A survey on the conjecturally minimal volume of klt ample (resp. Fano) varieties and specific classification of 3-folds canonical thresholds in the open interval (1/3,1/2)

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吳宏宜(Hung-Yi Wu) 查詢紙本館藏

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論文名稱

(A survey on the conjecturally minimal volume of klt ample (resp. Fano) varieties and specific classification of 3-folds canonical thresholds in the open interval (1/3,1/2))

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至系統瀏覽論文 (2025-8-1以後開放)

摘要(中)

這篇碩士論文主要分為兩個部分，前半部分為整理Totaro的文章，將作者沒有闡述清楚的細節補齊，用一些更為淺顯易懂的敘述方式讓它更好閱讀。
後半段則是從Totaro計算global log canonical threshold (glct)的地方出發，提出了一些相關問題，如：log canonical threshold (lct)及canonical threshold (ct)的計算。
由於lct的計算相對發展的完全，我們選擇了去研究ct的樣貌，此部分的主要結果為：刻畫三維光滑的代數簇在(1/3,1/2)開區間中所有會發生的ct。

摘要(英)

This thesis is mainly divided into two parts.
The first half is to sort out Totaro′s paper, fill in the details that the author did not explain clearly, and use some methods to make it easier to read.
The second half starts from Totaro′s calculations on the global log canonical threshold (glct) to ask some related issues, such as: the calculations of log canonical threshold (lct) and canonical threshold (ct).
Since the calculations of lct are relatively developed, we choose to compute the ct in dimension 3.
Finally, we describe completely the set of all values of ct in the smooth algebraic varieties in the open interval (1/3,1/2).

關鍵字(中)

★ 代數幾何

關鍵字(英)

★ Algebraic geometry

論文目次

1 Introduction P.01
2 Fundamental Knowledge of Algebraic Geometry P.04
3 Reorganize the proof of Theorem in [Totaro22] P.10
3.1 Main result of Totaro P.10
3.2 To check the fano variety in Theorem 3.3 is exceptional P.18
3.3 The work on bottom weight P.27
4 Specific classification of 3-fold canonical thresholds in (1/3,1/2) P31
References P49

參考文獻

V. Alexeev and W. Liu. Open surfaces of small volume. Algebr. Geom. 6 (2019), 312–327.

C. Birkar. Anti-pluricanonical linear systems on Fano varieties. Ann. of Math. 190 (2019), 345-463.

Jheng-Jie Chen. On threefold canonical thresholds. Advances in Mathematics Volume 404, Part B, 6 August 2022, 108447.

Jheng-Jie Chen, Accumulation points on 3-fold canonical thresholds.

Jheng-Jie Chen, private notes on classifying 3-folds canonical threshsolds.

A. R. Iano-Fletcher. Working with weighted complete intersections. Explicit birational geometry of 3-folds, 101-173. London Math. Soc. Lecture Notes Ser., 281, Cambridge Univ. Press, Cambridge (2000).

William Fulton, Algebraic curves. An introduction to algebraic geometry. Notes written with the collaboration of Richard Weiss. Mathematics Lecture Note Series. W. A. Benjamin, Inc., New York-Amsterdam, 1969. xiii+226 pp.

C. Hacon, J. McKernan, and C. Xu. ACC for log canonical thresholds. Ann. of Math. 180 (2014), 523–571.

Jingjun Han, Jihao Liu and Yujie Luo, ACC for minimal log discrepancies of terminal threefolds, arXiv: 2202.05287v2.

Robin Hartshorne, Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp. ISBN: 0-387-90244-9

Y. Kawamata, Divisorial contractions to 3-dimensional terminal quotient singularities, Higher-dimensional complex varieties (Trento, 1994), 241-246, de Gruyter, Berlin 1996.

J. Kollar. Singularities of the minimal model program. With the collaboration of Sandor Kovacs. Cambridge (2013).

Burt Totaro, Klt varieties with conjecturally minimal volume, to appear in International Mathematics Research Notices(IMRN) rnad047, arXiv: 2210.11354v2.

Y. Prokhorov, Gap conjecture for 3-dimensional canonical thresholds, J. Math. Sci. Univ. Tokyo 15 (2008), no. 4, 449-459.

D. A. Stepanov, Smooth three-dimensional canonical thresholds, (Russian) Mat. Zametki 90 (2011), no. 2, 285-299; translation in Math. Notes 90 (2011), no. 1-2, 265-278.

I. A. Cheltsov, Log canonical thresholds on hypersurfaces}, Mathematics. 192:8 1241-1257 (2001).

指導教授

陳正傑(Jheng-Jie Chen)

審核日期

2023-7-27

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