READING SEMINAR: BIRATIONAL GEOMETRY IN POSITIVE CHARACTERISTICS
We will focus on the minimal model program for 3-folds in positive characteristics and related topics. The goal is to understand basic knowledge on MMP and discuss some open questions.
Organizers: Meng Chen (Fudan), Lei Zhang (USTC), Chen Jiang (SCMS)
Seminar Room: SCMS 406
Schedule:
9.27 (Fri), 2pm-5pm, Jiang
10.25 (Fri), 2pm-5pm, Jiang
10.26 (Sat), 9am-12pm, Zhang
11.15 (Fri), 2pm-5pm, Zhang
12.6 (Fri), 2pm-5pm, Zhang & Jiang
12.13 (Fri), 2pm-5pm, TBA
Program:
1. MMP in dimension 3 in char p > 5 (Chen Jiang, 6h): Introduce MMP in dimension 3 (assuming existence of flips) and related topics.
References:
[Bir13] C. Birkar, Existence of flips and minimal models for 3-folds in char p, Annales scientifique de l’ENS 49 (2016), 169–212.
[BW17] C. Birkar and J. Waldlon, Existence of Mori fibre spaces for 3- folds in char p, Adv. Math. 313 (2017), 62–101.
2. Introduction to F-singularities (Lei Zhang, 3h): Introduce differ- ent types of singularities used in positive characteristics and comparing with singularities in characteristic 0.
References:
[Har98] N. Hara, Classification of two-dimensional F-regular and F-pure singularities, Adv. Math. 133 (1998), no. 1, 33–53.
[HW02] N. Hara and K.-i. Watanabe, Regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11.2 (2002), 363–392.
[Sch] Schwede’s notes.
3. Existence of flips in dimension 3 and p > 5 (Lei Zhang, 3h): Introduce the proof of existence of flips.
Reference:
[HX15] C. Hacon and C. Xu, On the three dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015), 711–744.
4. Nonvanishing and Abundance in dimension 3 (Zhang & Jiang): Introduce strategy of Nonvanishing and Abundance in dimension 3 in char 0 and char p.
References:
[Kol92] J. Kollar et al., Flips and abundance for algebraic threefolds, Asterisque No. 211 (1992).
[Zha17] L. Zhang, Abundance for 3-folds with non-trivial Albanese maps in positive characteristic, arXiv: 1705.00847 (2017).
[XZ19] C. Xu and L. Zhang, Nonvanishing for threefolds in characteristic p > 5, Duke Math. J., 168 (7) (2019),1269–1301.
5. Fujita conjecture in dimension 3 (TBA): The strategy to study Fujita conjecture in characteristic 0 and Seshadri constants.
References:
[Kaw97] Y. Kawamata, Fujita’s freeness conjecture for 3-folds and 4-folds, Math. Ann. 308.3 (1997), 491–505.
[EKL95] L. Ein, O. Kuchle, and R. Lazarsfeld, Positivity of ample line bundles, J. Differential Geom. 42.2 (1995), 193–219.
[Mur18] T. Murayama, Frobenius-Seshadri constants and characteriza- tions of projective space, Math. Res. Let. 25.3 (2018), pp. 905–936.