Exercise 111.39.3. Let $k$ be a field. Let $Z_1, Z_2 \subset \mathbf{P}^2_ k$ be irreducible closed subschemes of the form $V_{+}(F)$ for some homogeneous irreducible $F_ i \in k[X_0, X_1, X_2]$ of degree $> 0$. Show that $Z_1 \cap Z_2$ is not empty. (Hint: Use dimension theory to estimate the dimension of the local ring of $k[X_0, X_1, X_2]/(F_1, F_2)$ at $0$.)
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