Una revisión a las geometrías de Zariski uno–dimensionales
Resumen
En este artículo presentamos una introducción a las geometrías de Zariski y hacemos una revisión bibliográfica
actualizada de las recientes líneas de investigación en el área.
Referencias
- bibitem{zilberana} Abdolahzadi R. $&$ Zilber B. (2020). textit{Definability, interpretations and 'etale fundamental groups}. url{arXiv:1906.05052[math.LO]}.
- bibitem{voloch} Abramovich D. $&$ Voloch F. (1992). textit{Towards a proof of the Mordell-Lang conjecture in characteristic $p$}, Int. Math. Res. Not. 2, 103-115. url{https://doi.org/10.1155/S1073792892000126}.
- bibitem{albalahi} Albalahi A. (2019). textit{Zariski Geometries on Strongly Minimal Unars}. PhD thesis at University of East Anglia. url{https://ueaeprints.uea.ac.uk/id/eprint/72730}.
- bibitem{page1} Aschenbrenner M. url{https://www.math.ucla.edu/~matthias/223m.1.09s/}.
- bibitem{ax} Ax J. (1968). textit{The elementary theory of finite fields}, Annals of Mathematics, Second Series, textit{88} (2).: 239–271, url{doi:10.2307/1970573}, JSTOR 1970573.
- bibitem{baldwim-ii} Baldwin J. (1999). {it Review on A new strongly minimal set} by E. Hrushovski. The Journal of Symbolic Logic, 64, pp 904-905. url{https://doi.org/10.2307/2586508}.
- bibitem{baldwin71}
- Baldwin, J. T. $&$ A. H. Lachlan. (1971). textit{On Strongly Minimal Sets}. The Journal of Symbolic Logic, vol. 36, no. 1, pp. 79–96. url{https://doi.org/10.2307/2271517}.
- bibitem{bouscaren-group} Bouscaren E. (1989). textit{Model Theoretic Versions of Weil's Theeorem on PreGroups}. textit{The Model Theory of Groups}, 177-185, University of Notre Dame Press, Notre Dame, Indiana. url{https://projecteuclid.org/euclid.ndml/1175197778}.
- bibitem{bogomolov} F. A. Bogomolov, M. Korotaev $&$ Yu. Tschinkel. (2010). textit{A Torelli theorem for curves over finite fields}, Pure Appl. Math. Q., 1, pp. 245-294. url{https://doi.org/10.4310/PAMQ.2010.v6.n1.a7}.
- bibitem{eli} Bouscaren E. (1998). {it Introduction to Model theory}. In: Bouscaren E. (eds). {it Model Theory and Algebraic Geometry}. Lecture Notes in Mathematics, vol 1696. Springer, Berlin, Heidelberg. url{https://doi.org/10.1007/978-3-540-68521-0_1}.
- bibitem{boushr} Bouscaren E. (1998). textit{Proof of the Mordell-Lang conjecture for function fields}. In: Bouscaren E. (eds). Model Theory and Algebraic Geometry. Lecture Notes in Mathematics, vol 1696. Springer, Berlin, Heidelberg. url{https://doi.org/10.1007/978-3-540-68521-0_10}.
- bibitem{casanovas} Casanovas E. textit{The recent History of Model Theory}. (Universidad de Barcelona.) url{http://www.ub.edu/modeltheory/documentos/HistoryMT.pdf}.
- bibitem{chang-keisler} Chang C. and Keisler H. (1973). {it Model Theory}, Dover Books on Mathematics reprinting, 2013.
- bibitem{ugur} Efem U. (2013). {it Specializations and Algebraically closed fields}. url{arXiv:1304.3699v2[math.LO]}.
- bibitem{ugurphd} Efem U. (2017). textit{The Theory of Specializations.} PhD thesis, University of Oxford. url{https://ora.ox.ac.uk/objects/uuid:3c14ca5d-c3d7-4233-93d3-81a4e20c4d1f}.
- bibitem{efemzil} Efem, U., and Zilber, B. (2023). On the Theory of Specialisations of Regular Covers of Zariski Structures. arXiv preprint arXiv:2302.08542.
- bibitem{Gavrilovich} Gavrilovich, M. (2012). Covers of Abelian varieties as analytic Zariski structures. Annals of Pure and Applied Logic, 163(11), 1524-1548.
- bibitem{godel1930} G"odel K. (1930). textit{Die Vollständigkeit der Axiome des logischen Funktionenkalküls}. Monatshefte für Mathematik (in German). 37 (1): 349–360. url{doi:10.1007/BF01696781}.
- bibitem{godel1931} G"odel K. (1931). textit{Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I}, Monatshefte für Mathematik und Physik, v. 38 n. 1, pp. 173–198. url{doi:10.1007/BF01700692}.
- bibitem{grothen} Grothendieck A. (1966). textit{'el'ements de g'eom'etrie alg'ebrique. IV. 'etude locale des sch'emas et des morphismes de sch'emas. III.}, Inst. Hautes 'etudes Sci. Publ. Math., 28. url{http://www.numdam.org/item/PMIHES_1966__28__5_0/}.
- bibitem{harris} Harris J. (1992). {it Algebraic geometry: a first course}. Springer Graduated Texts in Mathematics vol. 133. url{https://doi.org/10.1007/978-1-4757-2189-8}.
- bibitem{harnik} Harnik $&$ Harrington. (1984). textit{Fundamentals of forking} in Annals of pure and applied logic, textit{26}, MRtextit{86}:03032. url{https://doi.org/10.1016/0168-0072(84).90005-8}.
- bibitem{hart} Hartshorne R. (2006). {it Algebraic Geometry}, Springer Graduated Texts in Mathematics 52. url{https://doi.org/10.1007/978-1-4757-3849-0}.
- bibitem{hilbertnul} Hilbert D. (1893). textit{Über die vollen Invariantensysteme}. Math. Ann. 42, pp. 313–373. url{https://doi.org/10.1007/BF01444162}.
- bibitem{zilber-hu-co} Hrushovski E. $&$ Zilber B. (1993). {it Zariski Geometries}, Bulletin of the American Mathematical Society, Vol. 28, No. 2. url{ https://doi.org/10.1090/S0273-0979-1993-00380-X}.
- bibitem{zilber-hu-to} Hrushovski E. $&$ Zilber B. (1996). {it Zariski Geometries}, Journal of the American Mathematical Society, Vol. 9, No. 1. url{http://www.jstor.org/stable/2152839}.
- bibitem{hr93} Hrushovski E. (1993). textit{A new strongly minimal set} in Annals of Pure and Applied Logic 62(textit{2}).:147-166. url{https://doi.org/10.1016/0168-0072(93).90171-9}.
- bibitem{hr966} Hrushovski E. (1996). textit{The Mordell-Lang conjecture for function fields}, JAMS textit{9}, 667–690. url{https://doi.org/10.1090/S0894-0347-96-00202-0}.
- bibitem{kangas-field} Kangas K. (2017).
- {it Finding a field in a Zariski-like structure}. url{arXiv:1502.03225v3[math.LO]}.
- bibitem{kangas-thesis} Kangas K. (2018). {it Finding groups in Zariski-like structures}. PhD thesis at Departament of Mathematics and Statistics at University f Helsinki. url{arXiv:1404.6811v1}.
- bibitem{Ko} Kollar, Janos. et al. (2021) {it Topological reconstruction theorems for varieties}. Preprint: arXiv:2003.04847v3
- bibitem{koning} K"onig D. (1927). textit{Über eine Schlussweise aus dem Endlichen ins Unendliche}, Acta Sci. Math. (Szeged). (in German). (3(2-3).).: 121–130.
- bibitem{lang} Lang S. (1955). textit{Introduction to Algebraic Geometry}. Dover Books on Mathematics reprinting, 2019.
- bibitem{los54} Loś J. (1954). textit{On the categoricity in power of elementary deductive systems and some related problems}, Colloquium Mathematicum, 3: 58–62, MR 0061561. url{http://eudml.org/doc/210012}.
- bibitem{jerzy} Lo's Jerzy (1955). textit{Quelques remarques, th'eorèmes et problèmes sur les classes d'efinissables d'algèbres}. In: textit{Mathematical interpretation of formal systems}, pp. 98–113. North-Holland Publishing Co., Amsterdam.
- bibitem{loweheim} L"owenheim L. (1915). textit{Über M"oglichkeiten im Relativkalkül}. Math. Ann. 76, 447–470. url{https://doi.org/10.1007/BF01458217}.
- bibitem{marker} Marker D. (2002). {it Model Theory: an introduction}. Springer-Verlag GTM, New York. url{https://doi.org/10.1007/b98860}.
- bibitem{za-marker} Marker D. (1998). {it Zariski geometries}. In: Bouscaren E. (eds). {it Model Theory and Algebraic Geometry}. Lecture Notes in Mathematics, vol 1696. Springer, Berlin, Heidelberg, 1998. url{https://doi.org/10.1007/978-3-540-68521-0_7}.
- bibitem{marjca} Marjca A. $&$ Toffalori C. (2003). {it A guide to classical and Mordern Model Theory}. Kluwer Acaemic Press. url{https://doi.org/10.1007/978-94-007-0812-9}.
- bibitem{maltsev} Maltsev A. (1936). textit{Untersuchungen aus dem Gebiete der mathematischen Logik}, Matematicheskii Sbornik, Novaya Seriya, 1(43). (3).: 323–336. url{http://mi.mathnet.ru/msb5392}.
- bibitem{mac71}
- Macintyre A. (1971). textit{On $aleph_{1}$-categorical theories of fields}. In: Fund. Math. 71.1,
- –25. url{http://eudml.org/doc/214316}.
- bibitem{morley} Morley M. (1965). textit{Categoricity in Power}, Transactions of the American Mathematical Society, Vol. 114, No. 2, 114 (2).: 514–538. url{https://doi.org/10.2307/1994188}.
- bibitem{nLab} nLab authors, {it Geometric Stability Theory}, url{http://ncatlab.org/nlab/show/geometric$%$20stability$%$20theory}, May-2020.
- bibitem{onshuus} Onshuus A. $&$ Zilber B. (2011). {it The first order theory of the universal specializations}, available at url{http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server/papers/355/355.pdf}.
- bibitem{pillay-review} Pillay, A. (1996). {it Review on Hrushovski Ehud and Zilber Boris. Zariski geometries}, Journal of the American Mathematical Society, vol. 9, pp. 1–56. The Journal of Symbolic Logic, 64, pp 906-908 url{doi:10.2307/2586511}.
- bibitem{pillay-one} Pillay A. (1983). {it An Introduction to Stability Theory}. Dover Books on Mathematics reprinting 2008.
- bibitem{pinzon} Pinz'on S. {it Completaciones y especializaciones de geometr'ias de Zariski}. (2016). MSc thesis, Departamento de Matem'aticas, Universidad de los Andes, Bogot'a-Colombia. url{https://repositorio.uniandes.edu.co/bitstream/handle/1992/13959/u754351.pdf?sequence=1&isAllowed=y}.
- bibitem{robinsonnul} Robinson A. (1963). textit{Introduction to model theory and the metamathematics of algebra},
- North-Holland, Amsterdam. MR textit{26}:4911.
- bibitem{ruiz} Ruiz C. textit{Zariski geometries and commutative Algebraic geometry}. Manuscript.
- bibitem{solanki2011} Solanki V. (2011). textit{Zariski Structures in Non-commutative Algebraic Geometry and Representation Theory}. PhD Thesis, University of Oxford.
- bibitem{sustretov} Sustretov D. (2013). textit{Lecture notes on Geometric Model Theory}, Moscow. url{http://people.mpim-bonn.mpg.de/sustretov/notes/hse.pdf}.
- bibitem{sustretov2012} Sustretov D. (2012). textit{Non-algebraic Zariski geometries}, PhD Thesis, University of Oxford.
- bibitem{shaf} Shafarevich I. (1994). {it Basic Algebraic Geometry}, Springer Verlag Ed 2, Vol 1. url{https://doi.org/10.1007/978-3-642-57908-0}.
- bibitem{shelah74} Shelah S. (1974). textit{Categoricity of uncountable theories}, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. of California, Berkeley, Calif., 1971). Proceedings of Symposia in Pure Mathematics, 25, Providence, R.I.: American Mathematical Society, pp. 187–203.
- bibitem{skolem34}
- Skolem Th. (1934). textit{Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschließlich Zahlenvariablen}. Fundamenta Mathematicae (in German). 23 (1).: 150–161. url{https://doi.org/10.4064/fm-23-1-150-161}.
- bibitem{skolem20} Skolem Th. (1920). textit{Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen}, Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse, 4: 1–36.
- bibitem{smith2008} Smith L. (2008). textit{Toric Varieties as Analytic Zariski Structures}, PhD Thesis, University of Oxford.
- bibitem{vaug}
- Vaught R. (1954). textit{Applications to the L"owenheim-Skolem-Tarski theorem to problems of completeness and decidability}, Indagationes Mathematicae, 16: 467–472, MR 0063993.
- bibitem{villaveces} Villaveces A. (2011) {it La Tricotom'ia de Zilber: una breve
- introducci'on geom'etrica.} EVM.
- bibitem{Vo}
- Voevodsky, V. A. (1991) {it 'etale topologies of schemes over fields of finite type over $mathbb{Q}$.} Mathematics of the USSR-Izvestiya {bf 37.3}: 511 pgs.
- bibitem{weil} Weil A. (1946). {it Foundations of Algebraic Geometry}. American Mathematical Society Colloquium Publications, 29, Providence, R.I.: American Mathematical Society, textit{MR} 0023093.
- bibitem{tent} Tent K. $&$ Ziegler M. (2012). textit{A Course in Model Theory}. Lecture Notes in Logic, vol. 40. Cambridge University Press, United Kingdom.
- bibitem{torres-thesis} Torres J. (2020) textit{Understanding Zariski geometries}. MSc Thesis. Universidad de Antioquia. Colombia.
- bibitem{torres-paper} Torres J; Hern'andez P $&$ Garc'ia D. textit{Some remarks on the notion of genus for Zariski geometries}. preprint.
- bibitem{Zanussi} Zanussi, M. (2021). Zariski Geometries and Quantum Mechanics. PhD thesis. Boise State University.
- bibitem{zilber2008} Zilber B. (2008). textit{A class of quantum Zariski geometries}. In Z. Chatzidakis, H. Macpherson, A. Pillay, and A. Wilkie, editors, textit{Model Theory with applications to algebra
- and analysis, I and II}. Cambridge University Press.
- bibitem{zilber} Zilber B. (2010). {it Zariski Geometries: Geometry from the Logicians point of view}. CUP. London Mathematical Society.
- bibitem{zilber93} Zilber B. (1993). {it Model theory and algebraic geometry}, In: Proc. 10th Easter Conference on Model Theory (wendisch Rietz, 1993). Seminarberichte 93, Humboldt Univ, Berlin, 93–117.
- bibitem{zilber83} Zilber B. (1984). textit{The structure of models of uncountably categorical theories}, Proc. Internat.
- Congr. Math. (Warsaw, 1983). vol. 1, North-Holland, Amsterdam, pp. 359–368.
- bibitem{zilber2012} Zilber B. (2014). textit{A curve and its abstract Jacobian}, Int. Math. Res. Not. IMRN. {bf 5}. 1425--1439.
Descargas
Los datos de descargas todavía no están disponibles.