This is an approximate order of topics that I will cover this semester. It will be updated as we go along. I will also try to include references to free online sources.

• Affine and projective varieties.
  notes by Andreas Gathmann
  https://agag-gathmann.math.rptu.de/class/alggeom-2002/alggeom-2002.pdf pages 8-17, 35-41.
• Hypersurfaces in affine and projective spaces. Bezout's theorem.
  notes by Andries Brouwer
  https://www.win.tue.nl/~aeb/2WF02/bezout.pdf
• Vector bundles, line bundles and maps to projective spaces. Segre and Veronese embeddings.
  notes by Alexey Zinger
  https://www.math.stonybrook.edu/~azinger/mat566-spr18/vectorbundles.pdf
  notes by Paul Hacking
  https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=551f937f989f1b26305974b0ee05018aa2f46bf0
• Divisors and divisor classes.
  blog by Charles Siegel
  https://rigtriv.wordpress.com/2008/04/16/weil-divisors-cartier-divisors-and-more-line-bundles/
• Elliptic curves.
  Book by Neal Koblitz, available at Math library
  Introduction to elliptic curves and modular forms
• Hurwitz formula and Hurwitz automorphism theorem.
  Thesis by Sanne Bosch
  https://studenttheses.uu.nl/bitstream/handle/20.500.12932/1166/The%20Riemann%20Hurwitz%20formula%20final%20(1).pdf?sequence=1
• Blowup construction. Weak Factorization theorem.
  Hartshorne's "Algebraic Geometry" book, pages 28-30. Also: https://en.wikipedia.org/wiki/Birational_geometry
• Action of finite groups on algebraic varieties and smooth manifolds. Clasification of finite subgroups of SL(2,C).
  lecture notes by Bernd Sturmfels
  https://www.imprs-mis.mpg.de/fileadmin/imprs/imprs-ringvorlesung-2018_june-19.pdf
  first few pages of notes of Igor Dolgachev
  https://dept.math.lsa.umich.edu/~idolga/McKaybook.pdf
• ADE singularities and their resolutions.
  blog by Steven Sam
  https://concretenonsense.wordpress.com/2010/02/01/resolving-du-val-singularities/
• Birational geometry, Kodaira dimension.
• Grassmannians.
• Abelian surfaces? K3 surfaces?
• Intersection theory?
• Cohomology, Riemann-Roch formulas?
• Language of schemes?

Projects.

• Modular curves. April 22
• Hodge theory. April 25
• Toric varieties. April 29