I want to prove the following:
Let $H_1,\dots,H_n$ be $n$ hyperplanes in $\mathbb{P}^n =\mathbb{P}^n \mathbb{C}$. Then $\cap_{i=1}^n H_i$ is not empty.
Please be noted that this is an exercise from William Fulton's Algebraic Curves, the section of projective Nullstensatz, Q4.12.
I know this statement is equivalent to the following:
Let $I$ be a homogeneous ideal in $\mathbb{C}[X_1,...,X_n]$. Let $M$ be the ideal generated by $X_1,...,X_n$. Then if $I$ contains $M^k$ for some $k$, $I$ can not be generated by less than $n$ forms of degree $1$.
I have tried to use induction on $n$ to prove the question, the case $n=0,1,2$ is obvious.
Assume the result for $n=l$. For $n=l+1$:
If $I$ is generated by $f_1,...,f_l$, then by the induction hypothesis, we may assume $X_1^k$ and $X_2^k$ cannot be generated by $f_1,...,f_{l-1}$ for all $k$, but I could not proceed to find a contradiction.
Any help will be appreciated.
Let me answer your algebraic reformulation of the question.
Since $I$ contains a power of $M$ we have $\sqrt I=M$. (In other words, $M$ is the only minimal ideal over $I$.) This shows $\operatorname{height}I=\operatorname{height}M=n$. If $I$ can be generated by $m<n$ elements then, by Krull Intersection Theorem, we get $n=\operatorname{height}I\le m$, a contradiction.