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Given a curve in projective $n$-space over an algebraically closed field and a homogeneous polynomial of degree $\geq 1$. Does it then always have a zero on the curve?

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Let $C$ be the given curve, and $f$ the homogeneous polynomial, and consider the hypersurface $V = \{f = 0\}$. The general theorem about intersections of projective varieties is Bezout's theorem. But there are weaker statements that suffice in this situation. The following can be found in Hartshorne, Algebraic geometry, Theorem I 7.2:

Theorem Let $Y, Z$ be varieties of dimensions $r,s$ in $\mathbb P^n$. If $r + s - n \geq 0$, then the intersection $Y \cap Z$ is nonempty.

In our situation, $r = \dim C = 1$ and $s = \dim V = n-1$, so $r + s - n = 0$, and hence $C \cap V$ is nonempty.

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