Prove that the intersection of $m\leq n$ projective varieties in $\mathbb{P}^n$ is never empty.

Question

This result is deemed trivial by my instructor but I am having troubles understanding it.

Equivalently we are claiming that the zero locus of $m\leq n$ homogeneous polynomials $f_1,\dots,f_m$ is nonempty in the projective space $\mathbb{P}^n$. I would reason trying to move the problem to the afffine $\mathbb{A}^{n+1}$ case. Being the polynomials homogeneous the affine zero locus $V_{\mathbb{A}^{n+1}}(f_1,...,f_m)$ surely contains the zero point. I think that if I am able to show it contains a nonzero point $p$ too, then it contains the whole line passing through zero and $p$,then I am done passing to the projective closure.

Am I correct?
Can someone help me?

Thanks!

Answer 1

Here's my comment developed in to an answer.

Hint: consider the dimension of $V_{\Bbb A^{n+1}}(f_1,\cdots,f_m)$.

Full solution under the spoiler text.

By the affine dimension theorem (Hartshorne I.7.1, for instance), the codimension of $V_{\Bbb A^{n+1}}(f_1,\cdots,f_m)$ is at most $m$, which means it has dimension at least $n+1-m$, which means that it is of dimension at least one. Since the defining polynomials are homogeneous, this means that it must contain a line through the origin, and therefore you are done. (For a proof of the dimension theorem if you don't have Hartshorne handy, check out Krull's Hauptidealsatz, which once translated in to geometry also gives you the result.)