Five Points Determine a Conic Algebraic Geometry $\mathbb{P}^1_k \times \mathbb{P}^1_k \not\cong \mathbb{P}^2_k$ Algebraic Geometry Bézout's Theorem Algebraic Geometry Rational Maps from Smooth Curves Are Morphisms Algebraic Geometry Hilbert's Nullstellensatz Algebraic Geometry Bezout's Theorem Algebraic Geometry Large Degree Implies Condition (*) Algebraic Geometry Genus Does Not Decrease Under Morphisms Algebraic Geometry