Book Review
The AMS does not provide abstracts of book reviews.
You may download the entire review from the links below.
Full text of review:
PDF
This review is available free of charge.
Book Information:
Authors:
David Mumford and
Tadao Oda
Title:
Algebraic geometry II
Additional book information:
Texts and Readings in Mathematics, Vol. 73,
Hindustan Book Agency,
New Delhi, India,
2015,
x+504 pp.,
ISBN 978-93-80250-80-9
M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75–95. MR 321934, DOI 10.1112/plms/s3-25.1.75
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
C. Chevalley, Sur la théorie des variétés algébriques, Nagoya Math. J. 8 (1955), 1–43 (French). MR 69544
R. Dedekind and H. Weber, Theorie der algebraischen Functionen einer Veränderlichen, J. Reine Angew. Math. 92 (1882), 181–290 (German). MR 1579901, DOI 10.1515/crll.1882.92.181
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240
J. Dieudonné, The historical development of algebraic geometry, Amer. Math. Monthly 79 (1972), 827–866. MR 308117, DOI 10.2307/2317664
Jean Dieudonné, History of algebraic geometry, Wadsworth Mathematics Series, Wadsworth International Group, Belmont, CA, 1985. An outline of the history and development of algebraic geometry; Translated from the French by Judith D. Sally. MR 780183
D. Gieseker, Lectures on moduli of curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69, Published for the Tata Institute of Fundamental Research, Bombay by Springer-Verlag, Berlin-New York, 1982. MR 691308
Alexander Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 221, 249–276 (French). MR 1611822
A. Grothendieck (with the collaboration of J. Dieudonné), Éléments de géométrie algébrique, I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960); II. Étude globale élémentaire de quelques classes de morphismes, ibid. 8 (1961); III-1. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 11 (1961); III-2. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 17 (1963); IV-1. Étude locale des schémas et des morphismes de schémas, ibid. 20 (1964); IV-2. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 24 (1965); IV-3. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 28 (1966); IV-4. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 32 (1967).
G. Halphen, Mémoire sur la classification des courbes gauches algébriques, J. Éc. Polyt. 52 (1882) 1–200.
Robin Hartshorne, Connectedness of the Hilbert scheme, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 5–48. MR 213368
Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, Berlin, Heidelberg, New York (1977).
A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), no. 1, 1–60 (German). MR 1510692, DOI 10.1007/BF01199469
Steven L. Kleiman, The Picard scheme, Alexandre Grothendieck: a mathematical portrait, Int. Press, Somerville, MA, 2014, pp. 35–74. MR 3287693
L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraische Grössen, J. Reine Angew. Math. 92 (1882), 1–122 (German). MR 1579896, DOI 10.1515/crll.1882.92.1
Mireille Martin-Deschamps and Daniel Perrin, Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n’est (presque) jamais réduit, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 6, 757–785 (French, with English summary). MR 1422990
Shigefumi Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593–606. MR 554387, DOI 10.2307/1971241
David Mumford, Pathologies of modular algebraic surfaces, Amer. J. Math. 83 (1961), 339–342. MR 124328, DOI 10.2307/2372959
David Mumford, Further pathologies in algebraic geometry, Amer. J. Math. 84 (1962), 642–648. MR 148670, DOI 10.2307/2372870
David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR 0209285
D. Mumford, Pathologies. III, Amer. J. Math. 89 (1967), 94–104. MR 217091, DOI 10.2307/2373099
D. Mumford, Algebraic Geometry I: Complex projective varieties, Grundlehren der Mathematischen Wissenschaft 221, Springer-Verlag, Berlin-New York (1976).
D. Mumford, Stability of projective varieties, Monegraphie de l’Enseignment Mathématique 24, L’Enseignment Mathématique, Geneva, 1977.
David Mumford, The red book of varieties and schemes, Second, expanded edition, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1999. Includes the Michigan lectures (1974) on curves and their Jacobians; With contributions by Enrico Arbarello. MR 1748380, DOI 10.1007/b62130
D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
David Mumford and Tadao Oda, Algebraic geometry. II, Texts and Readings in Mathematics, vol. 73, Hindustan Book Agency, New Delhi, 2015. MR 3443857
M. Noether, Zur Grundlegung der Theorie der Algebraischen Raumcurven, Verlag der Königlichen Akademie der Wissenschaften, Berlin, 1883.
Scott Nollet, The Hilbert schemes of degree three curves, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 3, 367–384 (English, with English and French summaries). MR 1443492, DOI 10.1016/S0012-9593(97)89925-9
Scott Nollet and Enrico Schlesinger, Hilbert schemes of degree four curves, Compositio Math. 139 (2003), no. 2, 169–196. MR 2025805, DOI 10.1023/B:COMP.0000005083.20724.cb
Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278 (French). MR 68874, DOI 10.2307/1969915
Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1–42 (French). MR 82175
F. Severi, Vorlesungen über algebraische Geometrie: Geometrie auf einer Kurve, Riemannsche Flächen, Abelsche Integrale, Teubner, Leipzig, 1921.
I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917
Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
Ravi Vakil, Murphy’s law in algebraic geometry: badly-behaved deformation spaces, Invent. Math. 164 (2006), no. 3, 569–590. MR 2227692, DOI 10.1007/s00222-005-0481-9
B. L. van der Waerden, Zur Nullstellentheorie der Polynomideale, Math. Ann. 96 (1927), no. 1, 183–208 (German). MR 1512314, DOI 10.1007/BF01209162
Bartel L. van der Waerden, Der Multiplizitätsbegriff der algebraischen Geometrie, Math. Ann. 97 (1927), no. 1, 756–774 (German). MR 1512387, DOI 10.1007/BF01447893
André Weil, On the Riemann hypothesis in functionfields, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 345–347. MR 4242, DOI 10.1073/pnas.27.7.345
André Weil, Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, Vol. 29, American Mathematical Society, New York, 1946. MR 0023093
André Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Publ. Inst. Math. Univ. Strasbourg, vol. 7, Hermann & Cie, Paris, 1948 (French). MR 0027151
André Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508. MR 29393, DOI 10.1090/S0002-9904-1949-09219-4
Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
Oscar Zariski, Some Results in the Arithmetic Theory of Algebraic Varieties, Amer. J. Math. 61 (1939), no. 2, 249–294. MR 1507376, DOI 10.2307/2371499
Oscar Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math. (2) 40 (1939), 639–689. MR 159, DOI 10.2307/1968949
Oscar Zariski, Pencils on an algebraic variety and a new proof of a theorem of Bertini, Trans. Amer. Math. Soc. 50 (1941), 48–70. MR 4241, DOI 10.1090/S0002-9947-1941-0004241-4
Oscar Zariski, Reduction of the singularities of algebraic three dimensional varieties, Ann. of Math. (2) 45 (1944), 472–542. MR 11006, DOI 10.2307/1969189
Oscar Zariski, The theorem of Bertini on the variable singular points of a linear system of varieties, Trans. Amer. Math. Soc. 56 (1944), 130–140. MR 11572, DOI 10.1090/S0002-9947-1944-0011572-3
Oscar Zariski, The concept of a simple point of an abstract algebraic variety, Trans. Amer. Math. Soc. 62 (1947), 1–52. MR 21694, DOI 10.1090/S0002-9947-1947-0021694-1
References
- M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75–95. MR 0321934, DOI 10.1112/plms/s3-25.1.75
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- C. Chevalley, Sur la théorie des variétés algébriques, Nagoya Math. J. 8 (1955), 1–43 (French). MR 0069544
- R. Dedekind and H. Weber, Theorie der algebraischen Functionen einer Veränderlichen, J. Reine Angew. Math. 92 (1882), 181–290 (German). MR 1579901, DOI 10.1515/crll.1882.92.181
- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 0340258
- P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 0262240
- J. Dieudonné, The historical development of algebraic geometry, Amer. Math. Monthly 79 (1972), 827–866. MR 0308117, DOI 10.2307/2317664
- Jean Dieudonné, History of algebraic geometry: An outline of the history and development of algebraic geometry, Wadsworth Mathematics Series, Wadsworth International Group, Belmont, CA, 1985. Translated from the French by Judith D. Sally. MR 780183
- D. Gieseker, Lectures on moduli of curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69, Published for the Tata Institute of Fundamental Research, Bombay; Springer-Verlag, Berlin-New York, 1982. MR 691308
- Alexander Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 221, 249–276 (French). MR 1611822
- A. Grothendieck (with the collaboration of J. Dieudonné), Éléments de géométrie algébrique, I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960); II. Étude globale élémentaire de quelques classes de morphismes, ibid. 8 (1961); III-1. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 11 (1961); III-2. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 17 (1963); IV-1. Étude locale des schémas et des morphismes de schémas, ibid. 20 (1964); IV-2. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 24 (1965); IV-3. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 28 (1966); IV-4. Étude cohomologique des faisceaux cohérents, Inst. Hautes Études Sci. Publ. Math. 32 (1967).
- G. Halphen, Mémoire sur la classification des courbes gauches algébriques, J. Éc. Polyt. 52 (1882) 1–200.
- Robin Hartshorne, Connectedness of the Hilbert scheme, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 5–48. MR 0213368
- Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977
- R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, Berlin, Heidelberg, New York (1977).
- A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), no. 1, 1–60 (German). MR 1510692, DOI 10.1007/BF01199469
- Steven L. Kleiman, The Picard scheme, Alexandre Grothendieck: a mathematical portrait, Int. Press, Somerville, MA, 2014, pp. 35–74. MR 3287693
- L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraische Grössen, J. Reine Angew. Math. 92 (1882), 1–122 (German). MR 1579896, DOI 10.1515/crll.1882.92.1
- Mireille Martin-Deschamps and Daniel Perrin, Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n’est (presque) jamais réduit, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 6, 757–785 (French, with English summary). MR 1422990
- Shigefumi Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593–606. MR 554387, DOI 10.2307/1971241
- David Mumford, Pathologies of modular algebraic surfaces, Amer. J. Math. 83 (1961), 339–342. MR 0124328, DOI 10.2307/2372959
- David Mumford, Further pathologies in algebraic geometry, Amer. J. Math. 84 (1962), 642–648. MR 0148670, DOI 10.2307/2372870
- David Mumford, Lectures on curves on an algebraic surface, With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. MR 0209285
- D. Mumford, Pathologies. III, Amer. J. Math. 89 (1967), 94–104. MR 0217091, DOI 10.2307/2373099
- D. Mumford, Algebraic Geometry I: Complex projective varieties, Grundlehren der Mathematischen Wissenschaft 221, Springer-Verlag, Berlin-New York (1976).
- D. Mumford, Stability of projective varieties, Monegraphie de l’Enseignment Mathématique 24, L’Enseignment Mathématique, Geneva, 1977.
- David Mumford, The red book of varieties and schemes, Second, expanded edition, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1999. Includes the Michigan lectures (1974) on curves and their Jacobians; With contributions by Enrico Arbarello. MR 1748380, DOI 10.1007/b62130
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- David Mumford and Tadao Oda, Algebraic geometry. II, Texts and Readings in Mathematics, vol. 73, Hindustan Book Agency, New Delhi, 2015. MR 3443857
- M. Noether, Zur Grundlegung der Theorie der Algebraischen Raumcurven, Verlag der Königlichen Akademie der Wissenschaften, Berlin, 1883.
- Scott Nollet, The Hilbert schemes of degree three curves, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 3, 367–384 (English, with English and French summaries). MR 1443492, DOI 10.1016/S0012-9593(97)89925-9
- Scott Nollet and Enrico Schlesinger, Hilbert schemes of degree four curves, Compositio Math. 139 (2003), no. 2, 169–196. MR 2025805, DOI 10.1023/B:COMP.0000005083.20724.cb
- Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278 (French). MR 0068874, DOI 10.2307/1969915
- Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 1–42 (French). MR 0082175
- F. Severi, Vorlesungen über algebraische Geometrie: Geometrie auf einer Kurve, Riemannsche Flächen, Abelsche Integrale, Teubner, Leipzig, 1921.
- I. R. Shafarevich, Basic algebraic geometry, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. MR 0366917
- Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
- Ravi Vakil, Murphy’s law in algebraic geometry: badly-behaved deformation spaces, Invent. Math. 164 (2006), no. 3, 569–590. MR 2227692, DOI 10.1007/s00222-005-0481-9
- B. L. van der Waerden, Zur Nullstellentheorie der Polynomideale, Math. Ann. 96 (1927), no. 1, 183–208 (German). MR 1512314, DOI 10.1007/BF01209162
- Bartel L. van der Waerden, Der Multiplizitätsbegriff der algebraischen Geometrie, Math. Ann. 97 (1927), no. 1, 756–774 (German). MR 1512387, DOI 10.1007/BF01447893
- André Weil, On the Riemann hypothesis in function fields, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 345–347. MR 0004242
- André Weil, Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, vol. 29, American Mathematical Society, New York, 1946. MR 0023093
- André Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Actualités Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, 1948 (French). MR 0027151
- André Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508. MR 0029393, DOI 10.1090/S0002-9904-1949-09219-4
- Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
- Oscar Zariski, Some Results in the Arithmetic Theory of Algebraic Varieties, Amer. J. Math. 61 (1939), no. 2, 249–294. MR 1507376, DOI 10.2307/2371499
- Oscar Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math. (2) 40 (1939), 639–689. MR 0000159, DOI 10.2307/1968949
- Oscar Zariski, Pencils on an algebraic variety and a new proof of a theorem of Bertini, Trans. Amer. Math. Soc. 50 (1941), 48–70. MR 0004241, DOI 10.2307/1989911
- Oscar Zariski, Reduction of the singularities of algebraic three dimensional varieties, Ann. of Math. (2) 45 (1944), 472–542. MR 0011006, DOI 10.2307/1969189
- Oscar Zariski, The theorem of Bertini on the variable singular points of a linear system of varieties, Trans. Amer. Math. Soc. 56 (1944), 130–140. MR 0011572, DOI 10.2307/1990280
- Oscar Zariski, The concept of a simple point of an abstract algebraic variety, Trans. Amer. Math. Soc. 62 (1947), 1–52. MR 0021694, DOI 10.2307/1990628
Review Information:
Reviewer:
Scott Nollet
Affiliation:
Texas Christian University, Fort Worth, Texas 76129
Email:
s.nollet@tcu.edu
Journal:
Bull. Amer. Math. Soc.
57 (2020), 133-143
DOI:
https://doi.org/10.1090/bull/1664
Published electronically:
January 30, 2019
Review copyright:
© Copyright 2019
American Mathematical Society