環面簇的纖維乘積的探討

A preparation step toward the study of toric fiber products

陳昆賢(Kun-Hsien Chen)

指導教授 ： 林惠雯

國立臺灣大學/理學院/數學研究所/碩士(2012年)

https://doi.org/10.6342/NTU.2012.02630

摘要

如何決定環面簇的纖維乘積的結構是環面幾何中一個重要但困難的問題。本文
以兩種不同的面相來探討這個問題。一種來自於基本的觀點，另一種採用組合交
換代數的技巧。對於第二種方式，我們將會研究在有相同 grading 的多項式環中
的兩個理想的纖維乘積。希望本文能夠對這個問題的研究有實質的幫助。

關鍵字

纖維乘積 ； 環面簇 ； 多分次 Segre 乘積

並列摘要

How to determine the toric structure of a fiber product
is an important but difficult problem in toric geometry.
In this article, we adopt two different approaches toward this topic.
One is from the fundamental point of view and the other uses the technique
in combinatorial commutative algebra.
For the latter one, we study the toric fiber product of two homogeneous
ideals with respect to the same multigrading.
We hope this article will be helpful to further study of toric fiber products.

並列關鍵字

fiber products ； toric varieties ； multigraded Segre products

參考文獻

Weronika Buczyńska. Phylogenetic toric varieties on graphs. preprints, April 2010. arXiv:1004.1183 [math.AG].

David Cox, John Little, and Hal Schenck. Toric Varieties. American Mathematical Society, 2011.

Alexander Engstrom, Thomas Kahle, and Seth Sullivant. Multigraded commutative algebra of graph decompositions. preprints, February 2011. arXiv:1102.2601 [math.AC].

Marco Lo Giudice. My way to algebraic geometry. 2005.

Robin Hartshorne. Algebraic Geometry, volume 52 of Graduate Texts in Mathematics. Spinger-Verlag, New York, 1977.