Here is the list of topics for the presentation. In place of a final exam, you should record a 10 minute presentation on a topic of your choosing (with suggestions below). The presentation is due on the last day of the finals period (April 24). I recommend a Zoom call with yourself as an easy way to record.
A lot of these are pretty advanced, so you can do a very broad overview, or specialize to something very specific (one theorem or example). Also, there are many places in Vakil where more details are needed, so you can always do a presentation filling in one of the many points left as exercises there. You’re not required to do one on the list; in fact, I encourage you to think of your own.
- (Kähler) differentials (Vakil, Chapter 21)
- Blow ups (Vakil, Chapter 22)
- Resolution of singularities (Vakil, 22.4.6)
- Resolution of singularities and birational classification for surfaces (Vakil 28.6, Hartshorne V.5-6)
- Hilbert syzygy theorem (Vakil 24.4)
- Cohen-Macaulay schemes (Vakil 26.1-2)
- Serre's criterion for normality (Vakil 26.3)
- Stein factorization (Vakil 28.4)
- Proof of Serre duality (Vakil 29)
- Calabi-Yau 3-folds (The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning)
- Elliptic curves
- Weil conjectures and étale cohomology (Hartshorne, Appendix C, Milne: Lectures on Etale Cohomology)
- Bézout's theorem (Hartshorne I.7, Eisenbud-Harris III.3.5)
- Hilbert schemes (Eisenbud-Harris VI.2.3)
- Diophantine geometry (Diophantine Geometry: An Introduction)
- Jacobians of curves (Part A; yes, go ahead and read it)
- Algebraic groups (Milne: Algebraic Groups)
- Intersection theory and Chow rings (Intersection Theory)
- Normal cones (Chapter 4-5)
- The Grothendieck-Riemann-Roch Theorem (Chapter 15)
- Local complete intersections and Gorenstein schemes (Hyman Bass and Ubiquity: Gorenstein Rings)
- Geometric Invariant Theory (A Brief Introduction to Geometric Invariant Theory)
- Algebraic stacks (Introduction to Algebraic Stacks)