Franz Lemmermeyer
# Algebraic Geometry

### Review Midterm 1

Here's what we covered so far:
- 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.