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    Hartshorne

    [Hart10] covers classical varieties, schemes, cohomology, curves, and surfaces, with exercises throughout. Chapter I treats affine and projective varieties, morphisms, rational maps, nonsingular varieties, nonsingular curves, and intersections in projective space. II.1–II.8 treat sheaves, schemes and their first properties, separated and proper morphisms, sheaves of modules, divisors, projective morphisms, and differentials. III.1–III.6 treat derived functors, sheaf cohomology, cohomology of noetherian affine schemes, Čech cohomology, the cohomology of projective space, and Ext; III.7 proves Serre duality and III.9 treats flat morphisms. IV.1–IV.3 treat the Riemann–Roch theorem, Hurwitz’s theorem, and embeddings of curves in projective space.

    Introductory and supplementary texts

    [Gath] develops classical and scheme-theoretic algebraic geometry together with motivating examples.

    [Vaki25] gives extended treatments of sheaves, \(\Proj\), affine-local arguments, and related constructions used in [Hart10].

    [Miln08] treats dimension, tangent spaces, smoothness, and finiteness results in the classical setting, without the scheme apparatus.

    Classical varieties and geometric examples

    [Harr10] collects concrete projective examples including rational normal curves, Veronese and Segre embeddings, determinantal varieties, secant and tangent varieties, and Grassmannians.

    [Shaf13] treats varieties in projective space: basic notions, local properties such as tangent spaces and singular points, divisors and differential forms, and intersection numbers.

    [Mumf94] develops the geometry of \(\Spec\), including generic points, examples such as \(\Spec \ZZ[x]\), and nilpotent structure. [EiHa00] includes worked \(\Spec\) and \(\Proj\) computations and further geometric examples.

    Commutative algebra

    [AtMa18] covers localization, integral dependence, the Nullstellensatz, dimension theory, and completion.

    [Eise95] covers regular sequences, depth, Cohen–Macaulay and regular rings, flatness, and the Koszul complex.

    By topic

    For toric varieties, [Fult93] develops divisors, cohomology, and resolution of singularities from fan combinatorics.

    For divisors and positivity, [Laza04] treats ample, very ample, nef, and big divisors and the standard examples separating these notions.

    For birational geometry — minimal models, the cone theorem, and singularities of pairs — [KoMo98] treats these topics in Chapters 1 and 2.

    The coherent-sheaf formalism used in Chapter III of [Hart10] is developed in [Serr55].

    Problems and solutions

    [07] transcribes questions and follow-up questions from past oral examinations and records the examiner for each entry.

    [Vaki07] contains algebraic-geometry problems graded by difficulty. [Borc12] assigns a selected sequence of exercises from [Hart10] and includes written solutions to the first several assignments. [voAr12] is a written solution set for a scheme-theory course, including computations with \(\Spec\) and Čech cohomology.

    Exercise selections from Hartshorne

    A selection of exercises from [Hart10] recommended by Elham Izadi, by chapter and section:

    • I.1: 1, 2, 4, 5, 6; 7, 8, 9, 10.

    • I.2: 2, 6, 7, 8, 10; attempt 9.

    • II.1: 1, 2, 3, 4, 5; 6, 7, 8, 17, 22; 9; 18.

    • II.2: 1, 2, 3, 5; 6, 7, 8, 9, 12; 14, 16, 17, 18, 19; and 11, which is harder (compare Stacks Project 32.14 on universally closed morphisms).

    • II.3: 1, 2, 3, 4, 5; 6, 7, 8, 9, 10.

    • II.4: 1, 2, 3, 4, 6.

    • II.5: 1, 3, 5; 2, 4, 7, 8, 9; 10, 11, 12, 13; 14, 15, 16, 17, 18. Many textbooks prove these results in the text.

    • II.6: 1, 8; 5, 10; and Nonzero maps of invertible sheaves on an integral scheme are injective.

    • II.7: 1, 2, 3; 4, 5, 7, 8, 9.

    • II.8: 1, 2, 3; 4, 5.

    Richard Borcherds’ Berkeley Math 256A (1994) assigned, in order: I.1.1–1.6; I.1.7, 1.8, 1.9, 1.11, 1.12; I.2.1, 2.3, 2.4, 2.5, 2.9; I.2.12–2.17; I.3.1, 3.2, 3.4, 3.5, 3.6, 3.8, 3.11, 3.13; I.3.15, 3.16, 3.17, 3.19, 3.21; I.4.1, 4.4, 4.5, 4.6, 4.10; I.5.1, 5.2, 5.3, 5.6 with V.5.8(b); I.5.8, 5.10, 5.12, 5.14, 5.15; I.6.1, 6.2, 6.6, 6.7; I.6.3, 6.4, 6.5 with IV.2.5; II.1.1, 1.3, 1.8, 1.13, 1.18; II.2.2, 2.5, 2.6, 2.9, 2.10, 2.11; II.3.1, 3.2, 3.3, 3.5, 3.9, 3.11, 3.15, 3.18. The course page has solutions to I.1.1–1.7, IV.4.1 and IV.4.3.

    Reference documents

    Solution collections for Hartshorne

    Course notes and exam papers

    Problem lists and course pages

    Foundational texts

    Lecture notes

    Advanced topics

    Stacks and moduli

    Problem sets

    Past examinations

    Original locations of the Hartshorne solution collections

    [Hart10]
    Hartshorne, R., Algebraic Geometry, Springer, 2010. https://doi.org/10.1007/978-1-4757-3849-0
    [Gath]
    Gathmann, A., Algebraic Geometry, 2021--2022
    [Vaki25]
    Vakil, R., The Rising Sea: Foundations of algebraic geometry, Princeton University Press, 2025. https://doi.org/10.2307/jj.30669912
    [Miln08]
    Milne, J. S., Algebraic Geometry, 2008
    [Harr10]
    Harris, J., Algebraic geometry: A first course, Springer, 2010. https://doi.org/10.1007/978-1-4757-2189-8
    [Shaf13]
    Shafarevich, I. R., Basic algebraic geometry 1: Varieties in projective space, Springer Berlin Heidelberg, 2013. https://doi.org/10.1007/978-3-642-37956-7
    [Mumf94]
    Mumford, D., The Red Book of varieties and schemes, Springer Berlin / Heidelberg, 1994. https://doi.org/10.1007/978-3-662-21581-4
    [EiHa00]
    Eisenbud, D. and J. Harris, The geometry of schemes, Springer, 2000. https://doi.org/10.1007/b97680
    [AtMa18]
    Atiyah, M. F. and I. G. Macdonald, Introduction to Commutative algebra, CRC Press, 2018. https://doi.org/10.1201/9780429493638
    [Eise95]
    Eisenbud, D., Commutative algebra with a View Toward algebraic geometry, Springer-Verlag, 1995
    [Fult93]
    Fulton, W., Introduction to toric varieties, Princeton University Press, 1993. https://doi.org/10.1515/9781400882526
    [Laza04]
    Lazarsfeld, R., Positivity in algebraic geometry. I, Springer-Verlag, 2004
    [KoMo98]
    Kollár, J. and S. Mori, Birational geometry of algebraic varieties, Cambridge University Press, 1998
    [Serr55]
    Serre, J.-P., Faisceaux algébriques cohérents, Annals of Mathematics 61 (1955), 197–278. https://doi.org/10.2307/1969915
    [07]
    Harvard University Department of Mathematics, Sample Questions from Past Qualifying Exams, 2007
    [Vaki07]
    Vakil, R., Foundations of Algebraic Geometry: Problem Sets, 2007
    [Borc12]
    Borcherds, R. E., Math 256A: Algebraic Geometry, 2012
    [voAr12]
    von Pippich, A. and A. Aryasomayajula, Solutions for Exercises, BMS Basic Course Algebraic Geometry, 2012

    Browse the problems on Algebraic Geometry

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