AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Introduction to Algebraic Geometry
About this Title
Steven Dale Cutkosky, University of Missouri, Columbia, MO
Publication: Graduate Studies in Mathematics
Publication Year:
2018; Volume 188
ISBNs: 978-1-4704-3518-9 (print); 978-1-4704-4670-3 (online)
DOI: https://doi.org/10.1090/gsm/188
MathSciNet review: MR3791837
MSC: Primary 14-01
Table of Contents
Download chapters as PDF
Front/Back Matter
Chapters
- A crash course in commutative algebra
- Affine varieties
- Projective varieties
- Regular and rational maps of quasi-projective varieties
- Products
- The blow-up of an ideal
- Finite maps of quasi-projective varieties
- Dimension of quasi-projective algebraic sets
- Zariskiâs main theorem
- Nonsingularity
- Sheaves
- Applications to regular and rational maps
- Divisors
- Differential forms and the canonical divisor
- Schemes
- The degree of a projective variety
- Cohomology
- Curves
- An introduction to intersection theory
- Surfaces
- Ramification and Ă©tale maps
- Bertiniâs theorem and general fibers of maps
- Shreeram Abhyankar, On the ramification of algebraic functions, Amer. J. Math. 77 (1955), 575â592. MR 71851, DOI 10.2307/2372643
- Shreeram Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic $p\ne 0$, Ann. of Math. (2) 63 (1956), 491â526. MR 78017, DOI 10.2307/1970014
- Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321â348. MR 82477, DOI 10.2307/2372519
- S. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1617523, DOI 10.1007/978-3-662-03580-1
- Shreeram Abhyankar, On Macaulayâs examples, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., Vol. 311, Springer, Berlin, 1973, pp. 1â16. Notes by A. Sathaye. MR 0466156
- Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, No. 43, Princeton University Press, Princeton, N.J., 1959. MR 0105416
- Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR 1075991, DOI 10.1090/surv/035
- Shreeram S. Abhyankar, Resolution of singularities and modular Galois theory, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 131â169. MR 1816069, DOI 10.1090/S0273-0979-00-00892-2
- Dan Abramovich, Kalle Karu, Kenji Matsuki, and JarosĆaw WĆodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531â572. MR 1896232, DOI 10.1090/S0894-0347-02-00396-X
- Kenji Matsuki, Correction: âA note on the factorization theorem of toric birational maps after Morelli and its toroidal extensionâ [Tohoku Math. J. (2) 51 (1999), no. 4, 489â537; MR1725624 (2000i:14073)] by D. Abramovich, Matsuki and S. Rashid, Tohoku Math. J. (2) 52 (2000), no. 4, 629â631. MR 1793939, DOI 10.2748/tmj/1178207758
- Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485â496. MR 146182, DOI 10.2307/2372985
- M. Artin, NĂ©ron models, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 213â230. MR 861977
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- Maurice Auslander, On the purity of the branch locus, Amer. J. Math. 84 (1962), 116â125. MR 137733, DOI 10.2307/2372807
- Maurice Auslander and D. A. Buchsbaum, Unique factorization in regular local rings, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 733â734. MR 103906, DOI 10.1073/pnas.45.5.733
- Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225, DOI 10.1007/978-3-642-57739-0
- Dave Bayer and David Mumford, What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (Cortona, 1991) Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp. 1â48. MR 1253986
- Arnaud Beauville, Complex algebraic surfaces, London Mathematical Society Lecture Note Series, vol. 68, Cambridge University Press, Cambridge, 1983. Translated from the French by R. Barlow, N. I. Shepherd-Barron and M. Reid. MR 732439
- E. Bertini, Introduzione alla geometria proiettiva degli iperspazi, Enrico Spoerri, Pisa, 1907.
- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405â468. MR 2601039, DOI 10.1090/S0894-0347-09-00649-3
- Bhargav Bhatt, Javier Carvajal-Rojas, Patrick Grant, Karl Schwede and Kevin Tucker, Ătale fundamental groups of strongly F-regular schemes, arXiv:1611.03884.
- Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207â302. MR 1440306, DOI 10.1007/s002220050141
- Nicolas Bourbaki, Commutative algebra. Chapters 1â7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. Translated from the French; Reprint of the 1989 English translation. MR 1727221
- Ana MarĂa Bravo, Santiago Encinas, and Orlando Villamayor U., A simplified proof of desingularization and applications, Rev. Mat. Iberoamericana 21 (2005), no. 2, 349â458. MR 2174912, DOI 10.4171/RMI/425
- A. Bravo and O. Villamayor U., Singularities in positive characteristic, stratification and simplification of the singular locus, Adv. Math. 224 (2010), no. 4, 1349â1418. MR 2646300, DOI 10.1016/j.aim.2010.01.005
- Egbert Brieskorn and Horst Knörrer, Plane algebraic curves, Modern BirkhÀuser Classics, BirkhÀuser/Springer Basel AG, Basel, 1986. Translated from the German original by John Stillwell; [2012] reprint of the 1986 edition. MR 2975988, DOI 10.1007/978-3-0348-5097-1
- M. P. Brodmann and R. Y. Sharp, Local cohomology, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 136, Cambridge University Press, Cambridge, 2013. An algebraic introduction with geometric applications. MR 3014449
- Winfried Bruns and JĂŒrgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Chris Christensen, Strong domination/weak factorization of three-dimensional regular local rings, J. Indian Math. Soc. (N.S.) 45 (1981), no. 1-4, 21â47 (1984). MR 828858
- I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54â106. MR 16094, DOI 10.1090/S0002-9947-1946-0016094-3
- V. Cossart and O. Piltant, Resolution of singularities of arithmetic threefolds, arXiv: 1412.0868.
- David Cox, John Little, and Donal OâShea, Ideals, varieties, and algorithms, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2007. An introduction to computational algebraic geometry and commutative algebra. MR 2290010, DOI 10.1007/978-0-387-35651-8
- Steven Dale Cutkosky, Local monomialization and factorization of morphisms, Astérisque 260 (1999), vi+143 (English, with English and French summaries). MR 1734239
- Steven Dale Cutkosky, Monomialization of morphisms from 3-folds to surfaces, Lecture Notes in Mathematics, vol. 1786, Springer-Verlag, Berlin, 2002. MR 1927974, DOI 10.1007/b83848
- Steven Dale Cutkosky, Resolution of singularities, Graduate Studies in Mathematics, vol. 63, American Mathematical Society, Providence, RI, 2004. MR 2058431, DOI 10.1090/gsm/063
- Steven Dale Cutkosky, Local monomialization of transcendental extensions, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1517â1586 (English, with English and French summaries). MR 2172273
- Steven Dale Cutkosky, Toroidalization of dominant morphisms of 3-folds, Mem. Amer. Math. Soc. 190 (2007), no. 890, vi+222. MR 2356202, DOI 10.1090/memo/0890
- Steven Dale Cutkosky, Resolution of singularities for 3-folds in positive characteristic, Amer. J. Math. 131 (2009), no. 1, 59â127. MR 2488485, DOI 10.1353/ajm.0.0036
- Steven Dale Cutkosky, A simpler proof of toroidalization of morphisms from 3-folds to surfaces, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 3, 865â922 (English, with English and French summaries). MR 3137475, DOI 10.5802/aif.2779
- Steven Dale Cutkosky, Counterexamples to local monomialization in positive characteristic, Math. Ann. 362 (2015), no. 1-2, 321â334. MR 3343880, DOI 10.1007/s00208-014-1114-7
- Steven Dale Cutkosky, Ramification of valuations and local rings in positive characteristic, Comm. Algebra 44 (2016), no. 7, 2828â2866. MR 3507154, DOI 10.1080/00927872.2015.1065845
- Steven Dale Cutkosky, Asymptotic multiplicities of graded families of ideals and linear series, Adv. Math. 264 (2014), 55â113. MR 3250280, DOI 10.1016/j.aim.2014.07.004
- Steven Dale Cutkosky and Olivier Piltant, Ramification of valuations, Adv. Math. 183 (2004), no. 1, 1â79. MR 2038546, DOI 10.1016/S0001-8708(03)00082-3
- S. D. Cutkosky and V. Srinivas, On a problem of Zariski on dimensions of linear systems, Ann. of Math. (2) 137 (1993), no. 3, 531â559. MR 1217347, DOI 10.2307/2946531
- Steven Dale Cutkosky and Hema Srinivasan, Factorizations of birational extensions of local rings, Illinois J. Math. 51 (2007), no. 1, 41â56. MR 2346185
- R. Dedekind and H. Weber, Theorie der algebraischen Functionen einer VerĂ€nderlichen, J. Reine Angew. Math. 92 (1882), 181â290 (German). MR 1579901, DOI 10.1515/crll.1882.92.181
- Richard Dedekind and Heinrich Weber, Theory of algebraic functions of one variable, History of Mathematics, vol. 39, American Mathematical Society, Providence, RI; London Mathematical Society, London, 2012. Translated from the 1882 German original and with an introduction, bibliography and index by John Stillwell. MR 2962951, DOI 10.1090/hmath/039
- P. Del Pezzo, Sulle Superficie di ordine $n$ immerse nello spazio di $n+1$ dimensioni, Rend. Circ. Mat. Palermo 1 (1886).
- Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89â133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
- David Eisenbud and Joe Harris, On varieties of minimal degree (a centennial account), Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3â13. MR 927946, DOI 10.1090/pspum/046.1/927946
- David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. MR 1730819
- Santiago Encinas and Herwig Hauser, Strong resolution of singularities in characteristic zero, Comment. Math. Helv. 77 (2002), no. 4, 821â845. MR 1949115, DOI 10.1007/PL00012443
- H. Flenner, L. OâCarroll, and W. Vogel, Joins and intersections, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR 1724388, DOI 10.1007/978-3-662-03817-8
- Gerhard Frey and Moshe Jarden, Approximation theory and the rank of abelian varieties over large algebraic fields, Proc. London Math. Soc. (3) 28 (1974), 112â128. MR 337997, DOI 10.1112/plms/s3-28.1.112
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
- A. M. GabriĂšlov, The formal relations between analytic functions, Funkcional. Anal. i PriloĆŸen. 5 (1971), no. 4, 64â65 (Russian). MR 0302930
- Roger Godement, Topologie algébrique et théorie des faisceaux, Publ. Inst. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). MR 0102797
- Shiro Goto, Koji Nishida, and Keiichi Watanabe, Non-Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsikâs question, Proc. Amer. Math. Soc. 120 (1994), no. 2, 383â392. MR 1163334, DOI 10.1090/S0002-9939-1994-1163334-9
- Hans Grauert, Ăber Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331â368 (German). MR 137127, DOI 10.1007/BF01441136
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- Alexander Grothendieck, Sur quelques points dâalgĂšbre homologique, Tohoku Math. J. (2) 9 (1957), 119â221 (French). MR 102537, DOI 10.2748/tmj/1178244839
- Alexander Grothendieck, Cohomologie locale des faisceaux cohĂ©rents et thĂ©orĂšmes de Lefschetz locaux et globaux $(SGA$ $2)$, Advanced Studies in Pure Mathematics, Vol. 2, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968 (French). AugmentĂ© dâun exposĂ© par MichĂšle Raynaud; SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois-Marie, 1962. MR 0476737
- A. Grothendieck and J. DieudonĆe, ĂlĂ©ments de GĂ©omĂ©trie AlgĂ©brique I, Publ. Math. I.H.E.S. 4 (1960).
- A. Grothendieck and J. DieudonĆe, ĂlĂ©ments de GĂ©omĂ©trie AlgĂ©brique II, Publ. Math. I.H.E.S. 8 (1961).
- A. Grothendieck and J. DieudonĆe, ĂlĂ©ments de GĂ©omĂ©trie AlgbÌrique III, Publ. Math. I.H.E.S. 11 (1961) and 17 (1963).
- A. Grothendieck and J. DiudonĆe, ĂlĂ©ments de GĂ©omĂ©trie AlgĂ©brique IV, Publ. Math. I.H.E.S. 20 (1964), 24 (1965), 28 (1966), 32 (1967).
- A. Grothendieck and J. A. DieudonnĂ©, ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 166, Springer-Verlag, Berlin, 1971 (French). MR 3075000
- A. Grothendieck, RevĂȘtements Ă©tales et Groupe Fondemental, Lecture Notes in Math. 224, Springer-Verlag, Heidelberg (1971).
- Alexander Grothendieck and Jacob P. Murre, The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics, Vol. 208, Springer-Verlag, Berlin-New York, 1971. MR 0316453
- Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- R. Hartshorne, Ample subvarieties of algebraic varieties, Notes written in collaboration with C. Musili, Springer-Verlag, Berlin, Heidelberg, New York, 1970.
- Herwig Hauser, On the problem of resolution of singularities in positive characteristic (or: a proof we are still waiting for), Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 1, 1â30. MR 2566444, DOI 10.1090/S0273-0979-09-01274-9
- JĂŒrgen Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970), 175â193. MR 269762, DOI 10.1007/BF01273309
- J. Herzog, A. Simis, and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984), no. 2, 661â683. MR 737891, DOI 10.1090/S0002-9947-1984-0737891-6
- Heisuke Hironaka, ON THE THEORY OF BIRATIONAL BLOWING-UP, ProQuest LLC, Ann Arbor, MI, 1960. Thesis (Ph.D.)âHarvard University. MR 2939137
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109â203; ibid. (2) 79 (1964), 205â326. MR 0199184, DOI 10.2307/1970547
- Heisuke Hironaka, Three key theorems on infinitely near singularities, SingularitĂ©s Franco-Japonaises, SĂ©min. Congr., vol. 10, Soc. Math. France, Paris, 2005, pp. 87â126 (English, with English and French summaries). MR 2145950
- W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry. Vol. I, Cambridge, at the University Press; New York, The Macmillan Company, 1947. MR 0028055
- Craig Huneke, Determinantal ideals of linear type, Arch. Math. (Basel) 47 (1986), no. 4, 324â329. MR 866520, DOI 10.1007/BF01191358
- Craig Huneke, On the symmetric and Rees algebra of an ideal generated by a $d$-sequence, J. Algebra 62 (1980), no. 2, 268â275. MR 563225, DOI 10.1016/0021-8693(80)90179-9
- Nathan Jacobson, Basic algebra. I, W. H. Freeman and Co., San Francisco, Calif., 1974. MR 0356989
- H. W. E. Jung, Darstellung der Functionen eies algebraishce korrspondenzen und der verallgemeinerte korrespondenprrinzp, Math. Ann. 28 (1887).
- Kalle Karu, Local strong factorization of toric birational maps, J. Algebraic Geom. 14 (2005), no. 1, 165â175. MR 2092130, DOI 10.1090/S1056-3911-04-00380-7
- Hiraku Kawanoue and Kenji Matsuki, Toward resolution of singularities over a field of positive characteristic (the idealistic filtration program) Part II. Basic invariants associated to the idealistic filtration and their properties, Publ. Res. Inst. Math. Sci. 46 (2010), no. 2, 359â422. MR 2722782, DOI 10.2977/PRIMS/12
- Kiran S. Kedlaya, On the algebraicity of generalized power series, Beitr. Algebra Geom. 58 (2017), no. 3, 499â527. MR 3683025, DOI 10.1007/s13366-016-0325-3
- Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293â344. MR 206009, DOI 10.2307/1970447
- JĂĄnos KollĂĄr, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
- JĂĄnos KollĂĄr and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- W. Krull, BeitrĂ€ge zur Arithmetik kommutativer IntegritĂ€tsbereiche. VI. Der allgemeine Diskriminantensatz. Unverzweigte Ringerweiterungen, Math. Z. 45 (1939), no. 1, 1â19 (German). MR 1545800, DOI 10.1007/BF01580269
- Franz-Viktor Kuhlmann, Valuation theoretic and model theoretic aspects of local uniformization, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, BirkhĂ€user, Basel, 2000, pp. 381â456. MR 1748629
- Hagen Knaf and Franz-Viktor Kuhlmann, Every place admits local uniformization in a finite extension of the function field, Adv. Math. 221 (2009), no. 2, 428â453. MR 2508927, DOI 10.1016/j.aim.2008.12.009
- Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
- Serge Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983. Reprint of the 1959 original. MR 713430
- S. Lang and A. NĂ©ron, Rational points of abelian varieties over function fields, Amer. J. Math. 81 (1959), 95â118. MR 102520, DOI 10.2307/2372851
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
- Robert Lazarsfeld and Mircea MustaĆŁÄ, Convex bodies associated to linear series, Ann. Sci. Ăc. Norm. SupĂ©r. (4) 42 (2009), no. 5, 783â835 (English, with English and French summaries). MR 2571958, DOI 10.24033/asens.2109
- Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Ătudes Sci. Publ. Math. 36 (1969), 195â279. MR 276239
- Joseph Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151â207. MR 491722, DOI 10.2307/1971141
- StanisĆaw Ćojasiewicz, Introduction to complex analytic geometry, BirkhĂ€user Verlag, Basel, 1991. Translated from the Polish by Maciej Klimek. MR 1131081, DOI 10.1007/978-3-0348-7617-9
- William S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, Inc., New York, 1967. MR 0211390
- Teruhisa Matsusaka, The theorem of Bertini on linear systems in modular fields, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 26 (1950), 51â62. MR 41483, DOI 10.1215/kjm/1250778055
- Hideyuki Matsumura, Geometric structure of the cohomology rings in abstract algebraic geometry, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 32 (1959), 33â84. MR 137709, DOI 10.1215/kjm/1250776697
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- Artibano Micali, Sur les algĂšbres universelles, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 2, 33â87 (French). MR 177009
- James S. Milne, Ătale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103â150. MR 861974
- J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167â212. MR 861976
- Shigefumi Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593â606. MR 554387, DOI 10.2307/1971241
- Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133â176. MR 662120, DOI 10.2307/2007050
- Shigefumi Mori, Flip theorem and the existence of minimal models for $3$-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117â253. MR 924704, DOI 10.1090/S0894-0347-1988-0924704-X
- David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
- D. Mumford, The red book of varieties and schemes, Springer-Verlag, 1980.
- David Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Ătudes Sci. Publ. Math. 9 (1961), 5â22. MR 153682
- 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
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
- James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- M. Nagata, Lectures on the fourteenth problem of Hilbert, Tata Institute of Fundamental Research, Bombay, 1965. MR 0215828
- Tadao Oda, Torus embeddings and applications, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay; Springer-Verlag, Berlin-New York, 1978. Based on joint work with Katsuya Miyake. MR 546291
- Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics, Vol. 169, Springer-Verlag, Berlin-New York, 1970 (French). MR 0277519
- D. Rees, On a problem of Zariski, Illinois J. Math. 2 (1958), 145â149. MR 95843
- Miles Reid, Canonical $3$-folds, JournĂ©es de GĂ©ometrie AlgĂ©brique dâAngers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den RijnâGermantown, Md., 1980, pp. 273â310. MR 605348
- Paul C. Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian, Proc. Amer. Math. Soc. 94 (1985), no. 4, 589â592. MR 792266, DOI 10.1090/S0002-9939-1985-0792266-5
- Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 538169
- Judith Sally, Regular overrings of regular local rings, Trans. Amer. Math. Soc. 171 (1972), 291â300. MR 309929, DOI 10.1090/S0002-9947-1972-0309929-3
- Judith D. Sally, Numbers of generators of ideals in local rings, Marcel Dekker, Inc., New York-Basel, 1978. MR 0485852
- Pierre Samuel, Algebraic theory of numbers, Houghton Mifflin Co., Boston, Mass., 1970. Translated from the French by Allan J. Silberger. MR 0265266
- Jean-Pierre Serre, Faisceaux algĂ©briques cohĂ©rents, Ann. of Math. (2) 61 (1955), 197â278 (French). MR 68874, DOI 10.2307/1969915
- Jean-Pierre Serre, GĂ©omĂ©trie algĂ©brique et gĂ©omĂ©trie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1â42 (French). MR 82175
- Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564, DOI 10.1007/978-1-4612-1035-1
- Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
- Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
- David L. Shannon, Monoidal transforms of regular local rings, Amer. J. Math. 95 (1973), 294â320. MR 330154, DOI 10.2307/2373787
- Aron Simis, Bernd Ulrich, and Wolmer V. Vasconcelos, Cohen-Macaulay Rees algebras and degrees of polynomial relations, Math. Ann. 301 (1995), no. 3, 421â444. MR 1324518, DOI 10.1007/BF01446637
- Ernst Snapper, Multiples of divisors, J. Math. Mech. 8 (1959), 967â992. MR 0109156, DOI 10.1512/iumj.1959.8.58062
- Bernard Teissier, Valuations, deformations, and toric geometry, Valuation theory and its applications, Vol. II (Saskatoon, SK, 1999) Fields Inst. Commun., vol. 33, Amer. Math. Soc., Providence, RI, 2003, pp. 361â459. MR 2018565
- Michael Temkin, Inseparable local uniformization, J. Algebra 373 (2013), 65â119. MR 2995017, DOI 10.1016/j.jalgebra.2012.09.023
- B. L. van der Waerden, Modern Algebra. Vol. I, Frederick Ungar Publishing Co., New York, N. Y., 1949. Translated from the second revised German edition by Fred Blum; With revisions and additions by the author. MR 0029363
- B. L. van der Waerden, Modern Algebra. Vol. II, translated from the second revised German edition by Fred Blum; with revisions and additions by the author, Frederick Ungar Publishing Co., New York, N. Y., 1950.
- Robert J. Walker, Reduction of the singularities of an algebraic surface, Ann. of Math. (2) 36 (1935), no. 2, 336â365. MR 1503227, DOI 10.2307/1968575
- André Weil, Foundations of algebraic geometry, American Mathematical Society, Providence, R.I., 1962. MR 0144898
- A. Weil, Variétés Abéliennes et Courbes Algébriques, Hermann, Paris, 1971.
- JarosĆaw WĆodarczyk, Decomposition of birational toric maps in blow-ups & blow-downs, Trans. Amer. Math. Soc. 349 (1997), no. 1, 373â411. MR 1370654, DOI 10.1090/S0002-9947-97-01701-7
- Oscar Zariski, Algebraic surfaces, Second supplemented edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 61, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford. MR 0469915
- Oscar Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math. (2) 40 (1939), 639â689. MR 159, DOI 10.2307/1968949
- Oscar Zariski, Local uniformization on algebraic varieties, Ann. of Math. (2) 41 (1940), 852â896. MR 2864, DOI 10.2307/1968864
- Oscar Zariski, Pencils on an algebraic variety and a new proof of a theorem of Bertini, Trans. Amer. Math. Soc. 50 (1941), 48â70. MR 4241, DOI 10.1090/S0002-9947-1941-0004241-4
- Oscar Zariski, Foundations of a general theory of birational correspondences, Trans. Amer. Math. Soc. 53 (1943), 490â542. MR 8468, DOI 10.1090/S0002-9947-1943-0008468-9
- Oscar Zariski, Reduction of the singularities of algebraic three dimensional varieties, Ann. of Math. (2) 45 (1944), 472â542. MR 11006, DOI 10.2307/1969189
- Oscar Zariski, The concept of a simple point of an abstract algebraic variety, Trans. Amer. Math. Soc. 62 (1947), 1â52. MR 21694, DOI 10.1090/S0002-9947-1947-0021694-1
- Oscar Zariski, A simple analytical proof of a fundamental property of birational transformations, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 62â66. MR 28056, DOI 10.1073/pnas.35.1.62
- Oscar Zariski, Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. Amer. Math. Soc. 5 (1951), 90. MR 41487
- Oscar Zariski, On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 791â796. MR 95846, DOI 10.1073/pnas.44.8.791
- Oscar Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, vol. 4, Mathematical Society of Japan, Tokyo, 1958. MR 0097403
- Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560â615. MR 141668, DOI 10.2307/1970376
- Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. 1, Graduate Texts in Mathematics, No. 28, Springer-Verlag, New York-Heidelberg-Berlin, 1975. With the cooperation of I. S. Cohen; Corrected reprinting of the 1958 edition. MR 0384768