Elliptic curves in positive characteristic, supersingular and ordinary curves, Weil conjecture for elliptic curves (or a subset of these topics). Reference: , V.2, V.3, possibly V.4.
Toric varieties: definition, classification in terms of fans, example of a toric complete nonprojective threefold. References: ,  , .
Newton polytopes and Bernstein-Khovanski-Kushnirenko bound for the number of solutions of a system of "sparse" polynomial equations. Reference: .
Hurwitz Theorem bounding number of automorphisms of an algebraic curve, examples of groups realizing the bound (Hurwitz groups), the Klein quartic. Reference: , p. 337, 348; , pp. 44-47.
Cremona group of birational automorphisms of the plane: examples of finite subgroups, some results on classification of conjugacy classes. Reference: .
Some results and example on the construction of Hilbert schemes parametrizing subvarieties in a projective space. Reference: .
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: , , .
28 bitangents to a quartic plane curve and theta characteristics (square roots of the canonical line bundle). References: , .
Flag varieties of classical simple algebraic groups and their basic properties, Bruhat decomposition, count of points over a finite field. Reference: .
Examples of noncatenary commutative rings. Reference: .
 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.
 Artin, On Isolated Rational Singularities of Surfaces. American Journal of Mathematics 88 (1966), 129136.
 Bonavero, Lectures on Toric Varieties, Lecture 14.
 Brion, Lectures on the Geometry of Flag Varieties, available at arxiv.org.
 Burban, Du Val Singularities (PDF).
 Cox, Little, O'Shea, Using Algebraic Geometry, Graduate Texts in Mathematics, 185. Springer, New York, 2005.
 Danilov, The Geometry of Toric Varieties. Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85-134, 247.
 Dolgachev, Classical Algebraic Geometry: A Modern View. Cambridge University Press, 2012.
 Durfee, Fifteen Characterizations of Rational Double Points and Simple Critical Points. L'Enseignement Mathematique. Revue Internationale. 25 (1979), 131163.
 Fulton, Introduction to Toric Varieties. Annals of Mathematics Studies, 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton, NJ, 1993.
 Griffiths, Harris, Principles of Algebraic Geometry. Wiley-Interscience, 1994.
 Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics. Springer, 1997. ISBN: 9780387902449. [Preview with Google Books]
 Heitmann, Examples of Noncatenary Rings. Trans. AMS, 247 (1979), 125-136.
 Nitsure, Construction of Hilbert and Quot Schemes, available at arxiv.org.
 Silverman, Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. Springer, Dordrecht, 2009.