Algebraic Geometry: Some References

Curves

• Clemens, A scrapbook of complex curve theory, AMS.
• Coolidge, Treatise on algebraic plane curves, Dover.
• Forster, Lectures on Riemann surfaces, Springer.
• Fulton, Algebraic curves, Benjamin/Cummings
• Miranda, Algebraic curves and Riemann surfaces, AMS. (Recommended!)
• Mumford, Curves and their Jacobians, Univ. of Michigan Press
• Narasimhan, Compact Riemann surfaces, Birkhauser.
• Walker, Algebraic curves, Princeton UP.

Varieties (before schemes)

• Dolgachev, Topics in classical algebraic geometry, lecture notes.
• Eisenbud, Geometry of syzygies.
• Griffiths and Harris, Principles of algebraic geometry, Wiley.
• Harris, Algebraic geometry: a first course, Springer.
• Hodge and Pedoe, Methods of algebraic geometry I & II, Cambridge UP.
• Kempf, Algebraic varieties, Cambridge UP. [On reserve in library]
• Mumford, Algebraic geometry I: complex projective varieties, Springer.
• Reid, Undergraduate algebraic geometry, Cambridge UP.
• Shafarevich, Basic algebraic geometry, volume I, Springer. [On reserve in library]

Schemes and their cohomology

• Bredon, Sheaf theory, McGraw-Hill and Springer.
• Eisenbud and Harris, Geometry of schemes, Springer. [On reserve in library]
• Eisenbud et al., Computations in algebraic geometry with Macaulay 2.
• Godement, Topologie algebrique et theorie des faisceaux, Hermann.
• Grothendieck, Elements de geometrie algebrique, Publ. Math. IHES: I,  II,  III.1,  III.2,  IV.1,  IV.2,  IV.3,  IV.4.
• Hartshorne, Algebraic geometry, Springer. [On reserve in library]
• Iversen, Cohomology of sheaves, Springer.
• D. Mumford, Red book of varieties and schemes, Springer.
• Serre, ``Faisceaux algebriques coherents'' (AKA ``FAC'').
• Shafarevich, Basic algebraic geometry, volume II, Springer. [On reserve in library]
• Swan, Theory of sheaves, U. Chicago Press.

Commutative algebra

• M. Atiyah and I. MacDonald, Introduction to commutative algebra, Addison-Wesley.
• D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer, 1995.
• Matsumura, Commutative ring theory, Cambridge UP.
• Nagata, Local rings, Interscience.
• Northcott, Ideal theory, Cambridge UP.

Surfaces

• Barth, Peters, and Van de Ven, Compact complex surfaces, Springer.
• Beauville, Complex algebraic surfaces, Cambridge UP.
• Manin, Cubic forms, North-Holland.
• Miranda, Overview of algebraic surfaces. Recommended as a fast survey without proofs.
• Morrison, Geometry of K3 surfaces.
• Reid, Chapters on algebraic surfaces.
• Segre, Nonsingular cubic surfaces, Oxford UP.
• Shafarevich et al., Algebraic surfaces, AMS 1967.

A sampling of topics in the geometry of curves

• Arbarello, Cornalba, Griffiths, and Harris, Geometry of algebraic curves, volume 1, Springer.
• Baez, Klein's quartic curve.
• Dijkgraaf, Faber, and van der Geer, eds., The moduli space of curves, Birkhauser.
• Eisenbud and Harris, Progress in the theory of complex algebraic curves, Bulletin of the AMS 21 (1989), no. 2, 205--232.
• Griffiths and Harris, On the variety of special linear systems on a general algebraic curve, Duke Math. J. 47 (1980).
• Harris, Curves in projective space, Presses de l'Universite de Montreal.
• Harris and Morrison, Moduli of curves, Springer.
• Lazarsfeld, A sampling of vector bundle techniques in the study of linear series, in Cornalba et al., Lectures on Riemann surfaces, World Scientific.
• Lehavi, ``Any smooth plane quartic can be reconstructed from its bitangents,'' preprint.
• Miranda, Linear systems of plane curves, Notices of the AMS.

Some papers on Jacobians and abelian varieties

• Andreotti, On a theorem of Torelli, Amer. J. Math. 80 (1958).
• Farkas, On the Schottky relation and its generalization to arbitrary genus, Ann. of Math. 92 (1970).
• Farkas and Rauch, Period relations of Schottky type on Riemann surfaces, Ann. of Math. 92 (1970).
• Kempf, On the geometry of a theorem of Riemann, Ann. of Math. 98 (1973).
• Matsusaka, On a theorem of Torelli, Amer. J. Math. 80 (1958).
• Matsusaka, On a characterization of a Jacobian variety, Memoirs Coll. Sci. Kyoto ser. A 23, 1959.
• Shokurov, Distinguishing Prymians from Jacobians, Invent. Math. 65 (1981).
• Welters, A criterion for Jacobian varieties, Ann. of Math. 120 (1984).

A few other recommendations

• Andreotti and Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. 69 (1959).
• Artin and Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972).
• Clemens and Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1972).
• Griffiths, Variations on a theorem of Abel, Invent. Math. 35 (1976).
• Iskovskih and Manin, Three-dimensional quartics and counterexamples to the Luroth problem, Math. USSR-Sb. 15 (1971).