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Lemma 59.67.7. Let $k$ be an algebraically closed field. Let $f_1, \ldots , f_ s \in k[T_1, \ldots , T_ n]$ be homogeneous polynomials of degree $d_1, \ldots , d_ s$ with $d_ i > 0$. If $s < n$, then $f_1 = \ldots = f_ s = 0$ have a common nontrivial solution.

Proof. This follows from dimension theory, for example in the form of Varieties, Lemma 33.34.2 applied $s - 1$ times. $\square$

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