二、申請補助經費：（一）請將本計畫申請書之第四項

(1)

表 C002 計畫主持人：謝世峰申請條碼編號： 98WFA0300349 共 1 頁第 1 頁

二、申請補助經費：

（一）請將本計畫申請書之第四項(表 C004)、第五項(表 C005)、第六項(表 C006)、第七項(表 C007)、

第八項(表 C008)所列費用個別加總後，分別填入「研究人力費」、「耗材、物品及雜項費用」、

「研究設備費」、「國外或大陸地區差旅費」及「出席國際學術會議差旅費」欄內。

（二）若有申請國際合作研究計畫費用者，請將表 I002 之「C 類經費合計」欄金額填入「國際合

作研究計畫國外學者來臺費用」欄內，「A 類經費與 B 類經費合計」欄金額填入「國際合作

研究計畫出國差旅費」欄內。

（三）管理費為申請機構配合執行本計畫所需之費用，其計算方式係依本會規定核給補助管理費之項目費用總和及各申請機構管理費補助比例計算後直接產生，申請人不須填寫「管理費」欄。

（四）「貴重儀器中心使用額度」係將第九項(表 C009)所列使用費用合計數填入。

（五）請依各年度申請博士後研究之名額填入下表。

（六）申請機構或其他單位（含產業界）提供之配合項目，請檢附相關證明文件。

金額單位：新台幣元執行年次

補助項目

第一年 (99 年 8 月~100

年 7 月)

第二年 (100 年 8 月

~101 年 7 月)

第三年 (101 年 8 月

~102 年 7 月)

第四年第五年

業務費 608,000 608,000 608,000

研究人力費 528,000 528,000 528,000

耗材及雜項費用 80,000 80,000 80,000 國際合作研究計畫

國外學者來臺費用 0 0 0

研究設備費 118,000 107,000 0 國外差旅費 100,000 100,000 100,000

國外或大陸地區差旅費 0 0 0

出席國際學術會議差旅費 100,000 100,000 100,000 國際合作研究計畫

出國差旅費 0 0 0

管理費 108,900 107,250 91,200 合計 934,900 922,250 799,200

貴重儀器中心使用額度 0 0 0

國內、外

地區共 0 名共 0 名共 0 名共名共名

博士後研究

大陸地區共 0 名共 0 名共 0 名共名共名

申請機構或其他單位（含產業界）提供之配合項目（無配合補助項目者免填）

配合單位名稱配合補助項目配合補助金額配合年次證明文件

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表 C003 計畫主持人：陳榮凱申請條碼編號： 96WFA0102166 共 1 頁第 1 頁

三、主要研究人力：

（一）請依照「主持人」、「共同主持人」、「協同研究人員」及「博士後研究」等類別之順序分別填寫。

類別姓名服務機構/系所職稱在本研究計畫內擔任之具體工作性質、項目及範圍

＊每週平均投入工作時數比率 ( % ) 主持人陳榮凱國立臺灣大學數

學系暨研究所

教授研究探討 100%

※註：每週平均投入工作時數比率係填寫每人每週平均投入本計畫工作時數佔其每週全部工作時間之比率，以百分比表示（例如：50%即表示該研究人員每週投入本計畫研究工作之時數佔其每週全部工時之百分五十）。

（二）如申請博士後研究，請另填表 CIF2101 及 CIF2102（若已有人選者，請務必填註人選姓名，

並將其個人資料表併同本計畫書送本會）。

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四、研究人力費：

（一）類別/級別欄請依專任助理(含碩士、學士、三專、五(二)專及高中職)、兼任助理(含博士生、

碩士生、大專學生、講師及助教)及臨時工等填寫。

（二）專任助理及兼任助理之每月工作酬金標準，不得超過本會補助專題研究計畫專任助理人員工作酬金參考表及本會補助專題研究計畫兼任助理人員工作酬金支給標準表之規定。

（三）申請專任助理者，除依工作月數填列工作酬金及至多 1.5 個月年終工作獎金外，須另填列

投保勞保及健保之「雇主應負擔之勞、健保費」（於線上填列工作酬金時，系統會自動列

入勞、健保費）。

（四）請分年列述。

金額單位：新台幣元第 1 年

（一）專任助理、講師及助教級兼任助理、臨時工資類別/級

別人數姓名工作月數

月支酬金

（含勞健保費）小計

請述明：1.最高學歷 2.曾擔任專題研究計畫專任助理之經歷 3.在本計畫內擔任之具體工作性質、項目及範圍

合計（一）

（二）博士班研究生、碩士班研究生及大專學生兼任助理級別／

姓名

人數

（1）

每人每月單元數(2)

獎助月數 (3)

小計 (4)＝

＄2000×(1)×(2)×(3)

在本研究計畫內擔任之具體工作性質、項目及範圍

合計（二）

總計（三）＝合計（一）＋合計（二）

第 2 年

月支酬金

合計（一）

（二）博士班研究生、碩士班研究生及大專學生兼任助理

(4)

級別／

姓名

人數

（1）

獎助月數 (3)

小計 (4)＝

＄2000×(1)×(2)×(3)

碩士班研究生研究助學金

3 3 12 216,000 協助執行計畫

大專學生研究助學金

2 2 12 96,000 協助執行計畫

博士班研究生獎助金(博士候選人)

1 13 12 312,000 協助執行計畫

合計（二） 624,000

總計（三）＝合計（一）＋合計（二） 624,000

第 3 年

月支酬金

合計（一）

（二）博士班研究生、碩士班研究生及大專學生兼任助理級別／

姓名

人數

（1）

獎助月數 (3)

小計 (4)＝

＄2000×(1)×(2)×(3)

碩士班研究生研究助學金

3 1 12 72,000 協助執行計畫

博士班研究生獎助金

2 13 12 624,000 協助執行計畫

合計（二） 696,000

總計（三）＝合計（一）＋合計（二） 696,000

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五、耗材及雜項費用：

（一）凡執行研究計畫所需之耗材及雜項費用，均可填入本表內。

（二）說明欄請就該項目之規格、用途等相關資料詳細填寫，以利審查。

（三）若申請單位有配合款，請於備註欄註明。

（四）請分年列述。

第 1 年金額單位：新台幣元

項目名稱說明單位數量單價金額備註

消耗性器材電腦耗材、投影片、文具

等

1 1 20,000 20,000

電腦使用費硬體維護、軟體更新 1 1 20,000 20,000

資料檢索費學術論文查詢 1 1 10,000 10,000

論文發表費論文發表所需之費用 1 1 10,000 10,000

電郵費國內外郵寄、通訊所需之

費用

1 1 10,000 10,000

打字費打字、印刷 1 1 10,000 10,000

合計 80,000

第2年

等

1 1 20,000 20,000

資料檢索費學術論文查詢 1 1 10,000 10,000

費用

1 1 10,000 10,000

打字費打字、印刷 1 1 10,000 10,000

合計 80,000

(6)

第3年

等

1 1 20,000 20,000

資料檢索費學術論文查詢 1 1 10,000 10,000

費用

1 1 10,000 10,000

打字費打字、印刷 1 1 10,000 10,000

合計 80,000

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六、研究設備費：

（一）凡執行研究計畫所需單價在新台幣一萬元以上且使用年限在二年以上之各項儀器、機械及資訊設備（含各項電腦設施、網路系統、週邊設備、套裝軟體：如作業系統軟體，以及後續超過 2 年效益之軟體改版、升級與應用系統開發規劃設計）等之購置裝置費用及圖書館典藏之分類圖書等屬之，此項設備之採購，以與本研究計畫直接有關者為限。各類研究設備金額請於金額欄內分別列出小計金額。

（二）購置設備單價在新臺幣二十萬元以上者，須檢附估價單。

（三）若申請機構及其他機構有提供配合款，請務必註明提供配合款之機構及金額。

（四）儀器設備單價超過六十萬元(含)以上者，請詳述本項設備之規格與功能(諸如靈敏度、精確度…等)，其他重要特性與重要附件，以及申購本設備對計畫執行之必要性。本項設備若獲補助，主持人應負維護保養之責，並且在不妨礙個人研究計畫或研究群計畫之工作下，同意提供他人共同使用，以避免設備閒置。

（五）請分年列述。

第 1 年金額單位：新台幣元經費來源

類別設備名稱

(中文/英文) 說明數量單價金額本會補助

經費需求

提供配合款之機構名稱及金額其他設備軟體

(Mathematica

、GrindEq)/

數值計算、文書編寫 1 20,000 20,000 20,000

儀器及資訊設備

電腦、螢幕及周邊/

高階數值計算。

參閱共同契約

LP5-970060 第二組 14 項

1 60,000 60,000 60,000

印表機/ 參閱共同契約

LP5-970061 第 1 組 6.02 項

1 38,000 38,000 38,000

合計 _118,000 118,000 第 2 年

經費來源

類別設備名稱

經費需求

提供配合款之機構名稱及金額其他設備軟體(matlab)/ 數值計算 1 12,000 12,000 12,000

筆記型電腦/ 高階計算、文書編輯。參閱共同契約 LP5-970060 第 3 組第 3 項

1 60,000 60,000 60,000

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表 C006 計畫主持人：陳榮凱申請條碼編號： 99WFA0102043 共 7 頁第 2 頁圖書設備圖書/ 詳見書單

#1. Flips for 3-folds and 4-folds (Oxford Lecture Series in Mathematics and Its Applications) [Hardcover]

Alessio Corti (Editor)

• Hardcover: 184 pages

• Publisher: Oxford University Press, USA (August 23, 2007)

• Language: English

• ISBN-10:

0198570619

• ISBN-13:

978-0198570615

$99.00

#2. Methods of Algebraic Geometry in Control Theory:

Multivariable Linear Systems and Projective Algebraic Geometry Part II [Hardcover]

Peter Falb

$109.00

• Hardcover: 404 pages

• Publisher:

Birkhäuser Boston； 1 edition (February 1, 2000)

• Language: English

• ISBN-10:

0817641130

• ISBN-13:

978-0817641139

1 6,500 6,500 6,500

電腦螢幕及硬碟/

電腦螢幕 15000 硬碟 8000

1 23,000 23,000 23,000

合計 _29,500 29,500 第 2 年

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七、赴國外或大陸地區移地研究差旅費：

（一）類別分為「實驗」、「研究」、「田野調查」等。

（二）請詳述預定各出國人員之出國行程、預估經費、天數及地點。

（三）生活費、機票費及其他費用之標準，請依照行政院頒布之「中央各機關（含事業機構）派赴國外進修、研究、實習人員補助項目及數額表」規定填列（網址 http://web1.nsc.gov.tw/public/Data/02912191571.pdf）。

（四）請將所列各項費用換算為新台幣，並註明估算匯率。

（五）請分年列述。

第 1 年金額單位：新台幣元申請補助費用

經費類別預估經費

詳述預定各出國人員之類別、出國行程、預估經費、天數及地點

(類別包括實驗、研究、田野調查)

赴國外 100,000

主持人預計於 2011 暑假或 2012 寒假訪問 Princeton 大學 Kollar 教授約 14 日

預估經費約 100000

赴大陸 0

合計 100,000

第2年

申請補助費用

經費類別預估經費

詳述預定各出國人員之類別、出國行程、預估經費、天數及地點

(類別包括實驗、研究、田野調查)

赴國外 100,000

主持人預計於 2012 暑假或 2013 寒假訪問 Utah 大學 Hacon 教授

約 14 日

赴大陸 0

合計 100,000

第3年

申請補助費用

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八、出席國際學術會議差旅費：

（一）計畫內之研究人員得申請本項經費。

（二）請詳述預定參加國際學術會議之性質、預估經費、天數及地點。

（三）請詳述申請人近三年參加國外舉辦之國際學術會議論文之發表情形。（包括會議名稱、時間、

地點、發表之論文題目、補助機構，及後續收錄於期刊或專書之名稱、卷號、頁數、出版日期）。

（四）請分年列述。

第 1 年

出席國際學術會議差旅費

博士生人數共 0 名金額 100,000 元

費用說明預訂於 2011 年前往德國 Oberwolfach 參加會議

預計 12 日

近三年論文發表情形

1. Oberwolfach Conference-Komplex Geometrie, Germany 08/2008 2. Analytic and algebraic aspects of the MMP and higher dimensional classification, Grenoble, France 11/2008

3. Classification of Algebraic Varieties, Schiermonnikoog, Netherlands 05/2009 (proceeding to appear)

4. Arithmetic Geometry and Moduli Spaces in Algebraic Geometry, Hangzhou, China 08/2009

5. Algebraic Geometry on Varieties and Manifolds, Fudan Univ., China 05/2010

6. 5th Pacific Rim Conference on Mathematics, Stanford, USA 06/2010 7. International Congress of Chinese Mathematicians, Beijing, China 12/2010 (proceeding to appear)

第 2 年

費用說明

預訂於 2012 年前往美國參加 Western Algebraic Geometry Symposium 預計 14 日

第 3 年

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博 0 士生人數共名 100,000 金額元

費用說明

預訂於 2014 年 6 月前往法國參加 Complex Algebraic Geometry 相關之會議

預計 14 日

第 4 年

費用說明

預訂於 2014 年 8 月前往韓國參加 ICM 及 Satellite Conference in Algebraic Geometry

預計 14 日

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十一、研究計畫中英文摘要：請就本計畫要點作一概述，並依本計畫性質自訂關鍵詞。

（一）計畫中文摘要。（五百字以內）

於本計畫中，我們打算探討高維度極小模型理論以及相關的雙有理幾何問題。第一年我們希望能夠處理極小模型理論中的基本雙有理映射的分解問題，希望能對三維時的基本雙有理映射有完全具體的描述。第二年我們希望能夠處理四維或更高維的商奇異點。第三年我們希望能夠討論 extremal neighborhood 上的局部不變量，以期進一步探討 flip 的存在性及有限性。於第四年，我們將利用所發展出來的工具與方法應用至各式各樣的雙有理幾何問題上。

表 C011 共頁第頁

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（二）計畫英文摘要。（五百字以內）

In this project, we propose to study various aspects circling around higher dimensional minimal model program and its birational geometry.

In the first year, we will mainly study the classification problem of elementary maps of birational maps in minimal model program. In the second year, we will study quotient terminal singularities in dimension 4 and higher. We shall try to work on local invariants on extremal neighborhood in higher dimensional settings. This is considered to be helpful to the problem of existence and termination of flips. In the fourth year, we will seek for various applications on explicit birational geometry of minimal varieties of the theory we developed in the past three years.

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十二、研究計畫內容：

（一）近五年之研究計畫內容與主要研究成果說明。（連續性計畫應同時檢附上年度研究進度報告）

1. Birational Geometry of threefolds.

Given a variety of general type, it is natural and important to ask when the pluricanonical map is birational. This was previously known only for curves and surfaces to be 3-canonical and 5-canonical respectively. Only quite recently, Hacon and McKernan show that there exist a constant cn, depending only on dimension n, such that the m-th canonical map is birational for m greater than cn. This leads to the boundedness for varieties of general type. However, their result is non-explicit.

The standard approach uses natural geometric fibration such as Albanese map or canonical map.

Therefore the remaining difficult cases are the varieties with small birational invariants that there is no natural fibration structure and their minimal model is singular. In my series of joint work with Meng Chen (cf. [1,2,5]) we obtained a breakthrough on threefolds of general type by the study of three dimensional terminal singularities. We introduced a notion called “packing of baskets of singularities”

(cf. [5]). We also introduce a canonical sequence of prime packings, which is some sort of

approximation of baskets of singularities. The theory of basket we developed led to many important effective results for threefolds of general type. For example, we prove that

Theorem

Let X be a complex projective threefold of general type, then 1. P12>0,

2. P24>1

3. the canonical volume Vol≥ 1/2660

4. m-th canonical map is birational for m ≥73.

The worst known example is the following:

Example

Let X=X46 be a general hypersurface of degree 46 in the weighted projective space P(4,5,6,7,23).

Then Vol=1/420 and 26-th canonical map is not birational.

One sees that the bound we obtained is not only the first explicit bound ever but also not too far from being sharp. We would like to mention that the techniques also applicable to Fano threefolds (cf.

[2]) and other threefolds, for example, weighted complete intersections. In fact, we have the following result for Fanos.

Theorem

Let X be a complex projective weak Q-Fano threefold, then 1 P-6>0,

2 P-8>1

3 the canonical volume Vol≥ 1/330, which is sharp due to the following example.

Example

Let X=X66 be a general hypersurface of degree 66 in the weighted projective space P(1,5,6,22,33).

Then –K³=1/330.

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With the better understand of three dimensional singularities, we the turn our attention to the singularities in birational maps in minimal model program. In [6], we prove that flips and divisorial contraction to a curve can be decomposed into simpler elementary maps by an inductive argument using partial resolution of singularities on extremal rays.

2. Varieties with Kodaira dimension zero.

The classification theory of varieties usually reduced to the study of varieties of the following three types: varieties of general type, varieties with Kodaira dimension zero and varieties with negative Kodaira dimension. For varieties with Kodara dimension zero, it is conjecture by Ueno that they are decomposed into product of abelian varieties and Calabi-Yau up to etale and birational morphisms. In [4], we gave a partial answer to Ueno’s conjecture. In fact, in our recent preprint, we proved the second statement of Ueno’s conjecture. We also have some application to the famous Iitaks’s Conjecture Cnm.

表C012 共頁第頁

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（二）研究計畫之背景及目的。請詳述本研究計畫之背景、目的、重要性及國內外有關本計畫之研究情況、重要參考文獻之評述等。本計畫如為整合型研究計畫之子計畫，請就以上各點分別述明與其他子計畫之相關性。

The most challenging problems in algebraic geometry are the existence of minimal model program and the birational geometry of such minimal models. In the case of dimension less than two, then all of these

problems are well-known. During the year of 80s and early 90s, various results along this direction are obtained. First of all, by Mori’s work (cf. [Mo82]), it is known that one needs to allow at least terminal singularities in order for minimal model program to work. Reid and Mori classified three dimensional terminal singularities as isolated cyclic quotient of compound DuVal singularities (cf. [Mo85, YPG]). Mori then studied extremal neighborhood extensively, and showed that the flip of a small extremal contraction always exists in dimension three (cf. [Mo88, KM92]). Together with Shokurov’s work on termination of flips, this proved the existence of minimal model in dimension three.

This approach turns out to be extremely difficult in higher dimension in many aspects. First of all, there is no known classification for terminal singularities in dimension four or higher. In fact, it’s known that the embedded dimension of four dimension terminal singularities could be arbitrarily large. Therefore, it is difficult to classify singularities as in three-dimensional case. On the other hand, the termination of flips relies on associating to individual singularities invariants, which is also difficult in higher dimensions.

The recent break rough of Birkar, Cascini, Hacon and McKernan [BCHM] and some of the subsequent works proceed in a different approach. The new approach is an inductive approach that reduces the existence of minimal model into a non-vanishing theorem. By using this approach, they successfully showed the existence of minimal model for varieties of log general type. Even though the existence of minimal model is quite promising, very little is known about explicit birational geometry of minimal models even in dimension three. For example, the classification of divisorial contractions to a point is classified by Hayakawa (cf. [HaI, HaII]) and Kawakita (cf. [Kk01], [Kk05]) only quite recently. However, there is almost no result for

divisorial contraction to a curve in dimension three. It is tempting to understand the birational maps explicitly.

Moreover, given a three dimensional terminal singularities, there is a deformation into terminal quotient singularities. The collection of these quotient singularities is called the “basket of singularities”. In [YPG], Reid derived a singular Riemann-Roch formula that the contribution of terminal singularities can be computed by its basket. By using this formula and introduced the concept of “packing of baskets”, we are able to derive many useful inequalities among Euler characteristics. These lead to various effective results of birational geometry of threefolds (cf. [CC1, CC2]). It is then interesting to see whether the singularities between elementary birational in minimal model program are connected by packing of baskets. By the classification of Kawakita and Hayakawa, the singularities between divisorial contraction to a point are connected by a packing. As a by product, we proved a factorization theorem for flips and divisorial contractions to a curve in [CH].

表C012 共 6 頁第 1 頁

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[BCHM] C. Birkar, P. Cascini, C. Hacon, J. McKernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., to appear. arXiv:math/0610203v2

[CC1] J. A. Chen, M. Chen, Explicit birational geometry of threefolds of general type, I, Ann. Sci. \'Ec.

Norm. Sup\'er (4) 43 (2010), no. 3, 365--394.

[CC2] J. A. Chen, M. Chen, Explicit birational geometry of threefolds of general type, II, J. of Diff.

Geom. (to appear). ArXiv: 0810.5044

[CH] J. A. Chen, C.D. Hacon, Factoring 3-fold flips and divisorial contractions to curves. Jour. Reine Angew. Math., to appear, arXiv 0910.4209

[HaI] T. Hayakawa, Blowing ups of 3-dimensional terminal singularities, Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, 515--570.

[HaII] T. Hayakawa, Blowing ups of 3-dimensional terminal singularities. II, Publ. Res. Inst. Math. Sci.

36 (2000), no. 3, 423--456.

[Kk01] M. Kawakita, Divisorial contractions in dimension three which contract divisors to smooth points, Invent. Math. 145 (2001), no. 1, 105--119.

[Kk05] M. Kawakita, Three-fold divisorial contractions to singularities of higher indices, Duke Math. J.

130 (2005), no. 1, 57--126.

[KM92] J. Koll\'ar S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), no. 3, 533--703.

[Mo82] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math. 116 (1982), 133--176.

[Mo85] S. Mori, On 3-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43--66.

[Mo88] S. Mori, Flip theorem and the existence of minimal models for 3-folds, J.Amer. Math. Soc. 1 (1988), no.1, 117--253.

[YPG] M. Reid, Young person's guide to canonical singularities, Proc. Symposia in pure Math. 46 (1987), 345-414.

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（三）研究方法、進行步驟及執行進度。請分年列述：1.本計畫採用之研究方法與原因。2.預計可能遭遇之困難及解決途徑。3.重要儀器之配合使用情形。4.如為整合型研究計畫，請就以上各點分別說明與其他子計畫之相關性。5.如為須赴國外或大陸地區研究，請詳述其必要性以及預期成果等。

（四）預期完成之工作項目、成果及績效。請分年列述：1.預期完成之工作項目。2.對於學術研究、

國家發展及其他應用方面預期之貢獻。3.對於參與之工作人員，預期可獲之訓練。4.預期完成之研究成果及績效（如期刊論文、研討會論文、專書、技術報告、專利或技術移轉等質與量之預期績效）5.本計畫如為整合型研究計畫之子計畫，請就以上各點分別說明與其他子計畫之相關性。

In this project, we propose to study various aspects circling around higher dimensional minimal model program and its birational geometry.

Year One:

In the first year, we will mainly study the classification problem of elementary maps of birational maps in minimal model program. It is known that most of the divisorial contraction to a point can be realized by certain weighted blowup. One might wonder whether all divisorial contractions to a point can be realized by weighted blowups or not. Once we have a better understanding of these maps, we are able to have explicit description of flips, flops and divisorial contraction to a curve in dimension three. An explicit description of these maps might leads to various applications in birational geometry of threefolds.

However, the difficulty is that a weighted blowup depends on embedding of a singularity and also on the choice of weights. Unfortunately, there is no canonical choice of coordinates and neither is canonical choice of weights. A possible way to attack the problem is to study the exceptional divisor and singularities on the exceptional divisor. The exceptional divisor itself is a singular Del Pezzo surface. There are too many Del Pezzo surfaces to be completely classified. We expect that one can at least study Del Pezzo surfaces with given anticanonical volume. This will provide certain information of the exceptional divisor, and hopefully to the weights if it is a weighted blowup.

We will study resolution of Gorenstein terminal singularities in dimension three as well. We expect that there exists a canonical way to construct a canonical sequence of partial resolution such that each map allows us to play the “two-rays game”. This will lead to a factorization of birational maps with only Gorenstein terminal singularities, and hopefully to a factorization of flop as well. We would like to remark that there is a recent result of Kawamata showing the any two birational minimal models are connected by a sequence of flops. Moreover, the quantum cohomology between simple flop can be computed due to the seminar article of Lee, Lin and Wang (cf. [LLW]). Therefore, it would be very interesting to factorize flops into elementary maps and simple flops.

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Year Two:

It is essential to study singularities in minimal model program. In the second year, we will study quotient terminal singularities in dimension 4 and higher. This type of singularities can be considered as toric

singularities and hence are considered to be relatively easy than terminal singularities in general. We are interesting in the singular Riemann-Roch formula that taking the contribution of quotient terminal singularities. We hope that there is a generalized notion of “baskets of singularities” such that there is an explicit form of Riemann-Roch formula. This formula will be extremely useful for various effective and boundedness problem as we did in dimension three. In particular, one can expect to classified quasi-smooth weighted complete intersections in dimension four and higher. This work will be a starting point to provide many examples in higher dimensions.

The difficulty is that singular locus might be of positive dimension. Even though we start with an isolated point, a partial resolution might create a variety with non-isolated singularities. Therefore, it seems that the geometry of the locus must come into play. Moreover, even we have a toric description of a quotient singularity; one can expect to have a resolution by appropriate subdivision of cones. However, there is no canonical way to subdivide a cone and the combinatorics involved could be very complicated. In general, the combinatorics of cones and lattices in dimension four and higher are a lot complicated than those of

dimension three or less.

A possible solution is to consider resolution, i.e. subdivision of cones, guided by Euler characteristics.

Recall that in the lower dimension case, we always have that Euler characteristic increased for each map of the resolution. In other words, the Poincare series increased in some suitable sense. Therefore, one can try to pick a subdivision among all subdivisions with largest Poincare series. This might provide a better way to study the quotient terminal singularities.

Another possible way-out is the choice of weights. Notice that a subdivision of cones corresponds to a weighted blowup. In the case of dimension three, there is a unique choice of weight such that the weighted blowup is an extremal extraction. Now we have many different choices of weight in higher dimensions.

We have one more remark. In dimension three, given a singularity P in X of index r, there is a canonical cover YÆX such that the preimage of P consists of a sinple point Q, which is Gorenstein. In other words, being Gorenstein is the same as having index one in dimension three. However, this is no longer the case in dimension four. Take a quotient singularity 1/5(1,4,2,3) for example, one imagines that it has index 5 but it is Gorenstein. Therefore, being Gorenstein seems not to be a good notion in higher dimension.

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Year Three:

In the third year, we will turn our attention to minimal model program in dimension four or higher. The classification of Mori and Kollar on extremal neighborhood depends on the careful studies of local invariants on extremal neighborhood. We shall try to work on this in higher dimensional settings. The main tasks are to formulate a partial ordering on the set of local invariants and to find a partial resolution of extremal

neighborhood into an extremal neighborhood with controllable singularities. This is considered to be helpful to the problem of existence and termination of flips as what we have done in [CH].

The difficulty is that so far we don’t have good description of the terminal singularities that might involve in a given extremal neighborhood. Therefore, one can not expect to have a complete classification of

extremal neighborhood according the type of singularities contained in it (cf. [Mo88], [KM92]). An even more serious problem is that in the situation of dimension 4 or higher, the singular locus might be positive dimensional. Could it happen that the whole extremal curve is contained in the singular locus? Or can one deform the extremal curve into another one which contained only isolated points?

A possible solution is instead of classifying extremal neighborhood by the type of singularities contained in it, one might be able to classify local invariants. One needs to study the extremal contraction when local invariants are trivial, as people did in threefolds. If all these can be worked out, is there a reasonable way to discuss the possible derived flip and divisorial contractions? A positive answer to the above question will be very useful to the existence and terminal of flips. We hope to at least work this out in dimension four by our explicit understanding of threefolds.

Year Four:

In the fourth year, we will seek for various applications on explicit birational geometry of minimal varieties of the theory we developed in the past three years. The first type of question is the effectiveness of pluricanonical maps. Recall in the case of surfaces, a beautiful theorem of classical Italian school asserts that Theorem

Let X be a complex algebraic surface , then k=-∞ iff P12=0,

k=0 iff P12=1,

k=1 iff P12=2 and K²=0 k=2 iff P12=2 and K²>0

In general with intermediate Kodaira dimension, there is a theoretic bound thanks to the result of Fujino-Mori. Together with Kawamata’s result on threefold with k=0, Viehweg and Zhang’s result on threefolds with k=2 and our previous result on threefolds of general type, the remaining non-explicit case is the case when k=1. By using the technique of explicit birational geometry, one expects to have a similar unified result for threefolds and even fourfolds as in the two-dimensional case.

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Another possible application is some explicit aspects of Fano-type varieties. For example, it is known that weak Q-Fano threefold is bounded. It is known that the lower bound of anticanonical volume is 1/330, which is sharp. The known example with highest anticanonical volume is 72. However, it is not known whether 72 is optimal or not. We expect that our explicit description of birational map will be helpful to determine the upper bound of anticanonical volume. We hope that our work in the first three years will make it possible to study Fano 4-folds. .

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十三、近三年內執行之研究計畫

（請務必填寫近三年所有研究計畫）

計畫名稱

（本會補助者請註明編號）

計畫內擔

任之工作起迄年月補助或委託機構執行情形經費總額

數統學門數學組研究發展及推動計畫

(100-2114-M-002-001-)

主持人 2011/1/1 ~ 2011/12/31

行政院國家科學委員會

執行中 238,000

以代數幾何方法研究控制理論及相關問題

(99-2221-E-019-003-)

共同主持人 2010/8/1 ~ 2011/7/31

執行中 614,000

中華民國數學會學術研究發展計畫

(99-2114-M-310-001-)

主持人 2010/1/1 ~ 2010/12/31

執行中 2,000,000

(99-2114-M-002-002-)

主持人 2010/1/1 ~ 2010/12/31

執行中 238,000

中華民國數學會期刊出版及學術推廣業務計畫

(98-2114-M-310-001-)

主持人 2009/1/1 ~ 2009/12/31

已結案 1,600,000

(98-2114-M-002-005-)

主持人 2009/1/1 ~ 2009/12/31

已結案 238,000

高維流形中的奇點之研究 (97-2115-M-002-013-MY3)

主持人 2008/8/1 ~ 2011/7/31

執行中 2,468,000 中華民國數學會學術研究發

展計畫

(97-2114-M-310-001-)

主持人 2008/1/1 ~ 2008/12/31

經費報銷審核中，報告

已繳

1,600,000

高維流形中的奇點之研究 (97-2115-M-002-013-MY3)

主持人 2008/8/1 ~ 2011/7/31

執行中 2,468,000 合計 11,464,000

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100 年度自然處專題計畫主持人近五年研究成果

（修正：99/10/07）

姓名：職稱：服務機關系所：

一、近五年內(2006/1/1～2010/12/31)已出版之最具代表性研究成果至多六篇，擇其中五篇電子檔上傳。

(請依序填寫：姓名,著作名稱,發表年份,期刊,卷數,頁次,IF,▲：被引用次數，並以＊號註記該篇所有的通訊作者)

1. J. A. Chen, M. Chen, D.Q. Zhang, The 5-canonical system on 3-folds of general type, J. Reine Angew.

Math., 603, (2007), 165-181.

2.

J. A. Chen, M. Chen, An optimal boundedness on weak $\bQ$-Fano threefolds, Adv. Math., 219, (2008), 2086-2104.

3. F. Campana, J. A. Chen, T. Peternell, On strictly nef divisors, Math. Ann., 342, (2008), 565-585 4. J. A. Chen, C. D. Hacon, On Ueno’s Conjecture K, Math. Ann., 345, (2009), 287-296. arXiv

0802.1060.

5. J. A. Chen, M. Chen, Explicit birational geometry of threefolds of general type I, Ann. Sci. Ecole Norm. Sup., (43) 2010, 365-394.. arXiv 0810.5041

6. J. A. Chen, C.D. Hacon, Factoring $3$-fold flips and divisorial contractions to curves. Jour. Reine Angew. Math., to appear, arXiv 0910.4209

二、近五年內研究成果統計表（請務必更新個人資料表 C302-C303，未來審查

時將以該表之內容為準）

2006 2007 2008 2009 2010 以上合計

2010 已接受但 2011 以後出版

統計類別 者

總篇數 IF 總和總篇數 IF 總和總篇數 IF 總和總篇數 IF 總和總篇數 IF 總和總篇數 IF 總和總篇數 IF 總和

SCI 期刊論文

（含共同作者）

1 0.443 3 2.133 3 2.845 2 1.51 1 1.109 10 8.037 3 3.012

SCI 期刊論文

（限通訊作者）

僅物理學門填寫 SCI 期刊論文（限 非屬通訊之第一作者）

說明：

1. SCI (Science Citation Index) 之Impact Factor 係以 2009 年版本為準。請至有購買 Journal Citation Reports on the Web（JCR Web）資料庫之各大學圖書館或財團法人國家實驗研究院科技政策研究與資訊中心(http://cdnet.stpi.org.tw/db_search/01_isi.htm)進行查詢。

2. IF 總和：係該年度論文所刊載期刊之 Impact Factor 總和。

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1. J.A. Chen, C. D. Hacon, An example of a surface of general type with pg=q=2 and K_X²=5, Pacific Jour.

Math., 223, (2006), 219-228.

2. J. A. Chen, M. Chen, D.Q. Zhang, The 5-canonical system on 3-folds of general type, J. Reine Angew.

Math., 603, (2007), 165-181.

3. J.A. Chen, C.D. Hacon, Pluricanonical systems on irregular 3-folds of general type, Math. Zeit, 255, (2007),343-355.

4 J. A. Chen, M. Chen, On projective threefolds of general type. Elec. Res. Announc. Math. Sci., 14, (2007), 69-73.

5. F. Campana, J. A. Chen, T. Peternell, On strictly nef divisors, Math. Ann., 342, (2008), 565-585.

6. J. A. Chen, M. Chen, An optimal boundedness on weak $\bQ$-Fano threefolds, Adv. Math., 219, (2008), 2086-2104.

7. J.A. Chen, M. Chen, The canonical volume of threefolds of general type with X<1, J. London Math.

Soc., 78, (2008), 693-706. arXiv 0704.1702

8. J. A. Chen, C. D. Hacon, On the geography of threefolds of general type, J. Alg., 321, (2009), 2500-2507. arXiv 0802.0884.

9. J. A. Chen, C. D. Hacon, On Ueno’s Conjecture K, Math. Ann., 345, (2009), 287-296. arXiv 0802.1060.

10. J. A. Chen, M. Chen, Explicit birational geometry of threefolds of general type, I, Ann. Sci. École Norm. Sup. (43) 2010, 365-394. arXiv 0810.5041

11. J. A. Chen, M. Chen, Explicit birational geometry of threefolds of general type, II, Jour.Diff. Geom, to appear, arXiv 0810.5044

12. J Chen, J. A. Chen, M. Chen, On quasismooth weighted complete intersections, Jour. Alg. Geom, to appear. arXiv 0908.1439

13. J. A. Chen, C. D. Hacon, Factoring 3-fold flips and divisorial contractions to curves. Jour. Reine Angew. Math., to appear, arXiv 0910.4209

14. J. A. Chen, C. D. Hacon, Kodaira dimension of irregular varieties, arXiv 1008.2404

三、近五年內獲獎情形及重要會議邀請演講至多五項。

1. 中華民國數學會青年數學家獎 2007 2. 國科會傑出研究獎 2009-2012

3. 世界華人數學家大會晨興數學獎銀獎 2010 4. 中華民國數學年會大會主講 Plenary speaker 2010 5. 世界華人數學家大會大會主講 Plenary speaker 2010

四、近五年內其他資料：擔任國際重要學術學會理監事、國際知名學術期刊編輯或評審委員等。

Referee for journals include: Duke Math Jour, Comp. Math., Math. Ann., Jour. Alg Geom., Nagoya Math. Jour., Math. Zeit., etc.

1

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五、請簡述上述代表性研究成果之個人重要貢獻（至多一頁）。

1. Birational Geometry of threefolds.

Given a variety of general type, it is natural and important to ask when the pluricanonical map is birational. This was previously known only for curves and surfaces to be 3-canonical and 5-canonical respectively. Only quite recently, Hacon and McKernan show that there exist a constant cn, depending only on dimension n, such that the m-th canonical map is birational for m greater than cn. This leads to the boundedness for varieties of general type. However, their result is non-explicit.

The standard approach uses natural geometric fibration such as Albanese map or canonical map.

Therefore the remaining difficult cases are the varieties with small birational invariants that there is no natural fibration structure and their minimal model is singular. In my series of joint work with Meng Chen (cf. [1,2,5]) we obtained a breakthrough on threefolds of general type by the study of three dimensional terminal singularities. We introduced a notion called “packing of baskets of singularities”

(cf. [5]). We also introduce a canonical sequence of prime packings, which is some sort of

approximation of baskets of singularities. The theory of basket we developed led to many important effective results for threefolds of general type. For example, we prove that P12>0, P24>1 and m-th canonical map is birational for m ≥73.

We would like to mentioned that the techniques also applicable to Fano threefolds (cf. [2]) and other threefolds, for example, weighted complete intersections.

With the better understand of three dimensional singularities, we the turn our attention to the singularities in birational maps in minimal model program. In [6], we prove that flips and divisorial contraction to a curve can be decomposed into simpler elementary maps by an inductive argument using partial resolution of singularities on extremal rays.

2. Varieties with Kodaira dimension zero.

The classification theory of varieties usually reduced to the study of varieties of the following three types: varieties of general type, varieties with Kodaira dimension zero and varieties with negative Kodaira dimension. For varieties with Kodara dimension zero, it is conjecture by Ueno that they are decomposed into product of abelian varieties and Calabi-Yau up to etale and birational morphisms. In [4], we gave a partial answer to Ueno’s conjecture. In fact, in our recent preprint, we proved the second statement of Ueno’s conjecture together with some application to the famous Iitaks’s Conjecture Cnm.

2

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行政院國家科學委員會專題研究計畫申請書

一、基本資料：申請條碼：94WFA0102432

94WFA0102432

本申請案所需經費 ( 單選 ) A 類(研究主持費及執行計畫所須經費)

計畫類別 ( 單選 ) 一般型研究計畫

研究型別個別型計畫

計畫歸屬自然處

申請機構 / 系所（單位）國立臺灣大學數學系暨研究所

本計畫主持人姓名陳榮凱職稱教授身分證號碼 *******681 中文代數流形的雙有理分類

本計畫名稱

英文 Birational Classification of Algebraic Varieties

整合型總計畫名稱

整合型總計畫主持人身分證號碼

全程執行期限自民國 95 年 08 月 01 日起至民國 98 年 07 月 31 日

學門代碼名稱(如為其他類，請自行填寫學門) 研究學門(請參考本申請書所

附之學門專長分類表填寫 )

M2 數學

研究性質基礎研究

本年度申請主持國科會各類研究計畫(含預核案)共 1 件。(共同主持之計畫不予計入) 本件在本年度所申請之計畫中優先順序(不得重複)為第 1 。

本計畫是否為國際合作計畫：否本計畫是否申請海洋研究船：否

本計畫是否有進行下列實驗：（勾選下列任一項，須附相關實驗之同意文件）

計畫連絡人姓名：陳榮凱電話：(公) (02)33662848 (宅/手機) 0927576278

通訊地址台北巿羅斯福路四段 1 號國立臺灣大學數學系暨研究所

傳真號碼 (02)23914439 E-MAIL：jkchen@math.ntu.edu.tw

申請人簽章：單位系所主管簽章：執行機關簽章：

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二、申請補助經費：

（一）請將本計畫申請書之第四項(表 C004)、第五項(表 C005)、第六項(表 C006)、第七項(表 C007)、

第八項(表 C008)所列費用個別加總後，分別填入「研究人力費」、「耗材、物品及雜項費用」、

「研究設備費」、「國外或大陸地區差旅費」及「出席國際學術會議差旅費」欄內。

（二）若有申請國際合作研究計畫費用者，請將表 I002 之「C 類經費合計」欄金額填入「國際合

作研究計畫國外學者來臺費用」欄內，「A 類經費與 B 類經費合計」欄金額填入「國際合作

研究計畫出國差旅費」欄內。

（三）管理費為申請機構配合執行本計畫所需之費用，其計算方式係依本會規定核給補助管理費之項目費用總和及各申請機構管理費補助比例計算後直接產生，申請人不須填寫「管理費」欄。

（四）「貴重儀器中心使用額度」係將第九項(表 C009)所列使用費用合計數填入。

（五）請依各年度申請博士後研究之名額填入下表。

（六）申請機構或其他單位（含產業界）提供之配合項目，請檢附相關證明文件。

金額單位：新台幣元執行年次

補助項目

第一年 (95 年 8 月~96

年 7 月)

第二年 (96 年 8 月~97

年 7 月)

第三年 (97 年 8 月~98

年 7 月)

第四年第五年

業務費 946,000 946,000 946,000

研究人力費 816,000 816,000 816,000

耗材及雜項費用 130,000 130,000 130,000 國際合作研究計畫

國外學者來臺費用 0 0 0

研究設備費 90,000 0 40,000 國外差旅費 150,000 150,000 150,000 國外或大陸地區差旅費 100,000 100,000 100,000 出席國際學術會議差旅費 50,000 50,000 50,000

國際合作研究計畫

出國差旅費 0 0 0

管理費 155,400 141,900 147,900 合計 1,341,400 1,237,900 1,283,900

貴重儀器中心使用額度 0 0 0

國內、外

地區共 1 名共 1 名共 1 名共名共名

博士後研究

大陸地區共 0 名共 0 名共 0 名共名共名

申請機構或其他單位（含產業界）提供之配合項目（無配合補助項目者免填）

配合單位名稱配合補助項目配合補助金額配合年次證明文件

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五、耗材及雜項費用：

（一）凡執行研究計畫所需之耗材及雜項費用，均可填入本表內。

（二）說明欄請就該項目之規格、用途等相關資料詳細填寫，以利審查。

（三）若申請單位有配合款，請於備註欄註明。

（四）請分年列述。

第 1 年金額單位：新台幣元

雜支影印、文具、資料檢索、

國內差旅等。

1 130,000 130,000

合計 130,000

第2年

國內差旅等。

1 130,000 130,000

合計 130,000

第3年

國內差旅等。

1 130,000 130,000

合計 130,000

(29)

六、研究設備費：

（一）凡執行研究計畫所需單價在新台幣一萬元以上且使用年限在二年以上之各項儀器、機械及資訊設備（含各項電腦設施、周邊設備及套裝軟體、程式設計費）等之購置、裝置費用及圖書館典藏之分類圖書等屬之，此項設備之採購，以與本研究計畫直接有關者為限。

各類研究設備金額請於金額欄內分別列出小計金額。

（二）購置設備單價在新臺幣二十萬元以上者，須檢附估價單。

（三）若申請機構及其他機構有提供配合款，請務必註明提供配合款之機構及金額。

（四）儀器設備單價超過六十萬元(含)以上者，請詳述本項設備之規格與功能(諸如靈敏度、精確度…等)，其他重要特性與重要附件，以及申購本設備對計畫執行之必要性。本項設備若獲補助，主持人應負維護保養之責，並且在不妨礙個人研究計畫或研究群計畫之工作下，同意提供他人共同使用，以避免設備閒置。

第 1 年金額單位：新台幣元經費來源

類別設備名稱

經費需求

提供配合款之機構名稱及金額儀器及資

訊設備

筆記型電腦及週邊配件/

筆記型電腦 ~70000 週邊配件 ~20000

1 90,000 90,000 90,000

合計 _90,000 90,000 第 2 年

經費來源

類別設備名稱

經費需求

提供配合款之機構名稱及金額

合計第 3 年

經費來源

類別設備名稱

經費需求

提供配合款之機構名稱及金額儀器及資

訊設備

電腦設備局部更新/

工作主機升級

~20000

列印單元升級

~15000

其他附件升級

~5000

1 40,000 40,000 40,000

合計 _40,000 40,000

(30)

七、赴國外或大陸地區差旅費：

（一）類別分為「實驗」、「研究」、「田野調查」等。

（二）請詳述預定各出國人員之出國行程、預估經費、天數及地點。

（三）生活費、機票費及其他費用之標準，請依照行政院頒布之「中央各機關（含事業機構）派赴國外進修、研究、實習人員補助項目及數額表」規定填列（網址 http://web.nsc.gov.tw/public/data/471214491471.doc）。

（四）請將所列各項費用換算為新台幣，並註明估算匯率。

第 1 年金額單位：新台幣元申請補助費用

經費類別預估經費

詳述預定各出國人員之類別、出國行程、預估經費、天數及地點

(類別包括實驗、研究、田野調查)

赴國外 100,000

擬利用寒暑假期間前往國外訪問研究 1 至 4 週。

可能地點有美國、德國、日本、法國等。

赴大陸 0

合計 100,000

第2年

申請補助費用

經費類別預估經費

詳述預定各出國人員之類別、出國行程、預估經費、天數及地點

(類別包括實驗、研究、田野調查)

赴國外 100,000

擬利用寒暑假期間前往國外訪問研究 1 至 4 週。

可能地點有美國、德國、日本、法國等。

赴大陸 0

合計 100,000

第3年

申請補助費用

赴大陸 0

(31)

我們計畫利用未來三年時間探討代數流形的雙有理分類理論。主要圍繞在以下的三個子計畫：1. k=0 的代數流形的結構。 2. 代數叢流形的結構。3.弧空間的幾何與奇點。

這個計畫的成功將對雙有理分類理論有具體且基本的貢獻。

(32)

In this three-years project, we are going to investigate birational classification theory of algebraic varieties. It will consist of the following three related subprojects: 1. Structure of varieties with k=0. 2. Algebraic fiber spaces. 3. Geometry of space of arcs and singularities.

We hope that people will better understanding of birational classification theory once we succeed in this project.

(33)

（一）近五年之研究計畫內容與主要研究成果說明。（連續性計畫應同時檢附上年度研究進度報告）

The classification of algebraic varieties is one of the main theme in algebraic geometry. Our research was mainly around the birational classification problem. From Iitaka’s point of view, the building blocks of varieties consists of the following three categories: varieties of general type, varieties with k=0, and varieties with k=-∞.

Varieties of general type.

It’s well-known that mKX defines a birational map for m sufficiently large. It’s natural to ask whether one can determine m effectively. For dim X =1,2, these bounds are known to be 3 and 5 respectively.

There is a recent work by Hacon and McKernan asserts that there is a theoretical bound m(n) depends on dim X =n. However, the number m(n) is far from being optimal in any way. On my joint work with Meng Chen and DeQi Zhang, we obtained that 5K is birational for minimal Gorenstein threefolds with at worst canonical singularities. This bound is optimal. In a joint previous joint work with Hacon, we show that 7KX is birational for irregular threefolds. Our method can also be applied to certain higher dimensional irregular varieties.

Vareities with k=0.

In a series of joint work with Hacon, we first characterize abelian varieties as the varieties with P2=1 and q= dim X. And then we can also characterize varieties with q= dim X and Pm is small. Basically, these are going to be certain covering over abelian varieties. We can completely describe varieties with P3 ≤ 4 and q= dim X.

Miscellaneous Results.

We have the following results which are also related to classification theory.

1. Irregularity of image of Iitaka fibration.

Iitaka fibration is obtained by pluricanonical maps. In general, there is no good control of the geometry of its image except dimension. We can determine the irregularity of the image via the cohomological support loci of the original variety.

2. Effective non-vanishing on surface.

It’s a conjecture by Kawamata that if D is nef and D-K is nef and big. Then D is effective. We prove a non-vanishing theorem for Q-divisors on surface.

3. Strictly nef divisors.

A strictly nef divisor is a divisor such that C.D > 0 for all curves C. A conjecture of Serrano asserts that K+tD is ample for t>dimX+1. In a joint work with Peternell and Campana, we verified this conjecture for threefolds unless one exception.

表C012 共頁第頁

二、申請補助經費： （一）請將本計畫申請書之第四項

100 年度自然處專題計畫主持人近五年研究成果

一、近五年內(2006/1/1～2010/12/31)已出版之最具代表性研究成果至多六篇，擇 其中五篇電子檔上傳 。

二、近五年內研究成果統計表（請務必更新個人資料表 C302-C303，未來審查

時將以該表之內容為準）

1 0.443 3 2.133 3 2.845 2 1.51 1 1.109 10 8.037 3 3.012

三、近五年內獲獎情形及重要會議邀請演講至多五項。

四、近五年內其他資料：擔任國際重要學術學會理監事、國際知名學術期刊編輯或 評審委員等。

五、請簡述上述代表性研究成果之個人重要貢獻（至多一頁） 。

*94WFA0102432*

二、申請補助經費：（一）請將本計畫申請書之第四項

一、近五年內(2006/1/1～2010/12/31)已出版之最具代表性研究成果至多六篇，擇其中五篇電子檔上傳。

四、近五年內其他資料：擔任國際重要學術學會理監事、國際知名學術期刊編輯或評審委員等。

五、請簡述上述代表性研究成果之個人重要貢獻（至多一頁）。

94WFA0102432