國立成功大學 應用數學研究所 碩士論文 格拉斯曼流形、五次三維流形以及相交理論 Grassmannians, quintic threefolds, and intersection theory 研 究 生：宋政蒲 指導教授：章源慶 中 華 民 國 一 ○ 二 年 六 月 摘要 在這篇論文裡，我們將會探討一些格拉斯曼流形與五次三維流形的 相交理論。我們也將介紹格羅莫夫-威騰不變量，以及利用大量子積的結 合性，推導康采維奇的著名公式：計算出通過(3d-1)個一般點的 d 次平面 有理曲線的數目。 關鍵字：格拉斯曼流形、五次三維流形、相交理論、穩定映射、 格羅莫夫-威騰不變量、康采維奇公式。 ABSTRACT . In this thesis, we will investigate the intersection theory of some Grassmannians and the quintic threefolds. We will also introduce the notion of Gromov-Witten invariants, and derive Kontsevich's celebrated formula for the number of plane rational curves of degree d passing through 3d 1 general points from the associativity of the big quantum product. − Keywords: Grassmannian, quintic threefold, intersection theory, stable map, Gromov-Witten invariant, Kontsevich's formula. 誌謝 首先要感謝家人，老爸不斷督促我，老媽也持續地給我鼓勵，讓我度過不少難 關，兩個哥哥雖然因不認同我的想法和做法，而給了我很多壓力，但我仍很感謝他們 讓我很有家的感覺，且因他們讓家裡有穩定的收入，我才能無後顧之憂地做研究。 還有興大保育社、成大野鳥社、墾丁解說志工認識的朋友，讓我在這三年可以到 處遊歷賞鳥，讓我在研究和學習遇到瓶頸時，能夠藉著大自然來抒發壓力和尋找靈感， 讓我的碩士生涯不是苦悶的三年，而是最充實的三年。特別感謝傻剛、婉瑄和國權， 陪我到處賞鳥到處遊玩，還常常討論許多我們關心的時事，也給了我很多觀點。 接下來是在應數所上認識的夥伴，416 的大家、學長們和學弟，常常舉辦賽局理 論研討會，讓我在學習和研究之間得以喘息。而他們也常常給我很多鼓勵和意見。特 別感謝孫維良，總是可以從他那邊聽到很多數學上有趣的事情，也是少數會和我談論 數學的人，同時在學術上，他也是我的楷模。另外還有黛玉，在我趕著論文，神經緊 繃而沮喪的時候，用簡短的一句話化解了我的緊張。以及呂庭維，在口試當天也幫我 跑東跑西的，還幫我煮了一壺讓夏杼教授讚譽有加的咖啡。 再來要感謝夏杼老師和陳若淳老師，同時也是我的口試委員。在口試前總是用玩 笑的口吻，跟我交談，減輕了我口試前的緊張。而在口試結束後也給予我許多鼓勵和 肯定，讓我能下更多的決心走向研究路線。 最後便是我的指導教授－章源慶老師，很感謝他在我一開始想請他當指導教授時， 沒多說甚麼就收了我當學生，也時常給予我指導，而不論我問甚麼樣的問題，總是會 告訴我線索或者相關書籍，讓我不至於沒有方向。在研究上也沒有給我太大的壓力， 且常常會鼓勵我或者是在我因瓶頸而沮喪時安慰我。非常感謝老師這兩年多的照顧！ 最後，要謝謝的人太多了，只能謝天又謝地了。 CONTENTS 1. The Grassmannians G 2; n 1 ::::::::::::::::::::::::::::::::::: 7 ( + ) 2. Quintic Threefold :::::::::::::::::::::::::::::::::::::::::::: 12 3. Lines On the Quintic Threefold ::::::::::::::::::::::::::::::::::: 13 4. Rational Curves of Degree d 2 On the Quintic Threefold ::::::::::::::::::: 16 ≥ 5. Gromov-Witten Invariant ::::::::::::::::::::::::::::::::::::::: 24 6. Kontsevich's Formula for Genus Zero Curves on P2 ::::::::::::::::::::::: 27 Bibliography ::::::::::::::::::::::::::::::::::::::::::::::::: 31 1. The Grassmannians G(2; n + 1) 7 1. THE GRASSMANNIANS G 2; n 1 ( + ) We would like to study the enumerative geometry of G 2; n 1 , the set of 2-dimensional subspaces Cn+1 Pn of , which is also the set of lines of the n dimensional( projection+ ) space . A typical chart of G 2; n 1 is ( + ) 1 0 a11 a1 n 1 ( − ) : 0 1 a a ⎛ 21 ⋯ 2(n−1) ⎞ ⎜ ⎟ So the dimension of G 2; n 1 is 2 ⎝n 1 . ⋯ ⎠ ( + ) ( − ) Cn+1 Now we fix a basis e1; : : : ; en+1 for , and let Vi be the linear space of e1; : : : ; ei . Then we have a collection{ of subspaces} { } −−⇀ Cn+1 0 V0 V1 Vn+1 ; which is called a flag V . { } = ⊂ ⊂ ⋯ ⊂ = Definition 1. For a flag V , we define the Schubert cycle σa;b n 1 a b 0 by P ( P− ≥ ≥ ≥ ) σa;b V ` G 2; n 1 ` Vn−a ; ` Vn+1−b : ( ) = { ∈ ( + ) T ∩ ( ) ≠ ∅ ⊂ ( )} When b is zero, we simply write σa;0 as σa. A chart of σa;b is given by 1 a11 a12 a1 n a 1 0 0 ( − − ) ; 0 1 a a 0 0 ⎛ 21 ⋯ 1(n+⋯1−b−2) ⋯ ⋯ ⎞ ⎜ ⎟ so the dimension of σ⎝a;b is n a 1⋯n ⋯1 b ⋯2 2n a b 2. ⋯ ⎠ ( − − ) + ( + − − ) = − − − 1. The Grassmannians G(2; n + 1) 8 The σa;b V is the closure of cellular decomposition , has codimension 2 n 1 2n a b 2 a b, 2 a b and σa;b V( ) H ( + ) G 2; n 1 . ( − )−( − − − ) = + The[ total( )] cohomology∈ ( (H∗ G+ 2)); n 1 is spanned by σa;b V n 1 a b 0 . Furthermore, if V ′ is another( ( flag,+ there)) exists a deforming{[ ( map)]TV− V≤ ′,≤ we≤ have} ′ → σa;b V σa;b V σa;b: [ ( )] = [ ( )] = Now we are ready to calculate σa;b σc;d. G(2;n+1) Because the dimension of G 2; nS 1 is 2 n ⋅ 1 , σa;b σc;d is not zero unless G(2;n+1) ( + ) ( − ) ⋅ c d 2 n 1 S a b : 1 ;+ if= c ( n− 1) −b( ; d+ )n 1 a: Theorem 1. σa;b σc;d G(2;n+1) ⎧ 0 ; otherwise: ⎪ = − − = − − S ⋅ = ⎨ Proof. ⎪ ⎩⎪ The number of σa;b σc;d is equal to G(2;n+1) S ⋅ ′ σa;b V σc;d V ; G(2;n+1) ′ S [ ( ) ∩ ′ ( )] where Vi intersect Vj transversely. (i.e. if i j n 2;Vi Vj .) Note that + < + ∩ = ∅ P P σa;b V ` G 2; n 1 ` Vn−a ; ` Vn+1−b ; σ (V ′) = {`∈ G(2; n+ 1) T `∩ P(V ′ ) ≠ ∅ ; ` ⊂ P(V ′ )}: c;d n−c n+1−d For a line L σ V ( σ) = {V ′∈ , L( must+ contain) T ∩ ( points) ≠p∅ P ⊂V ( and )}q P V ′ . a;b c;d n−a n−c Because L is in P V , the point q contained in P V ′ has to be in P V ∈ ( n)+∩1−b ( ) n−∈c ( ) ∈ n(+1−b ) Therefore, q P V ′ P V . Similarly, p P V P V ′ . (n−c ) n+1−b n(−a ) n+1−d ( ) Then P V ′ P V and P V P V ′ are not empty n−c∈ ( n)+1∩−b ( ) n−a n+1∈−d ( ) ∩ ( ) ( ) ∩ ( ) ( ) ∩ ( ) P V ′ P Vn 1 b n c n 1 b n 2 n 1 b c n−c + − : P V P V ′ n a n 1 d n 2 n 1 a d ( n−a) ∩ ( n+1−d) ≠ ∅ ⇒ − + + − ≥ + − − ≥ ⇒ ⇒ However, c d(must) ∩ equal( to 2 )n≠ ∅1 ⇒ a −b .+ Therefore,+ − ≥c +n 1 b and− −d ≥n 1 a. + ( − ) − ( + ) = − − = − − 1. The Grassmannians G(2; n + 1) 9 So we know that σa;b ; σc;d 0 unless c is equal to n 1 b and d is equal to n 1 a. How many lines are⟨ in σa;b ⟩V= σc;d V ′ ? The question− is− easy, there is a unique− line− containing both p and q. ( ) ∩ ( ) ∗ The dual basis to σa;b , the basis of H G 2; n 1 , is σn−b−1;n−a−1 . For a cohomology g H2d G 2; n 1 , g a σ , if we want to find a , we can calculate { } ( ( + ))i d;d−i{ } i d 0≤i≤ 2 [ ] ∈ ( ( + )) [ ] = Q g σn−1−(d−i);n−1−d. G(2;n+1) STheorem[ ] 2.⋅ Pieri's formula for G 2; n 1 , σa σb;c( + ) σd;e : {d+e=a+b+c;n−1≥d≥b≥e≥c} ⋅ = Q Proof. Because the codimension of σa σb;c is a b c, then σa σb;c can be written as ⋅ +m+d;e σd;e : ⋅ d+e=a+b+c Furthermore, Q md;e σa σb;c ; σn−1−d;n−1−b σa σb;c σn−1−e;n−1−d G(2;n+1) = ⟨ ⋅ ⟩ = ( ⋅ ) ⋅ ′ ∗ S σa V σb;c V σn−1−e;n−1−d V ; G(2;n+1) ′ ∗ = [ ( ) ∩ ( ) ∩ ( )] where Vi ;Vj ;Vk intersect transversely. S Suppose md;e is not zero, σ V ′ ` G 2; n 1 ` P V ′ ; ` P V ′ b;c n−b n+1−c σ ( ) =V{∗ ∈ (` G+2;) nT ∩1 (` P)V≠∗∅ ⊂ ; `( P V)}∗ n−1−e;n−1−d 1+e 2+d ` P V ′ ,L P V ∗ T n−b (2+d ) = { ∈ ( + ) ∩ ( ) ≠ ∅ ⊂ ( )} P V ′ P V ∗ n b 2 d n 2 d b (1). ∵ ∩ n(−b ) ≠ ∅2+d ⊂ ( ) ` P V ∗ ,L P V ′ ⇒ ( 1)+e∩ ( ) ≠ ∅ ⇒n+1−c− + + ≥ + ⇒ ≥ P V ∗ P V ′ 1 e n 1 c n 2 e c (2). ∵ ∩ 1(+e ) ≠ ∅n+1−⊂c ( ) Now we consider σ V , if line L is also contained to σ V , L must contain point s in P V . ⇒ ( ) ∩ ( a ) ≠ ∅ ⇒ + + + − ≥ + ⇒a ≥ n−a Here P V and P V ′ P V ∗ intersect transversely. n−a n(+1−)c 2+d ( ) ( ) If σ V σ V ′ σ V ∗ is nonempty set, then there exists a line L, L P V a( ) b;c ( n−1)−e;n∩ −1(−d ) n−a and L P V ′ P V ∗ . ( ) ∩n+1−c( ) ∩ 2+d ( ) ∩ ( ) ≠ ∅ ⊂ ( ) ∩ ( ) 1. The Grassmannians G(2; n + 1) 10 Therefore, the intersection P V and P V ′ P V ∗ , the dimension is n−a n+1−c 2+d n( 1 )c 2( d )n∩ 1( d) c 2; is nonempty. ( + − ) + ( + ) − ( + ) = − + So, d c 2 n a is greater than n 1. d c a 0 d c d e b c 0 b e (3). In conclusion,− + + from− (1)(2)(3) we can+ see⇒ that−b; c;− d;≥ e are⇒ related− − ( by+ − − ) ≥ ⇒ ≥ n 1 d b e c 0 : Thus we get a necessary condition of m−d;e ≥0,≥ now≥ we≥ calculate≥ md;e. ≠ Since P V ′ and P V ∗ intersect transversely, we can choose a local coordinate x ; x ; : : : ; x n+1−c 2+d 1 2 n+1 such that the points of P V ′ are represented by ( ) ( n+1)−c ( ) ( ) x1; x2;:::;:::;:::;xn+1−c; 0;:::; 0 and the points of P V ∗ are represented( by ) 2+d ( ) 0;:::; 0; yn−d;:::;:::;:::;yn; yn+1 : The points of P V ′ P (V ∗ can be represented by ) n+1−c 2+d ( ) ∩ ( ) 0;:::; 0; zn−d; : : : ; zn+1−c; 0;:::; 0 : A line L in σ V ′ σ ( V ∗ must contain a point p in) P V ′ and a point q in P V ∗ . b;c n−1−e;n−1−d n−b 1+e Therefore, the( line)∩L can be constructed( ) by ( ) ( ) a11 a1 n b 0 0 ::: 0 0 ( − ) : 0 0 ::: 0 0 a a ⎛ ⋯ 2(n+1−e) ⋯ 2(n+1) ⎞ ⎜ ⎟ But L is in P V ′ P V ∗ which is locally represented by n+1−c ⎝ 2+⋯d ⋯ ⎠ ( ) ∩ ( ) 0;:::; 0; zn−d; : : : ; zn+1−c; 0;:::; 0 : So the line L can be constructed( by ) 0 0 zn−d zn−b 0 0 0 0 0 0 0 0 z z 0 0 ⎛ ⋯ ⋯ ⋯ n⋯+1−e ⋯n ⋯+1−c ⋯ ⎞ ⎜ ⎟ ⎝ n⋯ d 1⋯zeros ⋯ ⋯ ⋯b e zeros ⋯ c⋯zeros⎠ ↑ ( − − ) ↑ ( − ) ↑ 1. The Grassmannians G(2; n + 1) 11 Look at the above matrix, the space it constructs has n d 1 b e c n a 1 zeros, and intersects P V only in one dimension n−a ( − − ) + ( − ) + = − − That means the point s contained in P V and the line L is unique.