Tags: #algebraic_geometry #learning #todo

References

Texts

Commutative Algebra

• Eisenbud, Commutative Algebra with a view toward algebraic geometry
• Atiyah-MacDonald, Commutative Algebra

AG

• Vakil, Ths Rising Sea
• Hartshorne, Algebraic Geometry
• Harris, Algebraic Geometry: A First Course
• Shafarevich, Basic Algebraic Geometry 1
• Mumford, The Red Book of Varieties and Schemes
• Fulton, Introduction to Toric Varieties
• Milne, Algebraic Geometry Notes

Notes

• Quick Review
• Short overview of a course
• Harvard 233A 2017 Course notes
• Olsson course notes
• UW Spring 2016 Course
• MIT OCW 2009, Kedlaya
• MIT OCW 2015
• Tripos Part III 2019 Course notes

Topics

• A Tripos Part 3 review: https://www.dpmms.cam.ac.uk/~pmhw/PtIIIAG2014.pdf

• DZB’s recommendation for Hartshorne:

  • 2.1 — 2.8
  • 3.1 — 3.5 and 3.9
  • 4.1 — 4.3
  • The statement of Serre duality from 3.6

• The Hartshorne speedrun, c/o Nat:

  • Hartshorne II. 1-5
  • Hartshorne II. 6-9
  • Hartshorne III. 1-7
  • Hartshorne III. 8-10
  • Hartshorne III. 11-12
  • Hartshorne IV. 1-4
  • Hartshorne IV. 5-6

Scheme and Varieties

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• Affine and projective varieties; regular functions and maps; cones and projections

• Projective space and Grassmannian

• Ideals of varieties; the Nullstellensatz

• Rational functions, rational maps and blowing up

• Dimension and degree of a variety; the Hilbert function and Hilbert polynomial

• Smooth and singular points of varieties; the Zariski tangent space; tangent cones; dual varieties

• Families of varieties (Chow varieties and Hilbert schemes)

• algebraic curves: genus; the genus formula for plane curves,

• the Riemann-Hurwitz formula. Riemann-Roch theorem.

Cohomology of Schemes

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Toric Varieties

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Curves

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Problems

AG

• What is the blowup of a map of schemes?
• What is Hurwitz’s theorem?
• What is a generic point? A special point?
  • What is a closed point?
  • What is a rational point?
  • What are the singular points of the scheme?
• Types of schemes
  • What is a normal scheme? Normalization?
  • What is a reduced scheme? Irreducible?
  • What is an integral scheme?
• Types of morphisms
  • What is a closed immersion? A closed embedding?
  • What is a separated morphism?
  • What is a proper morphism? Projective?
  • What is a projective morphism? Affine?
  • What is a flat morphism?
  • What is an etale morphism?
  • What is a projective morphism?
  • What is a finite morphism? Finite type?
  • What is a rational morphism?
  • What is a quasicompact scheme?
  • What is a ramified morphism?
  • What is the derivative of a morphism?
  • What is a birational map?
  • What is a ramified morphism?
    • What is an example of an unramified morphism?
  • What is qcqs? (Quasi-compact and quasi-separated)
• What is an O_X module?
  • What is a sheaf of O_x modules?
  • What is a quasicoherent O_x module?
  • What is a locally free O_X module?
  • What does it mean for an O_X module to be globally generated?
  • What is the Euler characteristic of an O_x modules? The Hilbert polynomial?
• What is the tangent space of a scheme?
• What is the Jacobi criterion?
• What is the ideal sheaf?
• What is the pullback/pushforward of a sheaf?
• What is the Leray acyclicity theorem?
• What is Grothendieck vanishing?
• What is Serre’s projection formula?
• What is Serre vanishing?
• What is the higher pushforward?
• What is the cohomology of O_X for X = P^n
• What is Serre’s finiteness theorem?
• What is GAGA?
• What is the canonical sheaf?
  • When is the canonical sheaf dualizing?
• What is a complete local intersection?
• What is the adjunction formula?
• What is Serre duality?
  • What is the dualizing sheaf?
• Divisors:
  • What does it mean for a divisor to be ample? Very ample? Effective? Principal?
  • What is a Cartier divisor? A Weil divisor?
  • What is the divisor class group?
  • What is the invertible sheaf associated to a divisor?
  • What is the Picard group?
• What is the Riemann-Roch theorem?
• What is Chevalley’s finiteness theorem
• What is the transcendence degree?
• What is the dual curve?
• What is the resultant?
• What is Hensel’s lemma?
• What is a node? A cusp?
• What is a constructible set?
• What does it mean to twist a sheaf?
• What is the Birkhoff-Grothendieck theorem?
• What is a resolution of singularities?
• What is a local system?
• What are the weak and hard Lefschetz theorems?
• What does it mean to be nef?

Commutative Algebra

Solutions

• Old Harvard quals (mixed topics) <https://www.math.harvard.edu/graduate/study-the-qualifying-exam/some-old-qualifying-exams/> Also a lot of old problems
• Hartshorne solutions:
  • Andrew Egbert
  • Bryden R Cais
  • Chris Lomont
  • ismail saglam and Chris
  • Jinhyun Park
  • Joe Cutrone and Nick Marshburn
  • Richard Borcherds
  • Steven V Sam
  • W Stein
• A problem set: http://sycon.rutgers.edu/~asbuch/alggeo_f14/sol01.pdf
• A problem set: https://www3.nd.edu/~jmiglior/S2018%20PS1%20solutions.pdf
• A problem set: https://faculty.math.illinois.edu/~rtramel/Math418Spring2016HW12Sol.pdf
• A collection of problem sets in schemes: http://didaktik1.mathematik.hu-berlin.de/files/solnsalgeom.pdf # Problems

Vakil’s problem sets: https://math.stanford.edu/~vakil/0506-216/216psjun2807.pdf

Hartshorne

Recommended problems from Hartshorne (c/o Elham Izadi)

• Chapter 1,
  • Section 1 #1,2,4,5,6
  • Section 1 #7,8,9,10 ; Chapter 1, Section 2 #1
  • Section 2 #2,6,7,8,10 (also do #9 but do not turn it in)
• Chapter 2
  • Section 1 #1,2,3,4,5
  • Section 1 #6,7,8,17,22
  • Section 1 #9
  • Section 2 #1,2,3,5
  • Section 2 #6,7,8,9,12
  • Section 3 #1,2,3,4,5
  • Section 3, #6,7,8,9,10
  • Section 4, #1,2,3,4,6
  • 11 if you want to try something a little harder (See 32.14 of the Stacks project, on universally closed morphisms)
• Chapter 2 (round 2)

  • Section 2 #14,16,17,18,19

  • Section 1 #18, Section 5 #1,3,5

  • Section 5 #2,4,7,8,9

  • Section 5 #10,11,12,13

  • Section 5 #14,15,16,17,18

    Note: these problems are especially hard but also especially important and interesting, do as much of them as you can, only please make sure you read them carefully and understand the statements, in many textbooks these are done in the text

  • Section 6 #1,8, Section 7 #1

    Also prove the following: If X is integral, then any nonzero morphism of invertible sheaves is injective, any generically injective morphism of locally free sheaves is injective (hint: first prove that a locally free sheaf has no torsion subsheaf, where by a torsion sheaf we mean a sheaf whose support has codimension > 0).

  • Section 6 #5,10,

  • Section 7 #2,3

  • Section 7 #4,5,7,8,9

  • Section 8 #1,2,3

  • Section 8 #4, 5

Quals and Prelims

• List of questions from Harvard: <https://www.math.harvard.edu/media/alggeom.pdf>
• Old Harvard quals (mixed topics): <https://www.math.harvard.edu/media/quals90-02.pdf>
• More old Harvard
• https://math.illinois.edu/system/files/2021-01/MATH 511 - August 2020.pdf
• https://math.illinois.edu/system/files/inline-files/comp511_may2017.pdf
• https://math.illinois.edu/system/files/inline-files/comp511_may2016.pdf
• https://www.mit.edu/~fengt/AGProblems.pdf
• MIT OCW 2009 PSets (Kedlaya?)
  • Problem set 1 (PDF)
  • Problem set 2 (PDF)
  • Problem set 3 (PDF)
  • Problem set 4 (PDF)
  • Problem set 5 (PDF)
  • Problem set 6 (PDF)
  • Problem set 7 (PDF)
  • Problem set 8 (PDF)
  • Problem set 9 (PDF)
  • Problem set 10 (PDF)
  • Problem set 11 (PDF)
  • Problem set 12 (PDF)
• MIT 2021 AG course psets
  • Problem Set 1 (pdf file, latex source) posted 8 Sep, due 17 Sep
  • Problem Set 2 (pdf file, latex source) posted 17 Sep, due 24 Sep
  • Problem Set 3 (pdf file, latex source) posted 24 Sep, due 1 Oct
  • Problem Set 4 (pdf file, latex source) posted 1 Oct, due 8 Oct
  • Problem Set 5 (pdf file, latex source) posted 8 Oct, due 15 Oct
  • Problem Set 6 (pdf file, latex source) posted 15 Oct, due 22 Oct
  • Problem Set 7 (pdf file, latex source) posted 22 Oct, due 29 Oct
  • Problem Set 8 (pdf file, latex source) posted 29 Oct, due 5 Nov
  • Problem Set 9 (pdf file, latex source) posted 8 Nov, due 13 Nov
  • Problem Set 10 (pdf file, latex source) posted 12 Nov, due 19 Nov
  • Problem Set 11 (pdf file, latex source) posted 19 Nov, due 24 Nov
  • Problem Set 12 (pdf file, latex source) posted 24 Nov, due 4 Dec
• https://www.maths.cam.ac.uk/postgrad/part-iii/files/misc/AlgebraicGeometryExercises.pdf
• Tripos Part III AG:
  • 2021: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2021/paper_113.pdf
  • 2019: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2019/paper_113.pdf
  • 2018: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2018/paper_113.pdf
  • 2017: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2017/paper_113.pdf
  • 2016: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2016/paper_113.pdf
• Tripos Part III CA:
  • 2021: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2021/paper_101.pdf
  • 2019: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2019/paper_148.pdf
  • 2018: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2018/paper_101.pdf
  • 2017: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2017/paper_128.pdf
  • 2016: https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2016/paper_101.pdf (home page is not an obvious link).