---
# System prepended metadata

title: 'NCTS 2022 USRP: Toric Varieties and WCIs'

---

NCTS 2022 USRP: Toric Varieties and WCIs
===
[Class Record](https://drive.google.com/file/d/1p5eaGzohU9ZgBQoiqwWSGa5IkgpKgPMX/view?usp=sharing) : 2022.7.15 

Goal
---
* Learn basic knowledge of toric varieties and weighted complete intersections
* A warm-up reading and a term project will be assigned for students to investigate
* Each participant will present one of the selected projects in the end of the program


Plan
---
1. Introduction to toric varies (construction, divisors, intersections)
1. Quotient singularities (and their resolutions) using the toric description 
1. Weighted projective spaces and weighted hypersurfaces
1. Weighted complete intersections
1. Research problems and further development

Instructor
---
[Jungkai Alfred Chen](http://www.math.ntu.edu.tw/~jkchen/), Professor, NTU/NCTS
[Ching-Jui Lai](https://sites.google.com/site/cjlai72/home?authuser=1), Associate Professor, NCKU
[Jheng-Jie Chen](http://w2.math.ncu.edu.tw/member/full/59), Assistant Professor, NCU

Teaching Assistant
---
[Yen-An Chen](https://www.math.utah.edu/~yachen/), Postdoc, NCTS
Shih-Hsin Wang, PhD, the University of Utah

Lecture (10:30 AM~12:30 PM)
---
### Week 1
* [Lecture 1 (7/04) | Affine toric varieties](https://drive.google.com/file/d/1gFoFHO-kBu5CcZfItTI2NQaLUL-pMfhG/view?usp=sharing) (Ray, [video](https://drive.google.com/file/d/1F3IN0zTv5ClMNbzHjp0f983ZHKkg-woc/view?usp=sharing))
* [Lecture 2 (7/06) | Fan](https://drive.google.com/file/d/1OMHurwKPtixfTnOuy9VZ_3EWQpEM43k9/view?usp=sharing) (Ray, [video](https://drive.google.com/file/d/1G10SebwHWSVhTC7ZbBEoPSVSJxZ5P6Z1/view?usp=sharing))
* [Lecture 3 (7/08) | Divisors](https://drive.google.com/file/d/1Bx2OLivwfc-NAT3_L_-3sErbRNwUxcxK/view?usp=sharing) (Ray, [video](https://drive.google.com/file/d/1U_dM8uq6UiISlnfzzOH77s0CZ8adzasA/view?usp=sharing))
### Week 2
* Lecture 4 (7/11) | Projective toric varieties and polytopes (Alfred, 實體/Online)
* Lecture 5 (7/13) | Positivity I: linear system (Alfred, 實體/Online)
* Lecture 6 (7/15) | Positivity II (Alfred, 實體/Online)
### Week 3
* Lecture 7 (7/18) | WPS (Alfred or Ray, 實體/Online)
* Lecture 8 (7/20) | Resolution of Toric singularities: Terminal and canonical singularities (Jheng-Jie)
* Assigning Project
### Week 4
* Lecture 9 (7/25) | Reid’s Terminal lemma and Iano-Fletcher’s WCI, I (Jheng-Jie)
* Lecture 10 (7/27) | Iano-Fletcher’s WCI, II (Jheng-Jie)
### Week 5
* Lecture 11 (8/01) | Batyrev’s mirror construction (Yen-An)
* Lecture 12 (8/03) | Varieties of general type and Noether type inequalities (Alfred)
### Week 6
* Lecture 13 (8/08) | Boundedness of Fano varieties (Ray)
* Lecture 14 (8/10) | Toric Fano three-folds with terminal/canonical singularities (Jheng-Jie)



TA section (2:00 PM~4:00 PM)
---
### Week 1
* 7/05 | HW1(1), (4)([video](https://drive.google.com/file/d/1-oMEHW0oaFWsc6jY3P73A8JJYUTdmxjX/view?usp=sharing))
* 7/07 | HW1(5) 1.2.1-4b
* 7/08 | HW1(5) 1.2.12-15, (6)
### Week 2
* 7/12 | HW1 (8), (9), (10)
* 7/14 | HW2 (1), (2), (3)
* 7/15 | HW1 (11), (12)
### Week 3
* 7/19 | Exercises
* 7/21 | Exercises
### Week 4
* 7/26 | Exercises
* 7/28 | Exercises


Student Presentation (2:00 PM~4:00 PM)
---
### Week 5
* 8/02 | Klt varieties of general type with small volume
* 8/04 | Varieties of general type with doubly exponential asymptotics
### Week 6
* 8/09 | Bounds for smooth Fano weighted complete intersections
* 8/11 | On minimal varieties growing from quasi-smooth weighted hypersurfaces


Activities
---
* 7/02 (10:30 AM~12:30 PM) | Assign projects and discussions
* 8/12 Final Presentation/Closing Ceremony

Homework
---
* Week 1 ([PDF](https://drive.google.com/file/d/1vVG9CLvA5MxuTMiM2_HksC377C16S1HN/view?usp=sharing), [Tex](https://drive.google.com/file/d/1dGauHGPROVaW4UijxqGq004FwXsDqzLP/view?usp=sharing))
* Week 2 ([PDF](https://drive.google.com/file/d/1J3KMF4c-UNifFwDGcKXGmiac8w0TIUrc/view?usp=sharing), [Tex](https://drive.google.com/file/d/1npelmecSXcdnltCK_3XgFqL8xgUCA8sN/view?usp=sharing))
* Week 3 ([PDF](https://drive.google.com/file/d/1nEzkqxcSETFNaqFLlc6udEp5wUn5La87/view?usp=sharing))

Projects
---

A. General type, Calabi-Yau, or Fano WCI with extremal numerical proper-ties
B. Classification of reflexive polytope [5, 12, 13, 14]: Gorenstein toric Fano varieties.
C. Boundedness of mildly singular Fano and Calabi-Yau WCI’s [6]

Reference
---
[1] D. Cox; J. Little; H. Schenck, Toric Varieties.

[2] T. Hosgood’s, [An introduction to varieties in weighted projective space](https://drive.google.com/file/d/1eKiBdhRqG3FqSGOXi2cLW1ShVjBXXoGi/view?usp=sharing), arXiv:1604.02441

[3] I. Dolgachev, [Weighted projective varieties](https://drive.google.com/file/d/13LAiiXG-RUNvFrmdN8pUZIaBsZsh-iPk/view?usp=sharing), LNM 956.

[4] A. R. Iano-Fletcher, [Working with weighted complete intersections.](https://drive.google.com/file/d/13LAiiXG-RUNvFrmdN8pUZIaBsZsh-iPk/view?usp=sharing)

[5] Batyrev, Victor V., [Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties.](https://drive.google.com/file/d/13LAiiXG-RUNvFrmdN8pUZIaBsZsh-iPk/view?usp=sharing) J. Algebraic Geom. 3 (1994), no. 3, 493–535

[6] Chen, Jheng-Jie, [Finiteness of Calabi-Yau quasi-smooth weighted complete intersections.](https://drive.google.com/file/d/1OMJxD5FlHMJ_Ps0VH2ANI01t-cl0qER5/view?usp=sharing) Int. Math. Res. Not. IMRN 2015, no. 12, 3793–3809

[7] Louis Esser, Burt Totaro, Chengxi Wang, [Varieties of general type with doubly exponential asymptotics](https://drive.google.com/file/d/1XcUmH5JYfFNARvYx-sHlxlTUvaEnnueb/view?usp=sharing), arXiv:2109.13383

[8] Burt Totaro, Chengxi Wang, [Klt varieties of general type with small volume](https://drive.google.com/file/d/13gxfAixpg1vAaKhqVeQjtfcvzhBo7Q-k/view?usp=sharing), arXiv:2104.12200

[9] Meng Chen, Chen Jiang, Binru Li, [On minimal varieties growing from quasismooth weighted hypersurfaces](https://drive.google.com/file/d/1l8ZQig6NIW3z5yTe0JU6KHm8QDEAxXxJ/view?usp=sharing), arXiv:2005.09828

[10] Kasprzyk, Alexander M. [Canonical toric Fano threefolds](https://drive.google.com/file/d/1l8ZQig6NIW3z5yTe0JU6KHm8QDEAxXxJ/view?usp=sharing). Canad. J. Math. 62 (2010), no. 6, 1293–1309.

[11] Kasprzyk, Alexander M. [Toric Fano three-folds with terminal singularities](https://drive.google.com/file/d/1l8ZQig6NIW3z5yTe0JU6KHm8QDEAxXxJ/view?usp=sharing). Tohoku Math. J.(2) 58 (2006), no. 1, 101–121.

[12] Kreuzer, M.; Skarke, H.: [On the classification of reflexive polyhedra](https://drive.google.com/file/d/1u2HMVH5WyaiU_PmWxGPmA4S567x9XkDr/view?usp=sharing). Commun. Math. Phys. 185, 495-508 (1997)

[13] Kreuzer, M.; Skarke, H.: [Classification of reflexive polyhedra in three dimensions](https://drive.google.com/file/d/1GWdtwaWkUEKH2IiPFmTkXiJj-XNb7ixv/view?usp=sharing). Adv. Theor. Math. Phys. 2, 853-871 (1998)

[14] Kreuzer, M.; Skarke, H.: [Complete classification of reflexive polyhedra in four dimensions](https://drive.google.com/file/d/1UaJWaEhYKxwmc40iCIx-NGOAKH-GZZ2T/view?usp=sharing). Adv. Theor. Math. Phys. 4, 1209-1230 (2000)

[15] Przyjalkowski, Victor; Shramov, Constantin, [Bounds for smooth Fano weighted complete intersections](https://drive.google.com/file/d/1V6DzcoJiTZ9ufCME4vju-dSVmrNpDIF4/view?usp=sharing). Commun. Number Theory Phys. 14 (2020), no. 3, 511–553.
[16] Alexeev, V., & Liu, W. (2016). [Open surfaces of small volume](https://drive.google.com/file/d/1cqq1NQyNxHYRZHQdD0-Q2fIuHAiXk0q4/view?usp=sharing). arXiv preprint arXiv:1612.09116.

[17] Catanese, F. (2022). [Manifolds with vanishing Chern classes: hyperelliptic Manifolds, Manifolds Isogenous to a Torus Product, and some questions by Severi/Baldassarri](https://drive.google.com/file/d/1VAYDEOvlrVjt3vkOKO3000k-Zoxx297P/view?usp=sharing). arXiv preprint arXiv:2206.02646.

[18] Catanese, F. (2022). [Pluricanonical Maps and the Fujita Conjecture](https://drive.google.com/file/d/1OipvBUL1IMQr6kIh-6LtmNe86VNnyN7o/view?usp=sharing). arXiv preprint arXiv:2206.05142.

[19] Lee, T. J. (2022). [Mirror duality between Calabi-Yau fractional complete intersections](https://drive.google.com/file/d/1ZvX8QWK8YuzNVDlMSmQrs8i_nJgaNtET/view?usp=sharing). arXiv preprint arXiv:2206.06571.

[20] BROWN, G., COATES, T., CORTI, A., DUCAT, T., HEUBERGER, L., & KASPRZYK, A. [COMPUTATION AND DATA IN THE CLASSIFICATION OF FANO VARIETIES.](https://drive.google.com/file/d/18tNfl8BpVAkXqvoGY999tZMydj9vSheW/view?usp=sharing)

[21] Coughlan, S. ; Pignatelli, R.: [SIMPLE FIBRATIONS IN (1, 2)-SURFACES](https://arxiv.org/pdf/2207.06845.pdf)
