---
sort: 002
title: "Resources: Problems and Solutions"
---

# Problems and Solutions

#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.

## Reading Guide

-   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

## PSets and Solutions

#resources/solutions

-   [ ] **Hartshorne solutions**:
    -   [ ] [Andrew Egbert](attachments/Andrew%20Egbert.pdf)
    -   [ ] [Bryden R Cais](attachments/Bryden%20R%20Cais.pdf)
    -   [ ] [Chris Lomont](attachments/Chris%20Lomont.pdf)
    -   [ ] [ismail saglam and Chris](attachments/ismail%20saglam%20and%20Chris.pdf)
    -   [ ] [Jinhyun Park](attachments/Jinhyun%20Park.pdf)
    -   [ ] [Joe Cutrone and Nick Marshburn](attachments/Joe%20Cutrone%20and%20Nick%20Marshburn.pdf)
    -   [ ] [Richard Borcherds](attachments/Richard%20Borcherds.pdf)
    -   [ ] [Steven V Sam](attachments/Steven%20V%20Sam.pdf)
    -   [ ] [W Stein](attachments/W%20Stein.pdf)
-   [ ] 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](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset01.pdf))
    -   [ ] Problem set 2 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset02.pdf))
    -   [ ] Problem set 3 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset03.pdf))
    -   [ ] Problem set 4 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset04.pdf))
    -   [ ] Problem set 5 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset05.pdf))
    -   [ ] Problem set 6 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset06.pdf))
    -   [ ] Problem set 7 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset07.pdf))
    -   [ ] Problem set 8 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset08.pdf))
    -   [ ] Problem set 9 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset09.pdf))
    -   [ ] Problem set 10 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset10.pdf))
    -   [ ] Problem set 11 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset11.pdf))
    -   [ ] Problem set 12 ([PDF](https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/assignments/MIT18_726s09_pset12.pdf))
-   [ ] [MIT 2021 AG course](https://math.mit.edu/classes/18.721/Fall2021/index.html) PSets
    -   [ ] \*\*Problem Set 1 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet1.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet1.tex))
    -   [ ] \*\*Problem Set 2 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet2.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet2.tex))
    -   [ ] \*\*Problem Set 3 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet3.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet3.tex))
    -   [ ] \*\*Problem Set 4 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet4.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet4.tex))
    -   [ ] \*\*Problem Set 5 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet5.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet5.tex))
    -   [ ] \*\*Problem Set 6 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet6.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet6.tex))
    -   [ ] \*\*Problem Set 7 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet7.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet7.tex))
    -   [ ] \*\*Problem Set 8 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet8.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet8.tex))
    -   [ ] \*\*Problem Set 9 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet9.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet9.tex))
    -   [ ] \*\*Problem Set 10 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet10.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet10.tex))
    -   [ ] \*\*Problem Set 11 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet11.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet11.tex))
    -   [ ] \*\*Problem Set 12 ([pdf file](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet12.pdf), [latex source](https://math.mit.edu/classes/18.721/Fall2021/ProblemSet12.tex))
-   [ ] <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](https://www.maths.cam.ac.uk/postgrad/part-iii/pastpapers/2016/index.html)).
-   [ ] 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/2021-01/MATH%20511%20-%20August%202020.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

1.  Joe Cutrone and Nick Marshburn, <http://www.math.northwestern.edu/~jcutrone/Work/Hartshorne%20Algebraic%20Geometry%20Solutions.pdf> (removed). Chapters 1 to 4, part.
2.  Steven V Sam, <http://math.mit.edu/~ssam/soln/> (removed), 2008. Chapter4 1 to 4, part.
3.  Andrew Egbert, <http://divisibility.files.wordpress.com/2013/09/fullhartshornesolutions1.pdf>, 2013. Chapters 1 to 5, all.
4.  Richard Borcherds, <https://math.berkeley.edu/~reb/courses/256A/>, 1994. Chapter 1 & 4, part.
5.  Bryden R. Cais, <http://math.arizona.edu/~cais/CourseNotes/AlgGeom04/Hartshorne_Solutions.pdf>, 2004. Chapter 2 to 4, part.
6.  ismail saglam and Chris, <http://algebraicgeometry.blogspot.de/>, 2008--2009. Chapter 2 and 3, all.
7.  Chris Lomont, <http://lomont.org/Math/Solutions.pdf>, 1999. Chapters 2 to 4, part.
8.  Jinhyun Park, <http://mathsci.kaist.ac.kr/~jinhyun/sol2/hart.html>. Chapters 1 to 5, part.
9.  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](http://math.berkeley.edu/~reb/courses/256A/1.1.pdf) [1.2.](http://math.berkeley.edu/~reb/courses/256A/1.2.pdf) [1.3.](http://math.berkeley.edu/~reb/courses/256A/1.3.pdf) [1.4.](http://math.berkeley.edu/~reb/courses/256A/1.4.pdf) [1.5.](http://math.berkeley.edu/~reb/courses/256A/1.5.pdf) [1.6.](http://math.berkeley.edu/~reb/courses/256A/1.6.pdf) [1.7.](http://math.berkeley.edu/~reb/courses/256A/1.7.pdf) [4.1.](http://math.berkeley.edu/~reb/courses/256A/4.1.pdf) [4.3.](http://math.berkeley.edu/~reb/courses/256A/4.3.pdf)