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Book Review

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MathSciNet review: 3196799
Full text of review: PDF This review is available free of charge.
Book Information:

Authors: David A. Cox, John B. Little and Henry K. Schenck
Title:Toric varieties
Additional book information: Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011, xxiv+841 pp., ISBN 978-0-8218-4819-7, US $95.00.

• D. N. Bernstein, The number of roots of a system of equations, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 1–4 (Russian). MR 0435072
• David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR 1189133, DOI 10.1007/978-1-4757-2181-2
• David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811, DOI 10.1007/978-1-4757-6911-1
• V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247 (Russian). MR 495499
• Michel Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup. (4) 3 (1970), 507–588 (French). MR 284446
• Mathias Drton, Bernd Sturmfels, and Seth Sullivant, Lectures on algebraic statistics, Oberwolfach Seminars, vol. 39, Birkhäuser Verlag, Basel, 2009. MR 2723140, DOI 10.1007/978-3-7643-8905-5
• Günter Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, vol. 168, Springer-Verlag, New York, 1996. MR 1418400, DOI 10.1007/978-1-4612-4044-0
• William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
• I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
• Ron Goldman and Rimvydas Krasauskas (eds.), Topics in algebraic geometry and geometric modeling, Contemporary Mathematics, vol. 334, American Mathematical Society, Providence, RI, 2003. Papers from the Workshop on Algebraic Geometry and Geometric Modeling held at Vilnius University, Vilnius, July 29–August 2, 2002. MR 2046835, DOI 10.1090/conm/334
• G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518
• Edgar Martínez-Moro, Carlos Munuera, and Diego Ruano (eds.), Advances in algebraic geometry codes, Series on Coding Theory and Cryptology, vol. 5, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. MR 2516523, DOI 10.1142/9789812794017
• Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
• Tadao Oda and Katsuya Miyake, Almost homogeneous algebraic varieties under algebraic torus action, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 373–381. MR 0379501
• Richard P. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925, DOI 10.1016/0001-8708(80)90050-X
• Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949, DOI 10.1090/ulect/008

References

• D. N. Bernstein, The number of roots of a system of equations, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 1–4 (Russian). MR 0435072 (55 \#8034)
• David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR 1189133 (93j:13031)
• David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811 (99h:13033)
• V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247 (Russian). MR 495499 (80g:14001)
• Michel Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup. (4) 3 (1970), 507–588 (French). MR 0284446 (44 \#1672)
• Mathias Drton, Bernd Sturmfels, and Seth Sullivant, Lectures on algebraic statistics, Oberwolfach Seminars, vol. 39, Birkhäuser Verlag, Basel, 2009. MR 2723140 (2012d:62004), DOI 10.1007/978-3-7643-8905-5
• Günter Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, vol. 168, Springer-Verlag, New York, 1996. MR 1418400 (97i:52012), DOI 10.1007/978-1-4612-4044-0
• William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037 (94g:14028)
• I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1994. MR 1264417 (95e:14045), DOI 10.1007/978-0-8176-4771-1
• Ron Goldman and Rimvydas Krasauskas (eds.), Topics in algebraic geometry and geometric modeling, Contemporary Mathematics, vol. 334, American Mathematical Society, Providence, RI, 2003, Papers from the Workshop on Algebraic Geometry and Geometric Modeling held at Vilnius University, Vilnius, July 29–August 2, 2002. MR 2046835 (2004j:00026)
• G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin, 1973. MR 0335518 (49 \#299)
• Edgar Martínez-Moro, Carlos Munuera, and Diego Ruano (eds.), Advances in algebraic geometry codes, Series on Coding Theory and Cryptology, vol. 5, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. MR 2516523 (2010a:94001)
• Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894 (88m:14038)
• Tadao Oda and Katsuya Miyake, Almost homogeneous algebraic varieties under algebraic torus action, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 373–381. MR 0379501 (52 \#406)
• Richard P. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925 (81f:52014), DOI 10.1016/0001-8708(80)90050-X
• Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949 (97b:13034)

Review Information:

Reviewer:Frank Sottile
Affiliation:Department of Mathematics, Texas A&M University
Email:sottile@math.tamu.edu

Journal:Bull. Amer. Math. Soc. 51 (2014), 505-510
DOI:https://doi.org/10.1090/S0273-0979-2013-01441-1
Published electronically:November 4, 2013
Review copyright:© Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.