Algebraic Geometry

Review Midterm 1

• Parametrization of conics using sweeping lines; parametrization of certain singular curves.
• Group law on nonsingular conics.
• Mason's Theorem; application to nonexistence of parametrizations.
• Affine and projective spaces, projective plane as affine plane plus points at infinity; homogenization; lifting affine rational maps to projective polynomial maps; projective closure of curves.
• Tangents, singular points, multiplicity; Euler's identity.
• Study's Lemma.
• Lines intersect curves of degree m in exactly m points; applications: singular conics are degenerate; irreducible singular cubics have exactly one singular point.
• Projective transformations preserve degree, singular points, multiplicities; all nondegenerate conics over an algebraically closed field are projectively equivalent.