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\section*{books in algebraic geometry}

\hypertarget{books_in_algebraic_geometry}{}\section*{{Books in algebraic geometry}}\label{books_in_algebraic_geometry}

We should limit to books which we can really recommend, either by their special content, approach or pedagogical value. Historically fine but outdated books are in a separate historical section below. The outstanding surveys may be added to the lists if they are not too specialized to minor directions: the subfields may be covered in separate lists. See also MathOverflow discussions \href{http://mathoverflow.net/questions/2446/best-algebraic-geometry-text-book-other-than-hartshorne}{best-algebraic-geometry-text-book-other-than-hartshorne}, \href{http://mathoverflow.net/questions/61515/life-after-hartshorne-the-book-not-the-person}{life-after-hartshorne-the-book}. To orient in the basic standard terminology, the wikipedia's \href{https://en.wikipedia.org/wiki/Glossary_of_algebraic_geometry}{Glossary of algebraic geometry} is decent.

\noindent\hyperlink{introductory_level_but_excellent_textbooks}{Introductory level but excellent textbooks}\dotfill \pageref*{introductory_level_but_excellent_textbooks} \linebreak
\noindent\hyperlink{schemes_standard_sources}{Schemes, standard sources}\dotfill \pageref*{schemes_standard_sources} \linebreak
\noindent\hyperlink{grothendieck_school_sources}{Grothendieck school sources}\dotfill \pageref*{grothendieck_school_sources} \linebreak
\noindent\hyperlink{background_in_commutative_algebra}{Background in commutative algebra}\dotfill \pageref*{background_in_commutative_algebra} \linebreak
\noindent\hyperlink{algebraic_groupsgroup_schemes_fouriermukai_transform}{Algebraic groups/group schemes; Fourier-Mukai transform}\dotfill \pageref*{algebraic_groupsgroup_schemes_fouriermukai_transform} \linebreak
\noindent\hyperlink{complexanalytic_approach}{Complex-analytic approach}\dotfill \pageref*{complexanalytic_approach} \linebreak
\noindent\hyperlink{mori_program_and_birational_geometry}{Mori program and birational geometry}\dotfill \pageref*{mori_program_and_birational_geometry} \linebreak
\noindent\hyperlink{arithmetic_aspects}{Arithmetic aspects}\dotfill \pageref*{arithmetic_aspects} \linebreak
\noindent\hyperlink{cohomology_theories_for_schemes}{Cohomology theories for schemes}\dotfill \pageref*{cohomology_theories_for_schemes} \linebreak
\noindent\hyperlink{modern_extensions_of_scheme_theory}{Modern extensions of scheme theory}\dotfill \pageref*{modern_extensions_of_scheme_theory} \linebreak
\noindent\hyperlink{algorithmic_and_computational_methods}{Algorithmic and computational methods}\dotfill \pageref*{algorithmic_and_computational_methods} \linebreak
\noindent\hyperlink{historically_important_but_now_outdated}{Historically important but now outdated}\dotfill \pageref*{historically_important_but_now_outdated} \linebreak


\hypertarget{introductory_level_but_excellent_textbooks}{}\subsection*{{Introductory level but excellent textbooks}}\label{introductory_level_but_excellent_textbooks}

\begin{itemize}%
\item [[Miles Reid]], \emph{Undergraduate algebraic geometry}, London Math. Soc. Student Texts \textbf{12}
\item [[Joe Harris]], \emph{Introductory algebraic geometry} (varieties)
\item [[Igor Shafarevich]], \emph{Basic algebraic geometry} (varieties and schemes)
\item [[Shigeru Mukai]], \emph{An introduction to invariants and moduli}, Cambridge Studies in Adv. Math. \textbf{81}
\item [[William Fulton]], \emph{Algebraic curves. An introduction to algebraic geometry}, 3rd ed. 2008 (varieties)
\item [[James Milne|J. S. Milne]], \emph{Algebraic geometry}, 2017 \href{http://www.jmilne.org/math/CourseNotes/AG.pdf}{pdf}

\end{itemize}
\hypertarget{schemes_standard_sources}{}\subsection*{{Schemes, standard sources}}\label{schemes_standard_sources}

\begin{itemize}%
\item [[Robin Hartshorne]], \emph{Algebraic geometry}, Springer
\item Qing Liu, \emph{Algebraic geometry and arithmetic curves}, 592 pp. Oxford Univ. Press 2002
\item D. Eisenbud, J. Harris, \emph{The geometry of schemes}, Springer Grad. Texts in Math.
\item [[David Mumford]], \emph{Red book of varieties and schemes} (cf. also unfinished sequel notes for the later part of Mumford's course, coauthored with Oda, \href{http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap1-6.pdf}{ch. 1-6 pdf}, \href{http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap7-8.pdf}{ch. 7-8 pdf})
\item Amnon Neeman, \emph{Algebraic and analytic geometry}, London Math. Soc. Lec. Note Series \textbf{345}
\item M. Demazure, P. Gabriel, \emph{Groupes algebriques}, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970
\item Ravi Vakil's Stanford \href{http://math.stanford.edu/~vakil/0708-216}{course notes}
\item William Fulton, \emph{Intersection theory}, Springer 1984
\item Ulrich G\"o{}rtz, Torsten Wedhorn, \emph{Algebraic geometry I. Schemes with examples and exercises}, Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010. viii+615 pp. \href{http://www.springerlink.com/content/kt5u74/#section=748613&page=1}{Springerlink book}

\end{itemize}
\hypertarget{grothendieck_school_sources}{}\subsection*{{Grothendieck school sources}}\label{grothendieck_school_sources}

\begin{itemize}%
\item [[EGA]]
\item [[SGA]]
\item [[FGA]], [[FGA explained]]
\item R. Hartshorne, \emph{Residues and duality}
\item A. Grothendieck et al. \emph{Dix exposes sur la cohomologie des schemas}, North Holland, Amsterdam, 1968.

\end{itemize}
\hypertarget{background_in_commutative_algebra}{}\subsection*{{Background in commutative algebra}}\label{background_in_commutative_algebra}

\begin{itemize}%
\item M. Atiyah, I. G. Macdonald, \emph{Introduction to commutative algebra}, 1969, 1994
\item H. Matsumura, \emph{Commutative algebra}, 2 vols.; see also the online summary notes by D. Murfet, \href{http://therisingsea.org/notes/Matsumura.pdf}{Matsumura.pdf}, \href{http://therisingsea.org/notes/Matsumura-Part2.pdf}{Matsumura-Part2.pdf}
\item D. Eisenbud, \emph{Commutative algebra: with a view toward algebraic geometry}, Grad. Texts in Math. 150, Springer-Verlag 1995.
\item James Milne, \emph{A primer of commutative algebra}, (online notes in progress) \href{http://www.jmilne.org/math/xnotes/ca.html}{webpage}, \href{http://www.jmilne.org/math/xnotes/CA.pdf}{pdf}

\end{itemize}
\hypertarget{algebraic_groupsgroup_schemes_fouriermukai_transform}{}\subsection*{{Algebraic groups/group schemes; Fourier-Mukai transform}}\label{algebraic_groupsgroup_schemes_fouriermukai_transform}

\begin{itemize}%
\item William C. Waterhouse, \emph{Introduction to affine group schemes}, GTM 66, Springer 1979
\item [[Armand Borel]], \emph{Linear algebraic groups}, Springer GTM, 2 editions
\item Tonny A. Springer, \emph{Linear algebraic groups}, Progress in Mathematics, 9 (2nd ed.), BBirkh\"a{}user Boston 1998, \href{http://www.ams.org/mathscinet-getitem?mr=1642713}{MR1642713}
\item [[David Mumford]], \emph{Abelian varieties}, Oxford Univ. Press 1970
\item A. Polishchuk, \emph{Abelian varieties, theta functions and the Fourier transform}, Cambridge Univ. Press 2003
\item [[M. Demazure]], [[P. Gabriel]], \emph{Groupes algebriques}, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 -- has functor of points point of view (listed also under scheme theory); for review see Bull. London Math. Soc. (1980) 12 (6): 476-478, \href{http://dx.doi.org/10.1112/blms/12.6.476b}{doi}
\item SGA III
\item J. S. Milne, \emph{Abelian varieties}, course notes, \href{http://www.jmilne.org/math/CourseNotes/AV.pdf}{pdf}
\item [[Daniel Huybrechts]], \emph{Fourier-Mukai transforms in algebraic geometry}, Oxford Mathematical Monographs. 2006. 307 pages.
\item Andr\'e{} Weil, \emph{Courbes alg\'e{}briques et vari\'e{}t\'e{}s ab\'e{}liennes}, Paris: Hermann 1971
\item C. Bartocci, Ugo Bruzzo, D. Hernandez Ruiperez, \emph{Fourier-Mukai and Nahm transforms in geometry and mathematical physics}, Progress in Mathematics \textbf{276}, Birkhauser 2009.

\end{itemize}
\hypertarget{complexanalytic_approach}{}\subsection*{{Complex-analytic approach}}\label{complexanalytic_approach}

\begin{itemize}%
\item P. Griffiths, J. Harris, \emph{Principles of algebraic geometry}
\item Phillip A. Griffiths, \emph{Introduction to algebraic curves}
\item [[Daniel Huybrechts]], \emph{Complex geometry - an introduction}, Springer (2004). Universitext. 309 pages.
\item [[Donu Arapura]], \emph{Algebraic geometry over the complex numbers}, Springer Universitext 223, 329 pp.

\end{itemize}
\hypertarget{mori_program_and_birational_geometry}{}\subsection*{{Mori program and birational geometry}}\label{mori_program_and_birational_geometry}

\begin{itemize}%
\item J\'a{}nos Koll\'a{}r, Shigefumi Mori, \emph{Birational geometry of algebraic varieties}, With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Math. 134 (1998), viii+254 pp.
\item Kenji Matsuki, \emph{Introduction to the Mori program}, Universitext. Springer 2002. xxiv+478 pp. \href{http://www.ams.org/mathscinet-getitem?mr=1875410}{MR2002m:14011}
\item Herbert Clemens, J\'a{}nos Koll\'a{}r, Shigefumi Mori, \emph{Higher-dimensional complex geometry}, Ast\'e{}risque \textbf{166} (1988), 144 pp. (1989).

\end{itemize}
\hypertarget{arithmetic_aspects}{}\subsection*{{Arithmetic aspects}}\label{arithmetic_aspects}

\begin{itemize}%
\item Goro Shimura, \emph{Abelian varieties with complex multiplication and modular functions}, Princeton Univ. Press 1997
\item Dale Husem\"o{}ller, \emph{Elliptic curves}, Graduate Texts in Mathematics. 111 (2nd ed.). Springer 2004, ISBN 0-387-95490-2.
\item Anthony Knapp, \emph{Elliptic curves}, Math Notes. 40. Princeton University Press 1992
\item Neal Koblitz, \emph{Introduction to elliptic curves and modular forms}, Graduate Texts in Mathematics. \textbf{97} (2nd ed.). Springer-Verlag 1993
\item Joseph H. Silverman, \emph{The Arithmetic of Elliptic Curves}, Graduate Texts in Math. \textbf{106}, Springer 1986; \emph{Advanced Topics in the Arithmetic of Elliptic Curves} Graduate Texts in Mathl. \textbf{151}, Springer 1994.
\item J. H. Silverman, John Tate, \emph{Rational Points on Elliptic Curves}, Springer 1992.
\item [[Gerd Faltings]], \emph{Lectures on arithmetic Riemann-Roch theorem}, Annals of Math. Studies \textbf{127}, Princeton Univ. Press 1992
\item S. Bosch, W. L\"u{}tkebohmert, M. Raynaud, \emph{N\'e{}ron models}, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer 1990. x+325 pp. \href{http://www.ams.org/mathscinet-getitem?mr=1045822}{MR91i:14034}

\end{itemize}
\hypertarget{cohomology_theories_for_schemes}{}\subsection*{{Cohomology theories for schemes}}\label{cohomology_theories_for_schemes}

\begin{itemize}%
\item J. S. Milne, \emph{Etale cohomology}, Princeton Univ. Press 1980, \href{http://books.google.com/books?id=uAT4T4BXC50C&lpg=PP1&dq=milne%20etale%20cohomology}{gbooks}
\item P. Berthelot, A. Ogus, \emph{Notes on crystalline cohomology}, Princeton Univ. Press 1978. vi+243, ISBN0-691-08218-9
\item \ldots{}list basic literature on [[motive]]s
\item [[Marc Levine]], \emph{Mixed motives}, Math. Surveys and Monographs \textbf{57}, Amer. Math. Soc. 1998, free \href{http://www.ams.org/online_bks/surv57/surv57.pdf}{pdf}
\item F. Hirzebruch, \emph{Topological methods in algebraic geometry}

\end{itemize}
\hypertarget{modern_extensions_of_scheme_theory}{}\subsection*{{Modern extensions of scheme theory}}\label{modern_extensions_of_scheme_theory}

These are advanced books or long foundational expositions.

\begin{itemize}%
\item D. Knutson, \emph{Algebraic spaces}, Springer 1971
\item [[Ofer Gabber]], Lorenzo Ramero, \emph{Almost ring theory}, arxiv and published
\item [[Jacob Lurie]], \emph{Derived algebraic geometry}, several issues, arxiv
\item [[Bertrand Toen]], [[Gabrielle Vezzosi]], HAG and DAG
\item [[Nikolai Durov]], \emph{A new approach to Arakelov geometry}, arxiv
\item something basic on log schemes, e.g. from Kato/Ogus/Olsson

\end{itemize}
\hypertarget{algorithmic_and_computational_methods}{}\subsection*{{Algorithmic and computational methods}}\label{algorithmic_and_computational_methods}

Things like Groebner bases, combinatorical methods with toric varieties etc.

\begin{itemize}%
\item David A. Cox, John B. Little, Don O'Shea, \emph{Ideals, varieties, and algorithms}

\end{itemize}
\hypertarget{historically_important_but_now_outdated}{}\subsection*{{Historically important but now outdated}}\label{historically_important_but_now_outdated}

While many of these fine books are still pleasure for some readers, they do not capture the modern viewpoint and have maybe too old notation to be regularly used. But one should be aware of them, and of sometimes unique material exposed there.

\begin{itemize}%
\item W. V. D. Hodge, Daniel Pedoe, \emph{Methods of algebraic geometry}, 3 vols. (see review by Coxeter in Bull. Amer. Math. Soc. \textbf{55}, 3, part 1 (1949), 315-316, \href{http://projecteuclid.org/euclid.bams/1183513548}{euclid})
\item F. Severi, Vol. I (1942): Serie, sistemi d'equivalenza e correspondenze algebriche sulle varieta algebriche. Vol. I I (1958) and I I I (1959): Geometria dei sistemi algebrici sopra una superficie e sopra una varieta algebrica.

\end{itemize}
category: reference

[[!redirects books in algebraic geometry]] [[!redirects book in algebraic geometry]] [[!redirects books on algebraic geometry]] [[!redirects book on algebraic geometry]] [[!redirects algebraic geometry books]] [[!redirects algebraic geometry book]]



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