Fans, the orbit-cone correspondence, and toric resolution of singularities.
The fan dictionary, the construction, the orbit-cone correspondence, smoothness and resolution, and a table of which fan supplies which counterexample.
Affine toric computations, dualising a cone by hand, the equations of \(X_\sigma\), the cone over the rational normal curve, and minimal resolution by convex hull.
Polytopes and divisors, the normal fan, \(\Cl\) and \(\Pic\) from the rays, the polytope \(P_D\) and its lattice points as sections, ampleness, and the reflexive polytopes behind the toric Fanos.
Toric surfaces and toric morphisms, the completeness criterion, the classification of smooth complete toric surfaces, intersection numbers, blowups as star subdivisions, and weighted projective space.
Criteria and numerical invariants, one table of the fan conditions for smooth, complete, projective and Fano, the Betti numbers from the cone counts, and the degree as a normalised volume.