From "Introduction to Schemes":
Ch. 11
Definition of the cotangent module/sheaf
Regular schemes
The cotangent sheaf of P^n
Exact sequences and computations involving Omega_X
Normal bundle and the adjunction formula
Ch. 14
Weil divisors
Cartier divisors
Invertible sheaves
How to convert between these types, and when they are equivalent
Computations of Pic, Cl,..
Linear systems
Important examples: Pn, P1xP1, quadratic cone
Ch. 16
Correspondence between sections of invertible sheaves and morphisms to projective space
Automorphisms of Pn
Projective embeddings
Ch. 17
Ample invertible sheaves
Serre's theorems
Ch. 18
p_a,p_g, chi(O_X)
Canonical divisor of a curve
Morphisms and divisors on curves
Hyperelliptic curves
Ch. 19
The Riemann-Roch theorem and Serre duality
Proof of Riemann-Roch assuming Serre duality
Ch. 20
Basepoint freeness and very ampleness criteria
Curves on a quadric surface
Curves of genus 0, 1, 2, 3, 4
From JVRs notes:
Riemann surfaces (naively)
The relation between the topological and algebraic genus
Ramification
The Riemann-Hurwitz formula
Classifying hyperelliptic curves via ramification points
Classifying genus 1 curves via the j-invariant
Classifying genus 2 curves
Definition of a moduli functor
Definition of scheme representing a moduli functor
Examples:
The Grassmannian
The Hilbert scheme
The functor F_g of curves
Algebraic groups, definition and examples (finite groups, GL_n)
Action of an algebraic group on a scheme, definition and examples
Definition of a categorical quotient
The non-representability of F_g
Coarse moduli space, definition
The existence theorem for M_g, without proof