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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