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Problems
These are the problems in Hartshorne split into categories of essential, useful, and interesting. Those who are taking a casual run through Hartshorne should consider working the essential problems. Those who want to dig in a bit deeper, working the useful and interesting problems is good.
Chapter 1
Section 1
Essential: 1.5, 1.6, 1.7, 1.8, 1.9, 1.10
Useful: 1.2, 1.3, 1.4
Interesting: 1.1, 1.11, 1.12
Section 2
Essential: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7
Useful: 2.8, 2.10, 2.11, 2.12, 2.14, 2.17
Interesting: 2.13, 2.15, 2.16