Resources: Problems and Solutions

#exercises #projects/exercises

Hartshorne Problems

• 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

Recommended problems from Hartshorne (c/o Elham Izadi):

Chapter 1

• Section 1
  • 1,2,4,5,6
  • 7,8,9,10
• Section 2:
  • 2,6,7,8,10
  • Attempt 9

Chapter 2

• Section 1
  • 1,2,3,4,5
  • 6,7,8,17,22
  • 9
  • 18
• Section 2
  • 1,2,3,5
  • 6,7,8,9,12
  • 14,16,17,18,19
  • 11 if you want to try something a little harder (See 32.14 of the Stacks project, on universally closed morphisms)
• Section 3
  • 1,2,3,4,5
  • 6,7,8,9,10
• Section 4
  • 1,2,3,4,6
• Section 5
  • 1,3,5
  • 2,4,7,8,9
  • 10,11,12,13
  • 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
  • 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 \(>0\).
  • 5,10,
• Section 7
  • 1,2,3
  • 4,5,7,8,9
• Section 8
  • 1,2,3
  • 4, 5
• The Hartshorne speedrun (c/o Nat Pacheco-Tallaj):
  • 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

PSets and Solutions

#solutions

• 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
• Vakil’s problem sets: https://math.stanford.edu/~vakil/0506-216/216psjun2807.pdf
• Something from Cambridge, I forget: https://www.maths.cam.ac.uk/postgrad/part-iii/files/misc/AlgebraicGeometryExercises.pdf
• 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
• AG problems from Tony Feng 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)
  • **Problem Set 2 (pdf file, latex source)
  • **Problem Set 3 (pdf file, latex source)
  • **Problem Set 4 (pdf file, latex source)
  • **Problem Set 5 (pdf file, latex source)
  • **Problem Set 6 (pdf file, latex source)
  • **Problem Set 7 (pdf file, latex source)
  • **Problem Set 8 (pdf file, latex source)
  • **Problem Set 9 (pdf file, latex source)
  • **Problem Set 10 (pdf file, latex source)
  • **Problem Set 11 (pdf file, latex source)
  • **Problem Set 12 (pdf file, latex source)
• https://hilbert.math.wisc.edu/wiki/index.php/Math_764_--_Algebraic_Geometry_II_--_Homeworks

Quals and Prelims

• Tripos Part III Algebraic Geometry:
  • 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 Commutative Algebra:
  • 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).
• List of questions from Harvard: <https://www.math.harvard.edu/media/alggeom.pdf>
• Old Harvard quals (mixed topics) <https://www.math.harvard.edu/graduate/study-the-qualifying-exam/some-old-qualifying-exams/>
• Old Harvard quals (mixed topics): <https://www.math.harvard.edu/media/quals90-02.pdf>
• 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