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, Griths, Harris, Geometry of Algebraic Curves, Vol. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267. Springer-Verlag, New York, 1985.
[2] Artin, On Isolated Rational Singularities of Surfaces. American Journal of Mathematics 88 (1966), 129136.
[3] Bonavero, Lectures on Toric Varieties, Lecture 14.
[4] Brion, Lectures on the Geometry of Flag Varieties, available at arxiv.org.
[5] Burban, 
Du Val Singularities (PDF).
[6] Cox, Little, O'Shea, Using Algebraic Geometry, Graduate Texts in Mathematics, 185. Springer, New York, 2005.
[7] Danilov, The Geometry of Toric Varieties. Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85-134, 247.
[8] Dolgachev, Classical Algebraic Geometry: A Modern View. Cambridge University Press, 2012.
[9] Durfee, Fifteen Characterizations of Rational Double Points and Simple Critical Points. L'Enseignement Mathematique. Revue Internationale. 25 (1979), 131163.
[10] Fulton, Introduction to Toric Varieties. Annals of Mathematics Studies, 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton, NJ, 1993.
[11] Griffiths, Harris, Principles of Algebraic Geometry. Wiley-Interscience, 1994.
[12] Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics. Springer, 1997. ISBN: 9780387902449. [Preview with Google Books]
[13] Heitmann, Examples of Noncatenary Rings. Trans. AMS, 247 (1979), 125-136.
[14] Nitsure, Construction of Hilbert and Quot Schemes, available at arxiv.org.
[15] Silverman, Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. Springer, Dordrecht, 2009.