \ldots, F_r \in \mathbb{C}[x_1, \ldots, x_n]$ homogeneous polynomials. Consider their zero set $V(F_1, ..., F_r)$ in $\mathbb{C}^n$ and let $Z_1, ..., Z_t$ be the irreducible components. How do I know that each $Z_i$ contains $(0,...,0)$? Thank you very much!

Possible duplicate of http://math.stackexchange.com/questions/524396/do-homogeneous-ideals-have-homogeneous-primary-decomposition — MooS, Apr 03 '17 at 18:11

score 1 · Answer 1 · answered Apr 03 '17 at 18:15

1

Homogenous ideals have a homogenous primary decomposition, see here (Theorem 10). Your desired statement is just the geometric version of this.

answered Apr 03 '17 at 18:15

MooS

32,317

score 1 · Accepted Answer · answered Apr 03 '17 at 18:15

1

Let $Z$ be an irreducible component and let $p=(a_1,\ldots, a_n)$ be a point on $Z$ not in any other component. If it is the origin, you have nothing to show. For any $\lambda\in\mathbb{C}$, $(\lambda a_1,\ldots, \lambda a_n)$ belongs to $V(F_i)$, since these are homogeneous. That is, you have a morphism $\mathbb{C}\to V(F_i)$ and since the former is irreducible, the image is contained in an irreducible component and now it is clear the image is contained in $Z$ and thus the origin belongs to $Z$.

answered Apr 03 '17 at 18:15

Mohan

18,858

Are you considering only one $V(F_i)$ or are you doing this for all of them? – Johnny T. Apr 04 '17 at 01:42
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All of them, I meant $V(F_1,\ldots,F_r)$. – Mohan Apr 04 '17 at 02:20

Question about irreducible components of affine varieties defined by homogeneous polynomials

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