#examples/exercises #projects/exercises

Advice from Juliette Bruce:

• Move quickly through some, skip a lot, and try to get to Ch. IV ASAP. Go back and learn the rest as needed.
• Exericses are great, but don’t try to do them all in order the first time. Get a list of recommended exercises from a trusted friend!
• Things to learn elsewhere:
  • Ch. I
  • Sections II.1 and II.2
• For Ch. II, see Vakil’s affine communication lemma.
• The heart and soul are in Ch. IV and V.

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

PSets and Solutions

#resources/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
• Dragos Oprea solutions https://mathweb.ucsd.edu/~doprea/145s08/
• 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

Solutions

Solutions to Algebraic Geometry by Robin Hartshorne

• Joe Cutrone and Nick Marshburn, http://www.math.northwestern.edu/~jcutrone/Work/Hartshorne%20Algebraic%20Geometry%20Solutions.pdf (removed). Chapters 1 to 4, part.
• Steven V Sam, http://math.mit.edu/~ssam/soln/ (removed), 2008. Chapter4 1 to 4, part.
• Andrew Egbert, http://divisibility.files.wordpress.com/2013/09/fullhartshornesolutions1.pdf, 2013. Chapters 1 to 5, all.
• Richard Borcherds, https://math.berkeley.edu/~reb/courses/256A/, 1994. Chapter 1 & 4, part.
• Bryden R. Cais, http://math.arizona.edu/~cais/CourseNotes/AlgGeom04/Hartshorne_Solutions.pdf, 2004. Chapter 2 to 4, part.
• ismail saglam and Chris, http://algebraicgeometry.blogspot.de/, 2008–2009. Chapter 2 and 3, all.
• Chris Lomont, http://lomont.org/Math/Solutions.pdf, 1999. Chapters 2 to 4, part.
• Jinhyun Park, http://mathsci.kaist.ac.kr/~jinhyun/sol2/hart.html. Chapters 1 to 5, part.
• W Stein, http://wstein.org/AG.html. Chapters 2 and 3, part.

To be added 10. Brian Osserman, 248A and 248B, https://www.math.ucdavis.edu/~osserman/classes/

More problems from https://math.berkeley.edu/~reb/courses/256A/: 1. August 28-30 1.1, 1.2, 1.3, 1.4, 1.5, 1.6 2. Sept 4-6 1.7, 1.8, 1.9, 1.11, 1.12 3. Sept 11, 13 2.1, 2.3, 2.4, 2.5, 2.9 4. Sept 18-20 2.12, 2.13, 2.14, 2.15, 2.16, 2.17 5. Sept 25-27 3.1, 3.2, 3.4, 3.5, 3.6, 3.8, 3.11, 3.13 6. Oct 2-4 3.15, 3.16, 3.17, 3.19, 3.21 7. Oct 9-11 4.1, 4.4, 4.5,4.6, 4.10 8. Oct 16-18 5.1, 5.2, 5.3, 5.6, chapter V exercise 5.8b 9. Oct 23-25 5.8, 5.10, 5.12, 5.14, 5.15 10. Oct 30-Nov 1 6.1, 6.2, 6.6, 6.7 11. Nov 6-8 6.3, 6.4, 6.5, Chapter IV Ex 2.5 12. Nov 13-15 Chapter II 1.1, 1.3, 1.8, 1.13, 1.18, 13. Nov 20 Chapter II 2.2, 2.5, 2.6, 2.9, 2.10, 2.11, 14. Nov 27-29 Chapter II 3.1, 3.2, 3.3, 3.5, 3.9, 3.11, 3.15, 3.18

Some solutions: 1.1 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 4.1. 4.3.