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題名 二維平滑熱帶環面法諾曲體之研究
On Two-Dimensional Smooth Tropical Toric Fano Varieties
作者 陳振偉
Chen, Chen Wei
貢獻者 蔡炎龍
Tsai, Yen Lung
陳振偉
Chen, Chen Wei
關鍵詞 熱帶環面法諾曲體
Tropical Toric Fano Varieties
日期 2011
上傳時間 30-Oct-2012 11:27:55 (UTC+8)
摘要 這篇論文裡，我們研究熱帶環面曲體，尤其是熱帶環面法諾曲體。如同古典代數幾何裡的情況一樣，要建構熱帶環面曲體，我們先從扇型開始建構。然而在某些結構裡沒辦法有熱帶化的對應，因此我們需要選一個適當的定義，這個定義必需可看成是古典情況類推而來的。在我們的論文中，使用我們認為合適的定義，計算所有平滑二維熱帶環面法諾曲體的情況，結果也證實非常類似古典的情形。
In this thesis, we survey and study tropical toric varieties with focus on tropical toric Fano varieties. To construct tropical toric varieties, we start with fans, just like the situation in classical algebraic geometry. However, some constructions does not make sense in tropical settings. Therefore, we need to choose a reasonable definition which give an analogue of a classical toric variety. In the end of this paper, we use the definition we choose, and explicitly calculate all smooth two-dimensional tropical toric Fano varieties which we found are very similar to classical cases.
參考文獻 Bibliography
[1] Danko Adrovic and Jan Verschelde. Tropical algebraic geometry in maple, a preprocessing algorithm for finding common factors to mul- tivariate polynomials with approximate coefficients. September 2008.
[2] T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas. Computing tropical varieties. J. Symbolic Comput., 42(1-2):54–73, 2007.
[3] E. Brugall ́e. Deformation of tropical hirzebruch surfaces. http: //www.math.jussieu.fr/~brugalle/articles/TropDegHirzebruch/ TropDeg.pdf.
[4] W. Bruns and J. Gubeladze. Polytopes, Rings, and K-Theory. Number 978 0-76352 in Springer Monographs in Mathematics. Springer, 2009.
[5] D.A. Cox, J.B. Little, and H.K. Schenck. Toric Varieties. Graduate Studies in Mathematics. American Mathematical Society, 2011.
95
[6] David Eisenbud and Joe Harris. The Geometry of Schemes. Springer, November 2001.
[7] Gu ̈nter Ewald. Combinatorial Convexity and Algebraic Geometry (Grad- uate Texts in Mathematics). Springer, October 1996.
[8] William Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton, NJ, 1993.
[9] Andreas Gathmann. Algebraic geometry, 2002. [10] Andreas Gathmann. Tropical algebraic geometry. Jahresber. Deutsch.
Math.-Verein., 108(1):3–32, 2006.
[11] I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Modern Birkha ̈user Classics. Birkh ̈auser, 2008.
[12] Branko Gru ̈nbaum. Convex Polytopes : Second Edition Prepared by Volker Kaibel, Victor Klee, and Gu ̈nter Ziegler (Graduate Texts in Math- ematics). Springer, May 2003.
[13] Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin. Tropical algebraic geometry, volume 35 of Oberwolfach Seminars. Birkh ̈auser Verlag, Basel, second edition, 2009.
[14] A Kasprzyk. Toric Fano Varieties and Convex Polytopes. PhD thesis, 2006.
96
[15] Diane Maclagan and Bernd Sturmfels. Introduction to Tropical Geome- try. October 9, 2009.
[16] G.G. Magaril-Ilyaev and V.M. Tikhomirov. Convex Analysis: Theory and Applications. Translations of Mathematical Monographs. American Mathematical Society, 2003.
[17] H. Meyer. Intersection Theory on Compact Tropical Toric Varieties. Sudwestdeutscher Verlag F R Hochschulschriften AG, 2011.
[18] G. Mikhalkin. Amoebas of algebraic varieties and tropical geometry. ArXiv Mathematics e-prints, February 2004.
[19] Grigory Mikhalkin. Enumerative tropical algebraic geometry in R2. J. Amer. Math. Soc., 18(2):313–377 (electronic), 2005.
[20] Grigory Mikhalkin. Tropical geometry and its applications. In Interna- tional Congress of Mathematicians. Vol. II, pages 827–852. Eur. Math. Soc., Zu ̈rich, 2006.
[21] Grigory Mikhalkin and Ilia Zharkov. Tropical curves, their jacobians and theta functions, November 2007.
[22] T. Oda. Convex bodies and algebraic geometry: an introduction to the theory of toric varieties : with 42 figures. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1988.
97
[23] Ju ̈rgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. In Idempotent mathematics and mathematical physics, volume 377 of Contemp. Math., pages 289–317. Amer. Math. Soc., Providence, RI, 2005.
[24] R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1970.
[25] K.E. Smith. An Invitation to Algebraic Geometry. Universitext (1979). Springer, 2000.
[26] David Speyer and Bernd Sturmfels. Tropical mathematics. Math. Mag., 82(3):163–173, 2009.
[27] Thorsten Theobald and Thorsten Theobald. Computing amoebas. Ex- perimental Math, 11:513–526, 2002.
[28] Gunter M. Ziegler. Lectures on Polytopes (Graduate Texts in Mathe- matics). Springer, November 1994.
描述 碩士
國立政治大學
應用數學研究所
96751007
100
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0096751007
資料類型 thesis
dc.contributor.advisor 蔡炎龍zh_TW
dc.contributor.advisor Tsai, Yen Lungen_US
dc.contributor.author (Authors) 陳振偉zh_TW
dc.contributor.author (Authors) Chen, Chen Weien_US
dc.creator (作者) 陳振偉zh_TW
dc.creator (作者) Chen, Chen Weien_US
dc.date (日期) 2011en_US
dc.date.accessioned 30-Oct-2012 11:27:55 (UTC+8)-
dc.date.available 30-Oct-2012 11:27:55 (UTC+8)-
dc.date.issued (上傳時間) 30-Oct-2012 11:27:55 (UTC+8)-
dc.identifier (Other Identifiers) G0096751007en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/54644-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 96751007zh_TW
dc.description (描述) 100zh_TW
dc.description.abstract (摘要) 這篇論文裡，我們研究熱帶環面曲體，尤其是熱帶環面法諾曲體。如同古典代數幾何裡的情況一樣，要建構熱帶環面曲體，我們先從扇型開始建構。然而在某些結構裡沒辦法有熱帶化的對應，因此我們需要選一個適當的定義，這個定義必需可看成是古典情況類推而來的。在我們的論文中，使用我們認為合適的定義，計算所有平滑二維熱帶環面法諾曲體的情況，結果也證實非常類似古典的情形。zh_TW
dc.description.abstract (摘要) In this thesis, we survey and study tropical toric varieties with focus on tropical toric Fano varieties. To construct tropical toric varieties, we start with fans, just like the situation in classical algebraic geometry. However, some constructions does not make sense in tropical settings. Therefore, we need to choose a reasonable definition which give an analogue of a classical toric variety. In the end of this paper, we use the definition we choose, and explicitly calculate all smooth two-dimensional tropical toric Fano varieties which we found are very similar to classical cases.en_US
dc.description.tableofcontents 1 Introduction 1
2 Background 4
2.1 Non-Archimedeanamoebas.................... 4
2.2 Semifield.............................. 9
2.3 TropicalSemifields ........................ 12
3 Toric variety and Fano variety 17
3.1 PolyhedralGeometry....................... 17
3.2 Fiberproductsofaffinevarieties................. 33
3.3 ToricVarieties........................... 38
3.4 Fanovarieties ........................... 48
4 Tropical Toric Variety 50
4.1 K(G,R,M)............................ 50
vi
4.2 TropicalToricVariety ...................... 67
4.3 Smooth two-dimensional tropical toric Fano varieties........................... 72
Bibliography
95		zh_TW
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0096751007en_US
dc.subject (關鍵詞) 熱帶環面法諾曲體zh_TW
dc.subject (關鍵詞) Tropical Toric Fano Varietiesen_US
dc.title (題名) 二維平滑熱帶環面法諾曲體之研究zh_TW
dc.title (題名) On Two-Dimensional Smooth Tropical Toric Fano Varietiesen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) Bibliography
[1] Danko Adrovic and Jan Verschelde. Tropical algebraic geometry in maple, a preprocessing algorithm for finding common factors to mul- tivariate polynomials with approximate coefficients. September 2008.
[2] T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas. Computing tropical varieties. J. Symbolic Comput., 42(1-2):54–73, 2007.
[3] E. Brugall ́e. Deformation of tropical hirzebruch surfaces. http: //www.math.jussieu.fr/~brugalle/articles/TropDegHirzebruch/ TropDeg.pdf.
[4] W. Bruns and J. Gubeladze. Polytopes, Rings, and K-Theory. Number 978 0-76352 in Springer Monographs in Mathematics. Springer, 2009.
[5] D.A. Cox, J.B. Little, and H.K. Schenck. Toric Varieties. Graduate Studies in Mathematics. American Mathematical Society, 2011.
95
[6] David Eisenbud and Joe Harris. The Geometry of Schemes. Springer, November 2001.
[7] Gu ̈nter Ewald. Combinatorial Convexity and Algebraic Geometry (Grad- uate Texts in Mathematics). Springer, October 1996.
[8] William Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton, NJ, 1993.
[9] Andreas Gathmann. Algebraic geometry, 2002. [10] Andreas Gathmann. Tropical algebraic geometry. Jahresber. Deutsch.
Math.-Verein., 108(1):3–32, 2006.
[11] I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Modern Birkha ̈user Classics. Birkh ̈auser, 2008.
[12] Branko Gru ̈nbaum. Convex Polytopes : Second Edition Prepared by Volker Kaibel, Victor Klee, and Gu ̈nter Ziegler (Graduate Texts in Math- ematics). Springer, May 2003.
[13] Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin. Tropical algebraic geometry, volume 35 of Oberwolfach Seminars. Birkh ̈auser Verlag, Basel, second edition, 2009.
[14] A Kasprzyk. Toric Fano Varieties and Convex Polytopes. PhD thesis, 2006.
96
[15] Diane Maclagan and Bernd Sturmfels. Introduction to Tropical Geome- try. October 9, 2009.
[16] G.G. Magaril-Ilyaev and V.M. Tikhomirov. Convex Analysis: Theory and Applications. Translations of Mathematical Monographs. American Mathematical Society, 2003.
[17] H. Meyer. Intersection Theory on Compact Tropical Toric Varieties. Sudwestdeutscher Verlag F R Hochschulschriften AG, 2011.
[18] G. Mikhalkin. Amoebas of algebraic varieties and tropical geometry. ArXiv Mathematics e-prints, February 2004.
[19] Grigory Mikhalkin. Enumerative tropical algebraic geometry in R2. J. Amer. Math. Soc., 18(2):313–377 (electronic), 2005.
[20] Grigory Mikhalkin. Tropical geometry and its applications. In Interna- tional Congress of Mathematicians. Vol. II, pages 827–852. Eur. Math. Soc., Zu ̈rich, 2006.
[21] Grigory Mikhalkin and Ilia Zharkov. Tropical curves, their jacobians and theta functions, November 2007.
[22] T. Oda. Convex bodies and algebraic geometry: an introduction to the theory of toric varieties : with 42 figures. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1988.
97
[23] Ju ̈rgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. In Idempotent mathematics and mathematical physics, volume 377 of Contemp. Math., pages 289–317. Amer. Math. Soc., Providence, RI, 2005.
[24] R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1970.
[25] K.E. Smith. An Invitation to Algebraic Geometry. Universitext (1979). Springer, 2000.
[26] David Speyer and Bernd Sturmfels. Tropical mathematics. Math. Mag., 82(3):163–173, 2009.
[27] Thorsten Theobald and Thorsten Theobald. Computing amoebas. Ex- perimental Math, 11:513–526, 2002.
[28] Gunter M. Ziegler. Lectures on Polytopes (Graduate Texts in Mathe- matics). Springer, November 1994.		zh_TW