Elliptic curves in positive characteristic, supersingular and ordinary curves, Weil conjecture for elliptic curves (or a subset of these topics). Reference: [15], V.2, V.3, possibly V.4.
Toric varieties: definition, classification in terms of fans, example of a toric complete nonprojective threefold. References: [7], [10] , [3].
Newton polytopes and Bernstein-Khovanski-Kushnirenko bound for the number of solutions of a system of "sparse" polynomial equations. Reference: [6].
Hurwitz Theorem bounding number of automorphisms of an algebraic curve, examples of groups realizing the bound (Hurwitz groups), the Klein quartic. Reference: [12], p. 337, 348; [1], pp. 44-47.
Cremona group of birational automorphisms of the plane: examples of finite subgroups, some results on classification of conjugacy classes. Reference: [8].
Some results and example on the construction of Hilbert schemes parametrizing subvarieties in a projective space. Reference: [14].
Rational double point surface singularities and finite subgroups in SL(2), classification in terms of simply-laced Dynkin graphs (An, Dn, E6, E7, E8). Refrences: [2], [5], [9].
28 bitangents to a quartic plane curve and theta characteristics (square roots of the canonical line bundle). References: [11], [8].
Flag varieties of classical simple algebraic groups and their basic properties, Bruhat decomposition, count of points over a finite field. Reference: [4].
Examples of noncatenary commutative rings. Reference: [13].
References
[1] Arbarello, Cornalba, Gri