The Stacks project

Exercise 111.57.3. Let $X = \mathbf{A}^2_\mathbf {C}$ where $\mathbf{C}$ is the field of complex numbers. A line will mean a closed subscheme of $X$ defined by one linear equation $ax + by + c = 0$ for some $a, b, c \in \mathbf{C}$ with $(a, b) \not= (0, 0)$. A curve will mean an irreducible (so nonempty) closed subscheme $C \subset X$ of dimension $1$. A quadric will mean a curve defined by one quadratic equation $ax^2 + bxy + cy^2 + dx + ey + f = 0$ for some $a, b, c, d, e, f \in \mathbf{C}$ and $(a, b, c) \not= (0, 0, 0)$.

1. Find a curve $C$ such that every line has nonempty intersection with $C$.

2. Find a curve $C$ such that every line and every quadric has nonempty intersection with $C$.

3. Show that for every curve $C$ there exists another curve such that $C \cap C' = \emptyset $.