Laboratory of
Birational Geometry
About the Laboratory
The laboratory was founded by the research grant MOST110-2123-M-002-005 supported by the Ministry of Science and Technology of Taiwan, with Jungkai Chen as the PI of the grant.
The laboratory aims to investigate some fundamental problems in birational geometry, based at National Taiwan University. We plan to develop some theory and techniques in higher dimension geometry including :singularities and explicit MMP, derived categories in MMP, motivic integration, pluricanonical maps, and explicit computations. We seek for a systematical approach for explicit birational classification theory, geometry of algebraic fiber spaces, and singularities in minimal model program. Also, we will also search for possible application to some real-world challenges.
With the collaboration of many more members in National Center for Theoretical Sciences (NCTS) and in other universities in Taiwan, we form some project-oriented research teams and run seminars and workshops.
Teams
Team 1: Varieties of general type (led by Jungkai Chen)
Team 2: Fano threefolds (led by Ching-Jui Lai)
Upcoming Events
Date Time Event
2024/05/06 14:00-16:00 Group Meeting
2024/05/10 15:30-17:00 NCTS Seminar in Algebraic Geometry
2024/05/13 14:00-16:00 Group Meeting
Group Meeting
2024.05.06 (Monday) 02:00PM-04:00PM
Speaker: 蘇品丞、張宏彬
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2024.05.10 (Friday) 03:30PM-05:00PM
Speaker: Zhan Li (Southern University of Science and Technology)
Title: On the relative Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces
Abstract: A fibration with a relatively trivial canonical divisor is called a Calabi-Yau fiber space. The Morrison-Kawamata cone conjecture relates the birational geometry of a Calabi-Yau fiber space to the convex geometry of a movable cone. Assuming the minimal model program conjectures, we show that the cone conjecture of the generic fiber implies the cone conjecture of the Calabi-Yau fiber space. As an application, the finiteness of minimal models of a Calabi-Yau fiber space in relative dimensions less or equal to 2 is established. This work is partially joint with Hang Zhao.
Location: Rm. 505, NCTS (Cosmology Bldg., NTU)
Group Meeting
2024.05.13 (Monday) 02:00PM-04:00PM
Speaker: TBC
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
Previous Events
NCTS Seminar in Algebraic Geometry
2024.05.03 (Friday) 03:30PM-05:00PM
Speaker: Kyoung-Seog Lee (POSTECH)
Title: Symmetric products, Jacobian and moduli space of vector bundles of a curve
Abstract: Symmetric products, Jacobians and moduli spaces of vector bundles on curves are fundamental objects in the study of algebraic curves. In this talk, I will explain how they are related in the level of their derived categories and motives. This talk is based on several joint works with I. Biswas, T. Gomez, H.-B. Moon and M. S. Narasimhan.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Mini-course on Non-Commutative Geometry
Date: 2024.04.17, 24, 29
Venue: Rm. 509, NCTS (Cosmology Bldg., NTU)
Speaker: Yujiro Kawamata (University of Tokyo & NCTS) (video)
Course Outline & Descriptions:
(1) I will explain deformations of sheaves over non-commutative (NC) base. The point is that there are more NC deformations than usual (C) deformations, and NC deformations yield more invariants. The moduli space of sheaves become larger in the sense that it has additional formal structure.
(2) I will explain deformations of a usual variety, based on commutative algebra, to NC varieties. Though the localizations of NC algebras are usually not possible, the deformations are constructed by gluing NC associative algebras. I will also consider NC deformations of McKay correspondence.
Online Meeting Link:
04.17 (Wednesday)
https://ntucc.webex.com/ntucc-en/j.php?MTID=m6db93858157c1e683347bf7b9f533fcd
04.24 (Wednesday)
https://ntucc.webex.com/ntucc-en/j.php?MTID=m3dda794d7f709605e14679428ee305f4
04.29 (Monday)
https://ntucc.webex.com/ntucc-en/j.php?MTID=mc24ffd50fc000636a1c5eef25d4136bd
Registration:
https://forms.gle/UmbeVZRMYktHxKfA9
For more information, please click here.
NCTS Seminar in Algebraic Geometry
2024.04.26 (Friday) 03:30PM-05:00PM
Speakers: Roberto Pignatelli (University of Trento) (video)
Title: Fibrations in (1,2)-Surfaces
Abstract: The content of this seminar stems from an ongoing collaboration with S. Coughlan, Y. Hu, and T. Zhang. By "(1,2)-surfaces" we denote complex algebraic surfaces with canonical singularities, ample canonical system, volume 1, and geometric genus 2. This is a class of surfaces that played a significant role in the theory of surfaces of general type in the last century and has shown in this century to also play an important role in the theory of 3-dimensional varieties, particularly in the recent proof of the 3-dimensional Noether inequality obtained by J. Chen, M. Chen, and C. Jiang. In fact, 3-dimensional varieties fibred in surfaces of type (1,2) play a role in this proof similar to that played by fibrations in curves of genus 2 in lower dimensions. In this seminar, I will introduce the concept of "simple" fibration in surfaces of type (1,2) and explain how through this concept we obtained a complete classification of 3-folds that satisfy the equality in the aforementioned inequality. I will also discuss analogies and differences with the 2-dimensional case. Finally, I will mention some open problems that we are currently investigating.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
Group Meeting
2024.04.22 (Monday) 10:00AM-11:00AM
Speaker: 中橋一萌(video)
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2024.04.19 (Friday) 03:30PM-05:00PM
Speakers: Iacopo Brivio (Harvard University) (video)
Title: Antiitaka conjecture for tame fibrations in positive characteristic
Abstract: A famous conjecture by Iitaka predicts that, given a fibration $f\colon X\to Y$ of smooth complex varieties with general fiber $F$, then the inequality $\kappa(K_X)\geq \kappa(K_F)+\kappa(K_Y)$ holds. It was recently shown by Chang that, when the stable base locus $\mathbb{B}(-K_X)$ is $f$-vertical, then a similar inequality holds for the anticanonical divisor: $\kappa(-K_X)\leq \kappa(-K_F)+\kappa(-K_Y)$. Over fields of positive characteristic it is known that both Iitaka's conjecture and Chang's theorem can fail. However the expectation is that, when $F$ is sufficiently well-behaved with respect to the Frobenius morphism, then these results should hold. In this talk I will show that this is the case for Chang's theorem. This is based on joint work with M. Benozzo and C.-K. Chang.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2024.04.12 (Friday) 03:30PM-05:00PM
Speakers: Jie Liu (AMSS) (video)
Title: Kawamata-Miyaoka type inequality for canonical Q-Fano varieties
Abstract: An irreducible normal projective variety is called Q-Fano if it has Q-factorial singularities, with Picard number one and ith ample anti-canonical divisor. According to the minimal model program, Q-Fano varieties with canonical singularities are one of the building blocks of algebraic varieties. In this talk I will show that the second Chern class of a canonical Q-Fano variety is effectively bounded from below by its first Chern class. This is joint work with Haidong Liu.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
Group Meeting
2024.03.11 (Monday) 2:00PM-4:00PM
Speakers: 張繼剛(video)、蘇品丞
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2024.03.01 (Friday) 03:30PM-05:00PM
Speaker: Martin Kalck (University of Graz) (video)
Title: Derived categories of singular varieties
Abstract: We will discuss derived categories of some (singular) projective varieties.
A particularly nice case are surfaces with cyclic quotient singularities with connections to the Markov Diophantine equation.
If time permits, we will discuss higher dimensional examples building on ongoing joint work with Yujiro Kawamata & Nebojsa Pavic.
Location: Rm. 505, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2024.02.23 (Friday) 11:00AM-12:00PM
Speaker: Fei Yu Chen (University of California, Berkeley) (video)
Title: Deformation theory and coalgebras
Abstract: Deformation theory aims to classify infinitismal deformations of mathematical objects. For many geometric objects, this comes out to understanding the Lie algebra structure on their tangent spaces. A key technique in proving these arguments is Koszul duality, which gives the cotangent complex of an algebra the extra structure of a Lie coalgebra, then you take a linear dual to get the Lie algebra structure on the tangent space. My work currently aims to understand the role of coalgebras in deformation theory. We develop a theory of a tangent complex and a coalgebraic version of formal moduli problems, and relate them to the algebraic theories. On the way we also need to understand the problem of rectification of coalgebras: relating the model structure of coalgebras with the infinity categorical definition of coalgebras, which unlike with algebras is not true in many cases.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2024.02.19 (Monday) 04:00PM-05:00PM
Speaker: Ivan Cheltsov (The University of Edinburgh) (video)
Title: K-moduli of Fano threefolds of Picard rank 4 and degree 24.
Abstract: Smooth Fano threefolds of Picard rank 4 and degree 24 are divisors in $(\mathbb{P}^1)^4$ of degree $(1,1,1,1)$. All of them are K-stable by a theorem of Grisha Belousov and Costya Loginov from Moscow. It is natural to expect that all singular K-polystable limits of these Fano threefolds are also divisors in $(\mathbb{P}^1)^4$ of degree $(1,1,1,1)$, and the corresponding K-moduli space can be obtained as a natural GIT quotient, which is classically known (but not well known) to be isomorphic to the weighted projective space $\mathbb{P}(1,3,4,6)$. Surprisingly, this is not the case - some singular K-polystable limits of smooth Fano threefolds of Picard rank 4 and degree 24 are not divisors in $(\mathbb{P}^1)^4$ of degree $(1,1,1,1)$. In this talk, I will find all singular K-polystable limits of smooth Fano threefolds of Picard rank 4 and degree 24, and show that the corresponding K-moduli space is a weighted blow up of $\mathbb{P}(1,3,4,6)$ at a smooth point with weights $(1,2,3)$. This is a joint work with Maksym Fedorchuk (Boston), Kento Fujita (Osaka) and Anne-Sophie Kaloghiros (London).
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2024.02.19 (Monday) 02:30PM-03:30PM
Speaker: Kento Fujita (Osaka University) (video)
Title: On the Calabi problem for Fano varieties
Abstract: I will give a survey on the recent progresses on the K-stability of Fano varieties. Especially, I will present the history of the Calabi problem, the original definition of K-stability, a valuative criterion for the K-stability of Fano varieties, Abban--Zhuang's theory, 3-dimensional application, and so on.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
Mini-Arbeitsgemeinschaft 2024
Severi–Brauer varieties
2024.01.26 - 2024.01.29
Organizers:
Jungkai Chen (NTU), Hsueh-Yung Lin (NTU)
Agenda:
Jan. 26 (Fri.)
• 14:00 – 15:00 Introduction [Volunteer: Hsueh-Yung Lin]
• 15:15 – 16:15 §1. Geometrically split vector bundles [Volunteer: Hung-Pin Chang]
Jan. 27 (Sat.)
• 08:30 – 09:30 §1. + part of §2. [Volunteer: Kuan-Wen Lai]
• 09:45 – 10:45 §2. Tangent bundles of Severi–Brauer varieties [Volunteer: Yi-Hung Chen]
• 11:00 – 12:00 §3. Non-minimal Severi–Brauer varieties [Volunteer: Chia-Hsiang Chen]
• 14:00 – 15:00 §4. Twisted linear systems, I [Volunteer:Yi-Teng Wu]
• 15:15 – 16:15 §4. Twisted linear systems, II [Volunteer: Chi-Kang Chang]
Jan. 28 (Sun.)
• 08:30 – 09:30 §5. The Brauer group [Volunteer: Jun Ting Lin]
• 09:45 – 10:45 §6. The index of a Severi–Brauer variety [Volunteer:Hajime Nakahashi]
• 11:00 – 12:00 §7. Construction of cyclic Severi–Brauer varieties [Volunteer: Yi Kung]
Jan. 28 (Mon.)
• 08:30 – 09:30 SOD for Severi–Brauer varieties [Volunteer: Chih-Huan Chang]
• 09:45 – 10:45 Period-index problem [Volunteer: Pedro Nunez]
Location: Hualien
NCTS Seminar in Algebraic Geometry
2024.02.02 (Friday) 03:30PM-05:00PM
Speaker: Johan Commelin (Utrecht University) (video)
Title: Aspects of the Liquid Tensor Experiment
Abstract: The Liquid Tensor Experiment is a project that formally verified the main theorem of liquid vector spaces by Clausen and Scholze, following up on a challenge by Scholze on Buzzard's XenaProject blog.
In this talk I will explain the thrust of this main theorem and share some mathematical insights gained from my protracted battle with its proof.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2024.01.08 (Monday) 03:30PM-05:00PM
Speaker: Weichung Chen (The University of Tokyo) (video)
Title: Ascending Chain Condition for Minimal Log Discrepancies for Bounded Generalized Pairs
Abstract: Minimal log discrepancy (mld) is an invariant of variety or log pair singularities that measures how singular they are. The ascending chain condition (acc) for mlds of singularities of a fixed dimension is conjectured to hold by Shokurov. On the other hand, generalized pairs play an important role in Birkar's proof of the Borisov-Alexeev-Borisov conjecture (BBAB theorem), which says that Fano varieties of a fixed dimension with a fixed lower bound of mlds form a bounded family. We show that the ACC conjecture holds for bounded families of generalized pairs. Applying BBAB theorem, we show that the ACC conjecture holds for generalized Fano pair.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2024.01.08 (Monday) 01:30PM-03:00PM
Speaker: Ryo Yamagishi (University of Bath) (video)
Title: Hilbert schemes of points on canonical surfaces
Abstract: The celebrated result of Fogarty claims that the Hilbert scheme of n points on a smooth surface is a smooth variety of dimension 2n. In this talk, as an analogue of this result, I will show that the Hilbert scheme of n points on a normal surface with canonical singularities is a normal variety of dimension 2n. The proof is done by realizing the Hilbert scheme for a specific surface singularity as a Nakajima quiver variety. This talk is based on joint work with Alastair Craw.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
Student Seminar
2024.01.04 (Thursday) 10:00AM-11:00AM
Speaker: 張繼剛 (video)
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
Student Seminar
2024.01.03 (Wednesday) 01:20PM-03:10PM
Speaker: 張志煥 (video)、李永丞
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
Student Seminar
2023.12.27 (Wednesday) 01:20PM-03:10PM
Speaker: 中橋一萌 (video)
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.12.22 (Friday) 03:30PM-05:00PM
Speaker: Zhi Jiang (Fudan University) (video)
Title: Surfaces of general type with p_g=q=2
Abstract: The classification of surfaces of general type with p_g=q=2 has attracted lots of efforts in the past two decades. In this talk we explain a fine classification theory of these surfaces whose canonical volumes are 5 or 6. Our method is based on the Chen-Jiang decomposition, cohomological rank functions, and the symmetry of finite Heisenberg groups. This is a joint work with Jiabin Du and Guoyun Zhang.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.12.15 (Friday) 03:30PM-05:00PM
Speaker: Kentaro Mitsui (Kobe University) (video)
Title: Deformation of α_p-actions to Z/pZ-actions on Surfaces
Abstract: We study α_p-actions and Z/pZ-actions on surfaces and their quotient singularities in positive characteristic p. We can connect some singularities such as non-taut rational double points via deformation of these actions. We also try to classify additive derivations, which correspond to α_p-actions. This is joint work with Hiroyuki Ito and Nobuo Sato.
Location: Rm. 440, NCTS (Astro-Math Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.12.04 (Monday) 04:00PM-05:30PM
Speaker: Cécile Gachet (Humboldt-Universität zu Berlin) (video)
Title: Finite quotients of abelian varieties with a Calabi-Yau manifold as a resolution
Abstract: Let G be a finite group acting on an abelian variety A. We say that G acts freely in codimension k if the fixed locus by every element of G has codimension at least k+1. In particular, if G acts freely in codimension 1 on A, then the quotient A/G is a potentially singular variety with trivial canonical class; the higher the k such that G acts freely in codimension 2, the less singular A/G is.
In this talk, I explain how to produce examples and show non-existence results for "particularly nice" desingularizations for such quotients A/G. Here, "particularly nice" ranges from resolutions that still have trivial canonical class (the so-called crepant resolutions) to resolutions that are Calabi--Yau manifolds. This can be thought of as a nice way to produce new Calabi-Yau varieties whose geometry may reflect that of the initial abelian variety A.
I will spend some time at first explaining local phenomena when investigating crepant resolutions of a given quotient singularity, recalling notably some early successes of the McKay correspondence due to Y. Ito and M. Reid. I will then describing results of K. Oguiso for the 3-dimensional case, and present some recent results in the 4- and higher dimensional cases.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.11.24 (Friday) 03:30PM-05:00PM
Speaker: Alexandra Kuznetsova (Steklov Math. Institute) (video)
Title: Automorphisms of families in abelian varieties
Abstract: We consider a fibration X whose general fiber is an abelian variety and fix a fiberwise birational automorphism f of X. Then f restricts into a regular automorphism of a general fiber of the fibration since abelian varieties do not contain rational curves. Thus, f induces a family of regular automorphisms of abelian varieties. However sometimes f does not induce a regular automorphism on any birational model of the whole variety X i.e. there is no regularization of f. In my talk I will show that the existence of the regularization depends on the dynamical properties of f. The talk is based on the joint work with Charles Favre.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.11.17 (Friday) 03:30PM-05:00PM
Speaker: Chih-Wei Chang (NTU) (video)
Title: Tropical Linear Series and Their Applications
Abstract: Let K be an algebraically closed valued field with valuation ring R, and let C be a curve over K. A semistable model of C over Spec R gives rise to a metric graph \Gamma. Baker proved the Specialization Lemma relating ranks of divisors on curves to those on graphs, envisioning tropical methods to prove results about divisors on algebraic curves, such as the Brill-Noether and Gieseker-Petri Theorems. David Jensen and Sam Payne defined the notions of tropical linear series and tropical dependence on graphs, which are closely related to problems concerning possibly incomplete linear series on algebraic curves, such as the maximal rank theorem. In this talk, I will review the story and discuss our recent progress in this direction. Based on joint work with David Jensen, Sam Payne, Dagan Karp, Jayden Wang, and Hernan Iriarte.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.11.10 (Friday) 03:30PM-05:00PM
Speaker: Kenta Hashizume (Niigata University) (video)
Title: Minimal model theory for log canonical pairs and log canonical loci
Abstract: The minimal model theory is a fundamental tool to construct a variety with good geometrical properties. The theory is indispensable to recent development of birational geometry. In this talk, I will start with the motivation of the minimal model theory, and then I will explain the recent development of the minimal model theory for log canonical pairs. After that, I will introduce a recent result on minimal model program for normal pairs along log canonical loci.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.10.27 (Friday) 03:30PM-05:00PM
Speaker: Ruijie Yang (Humboldt-Universität zu Berlin) (video)
Title: The geometric Riemann-Schottky problem and Hodge theory
Abstract: It is a classical problem in algebraic geometry, dated back to Riemann, to characterize Jacobians of smooth projective curves among all principally polarized abelian varieties. In 2012, Casalaina-Martin proposed a conjecture in terms of singularities of theta divisors. In this talk, I will present a partial solution of this conjecture using Hodge theory and D-modules. We also show that this conjecture can be deduced from a conjecture of Pareschi and Popa on GV sheaves and minimal cohomology classes.
To achieve this, we develop a theory of higher multiplier ideals, which is a Hodge-theoretic refinement of the theory of multiplier ideals in birational geometry. This new theory relies on the Hodge theory of Kashiwara-Malgrange V-filtration, Sabbah-Schnell’s theory of complex mixed Hodge modules and the language of twisted D-modules by Beilinson and Bernstein. This is based on the joint work with Christian Schnell.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.10.20 (Friday) 03:30PM-05:00PM
Speaker: Pedro Núñez (NTU) (video)
Title: Indecomposability of derived categories of hyperelliptic manifolds
Abstract: The MMP aims at classifying algebraic varieties up to birational equivalence, and derived categories are rich homological invariants of algebraic varieties that see a lot of their birational geometry. The basic operations of the MMP are reflected as semiorthogonal decompositions in the derived category, and minimal varieties of non-zero geometric genus are conjectured to have indecomposable derived categories. Hyperelliptic surfaces are an interesting example, because they show that the converse of the conjecture does not hold. In this talk, we discuss a generalization of the indecomposability of hyperelliptic surfaces to higher dimensional hyperelliptic manifolds, which is current joint work in progress with Pieter Belmans and Andreas Demleitner.
The talk will be roughly divided into three parts. First, an introductory part in which derived categories are briefly introduced, focusing on their relation to birational geometry. Then, an overview of semiorthogonal decompositions and their relation to the MMP. And lastly, a more detailed discussion of the semiorthogonal indecomposability question.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.10.13 (Friday) 03:30PM-05:00PM
Speaker: Keita Goto (NTU) (video)
Title: An Introduction to Berkovich Geometry and Kontsevich-Soibelman Conjecture
Abstract: Originally, Berkovich Geometry was known as one of the theories studying Non-Archimedean Geometry which is a branch to study analytic spaces over a non-Archimedean field.
However, a framework of Berkovich Geometry enables us to study analytic spaces over not only a non-Archimedean field but also a certain Banach ring.
In this talk, I'll introduce such a concept and fundamental facts of Berkovich Geometry.
Besides, I'll explain Kontsevich-Soibelman Conjecture as its application.
An SYZ fibration is often studied in the SYZ mirror symmetry context.
This conjecture predicts a certain equivalence between SYZ fibration and a non-Archimedean analog of SYZ fibration.
The latter half of my talk will be devoted to describing my joint work with Yuji Odaka, which partially proves this conjecture.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.10.06 (Friday) 03:30PM-05:00PM
Speaker: Sho Tanimoto (Nagoya University) (video)
Title: Sections of Fano fibrations over curves
Abstract: Manin’s conjecture predicts the asymptotic formula for the counting function of rational points on a smooth Fano variety, and it predicts an explicit asymptotic formula in terms of geometric invariants of the underlying variety. When you count rational points, it is important to exclude some contribution of rational points from an exceptional set so that the asymptotic formula reflects global geometry of the underlying variety. In Part I of my talk, I will describe birational geometry of these exceptional sets using higher dimensional algebraic geometry such as the minimal model program. In Part II, I will discuss applications of the study of exceptional sets to moduli spaces of sections of Fano fibrations. Then in Part III I will discuss the case of del Pezzo fibrations in more details. Part I is based on joint work with Brian Lehmann and Akash Kumar Sengupta. Part II is based on joint work with Brian Lehmann and Eric Riedl. Part III is based on joint work with Brian Lehmann.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
Student Seminar
2023.10.02 (Monday) 01:20PM-03:10PM
Speaker: 蘇品丞 (video)、張宏彬
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
NCTS Seminar in Algebraic Geometry
2023.09.22 (Friday) 03:30PM-05:00PM
Speaker: Kuan-Wen Lai (Academia Sinica) (video)
Title: Bijective Cremona Transformations of the Plane
Abstract: Cremona transformations are invertible maps from a projective space to itself defined by polynomials. We say a Cremona transformation is "bijective" if it induces a bijection on the set of rational points. For example, over a finite field, the set of rational points is finite, thus every such transformation induces a permutation in the usual sense, and it is natural to ask whether one can realize all possible permutations among these points this way. In this talk, I will present the answer for projective planes.
At least half of this talk is intended for a general audience in mathematics. This is joint work with Shamil Asgarli, Masahiro Nakahara, and Susanna Zimmermann, published on Selecta Mathematica, New Series 28, 53 (2022).
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2023 NCTS Mini-courses and Conference:
Modern Perspectives on Birational Geometry
Modern Perspectives on Birational Geometry
Date: 2023.07.29-08.04
Venue: Rm. 515, NCTS (Cosmology Bldg., NTU)
Invited Speakers:
Mini-courses Lecturers:
Dan Abramovich (Brown University)
Shigeru Mukai (RIMS, Kyoto university)
Yuri Prokhorov (Steklov Institute)
Conference Speakers:
Hamid Abban (University of Nottingham)
Caucher Birkar* (Tsinghua University)
Liana Heuberger (University of Bath)
Dmitry Kaledin (HSE University & Steklov Institute)
Konstantin Loginov (HSE University & Steklov Institute)
Steven Lu (Université du Québec à Montréal)
Gebhard Martin (University of Bonn)
Mirko Mauri (IST Vienna)
Keiji Oguiso (University of Tokyo)
Jihun Park (Pohang University of Science and Technology)
Marco Rampazzo (University of Bologna)
Francesco Russo (University of Catania)
Julia Schneider (EPFL)
Nicholas Shepherd-Barron (King's College London)
Constantin Shramov (HSE University & Steklov Institute)
Jenya Tevelev** (UMass Amherst)
Jaroslaw Wlodarczyk (Purdue University)
* online, ** TBC
For more information, please click here.
2023.07.21 (Friday)
03:30PM-05:00PM
Speaker: 黃建順 (video)
Reference: (Johannes Krah, arXiv:2304.01269) A Phantom on a Rational Surface.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2023.06.30 (Friday)
03:30PM-05:00PM
Speaker: 蘇品丞 (video)
Reference: (Coughlan and Pignatelli, arXiv:2207.06845) Simple fibrations in (1, 2)-surfaces
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
Workshop on Derived Categories
2023.06.20 - 2023.06.21
The goal of this workshop is to study the derived categories, with homological projective duality as the goal.
Organizer:
Jungkai Chen (NTU)
Agenda:
June 20 (Tue.)
12:40-13:40 Beilinson’s theorem 林俊廷
13:50-14:50 Orlov’s theorem on projective bundles 張宏彬
15:00-16:00 Orlov’s theorem on blowups 陳毅鴻
16:15-18:15 Some hiking
18:30- Dinner
20:00- Survey and free discussion
June 21 (Wed.)
08:30-09:30 Homological projective duality, I 蘇品丞
09:40-10:40 Even dimensional quadrics 中橋一萌
10:50-11:50 Pfaffian examples 黃建順
11:50-12:50 Lunch
12:50-13:50 Derived categories of moduli spaces of vector bundles on curves 張志煥
14:00-15:00 Grothendieck duality theorem, after Neeman 姚皓勻
Location: Xitou
2023.06.16 (Friday)
03:30PM-05:00PM
Speaker: Yi-Hong Chen (video)
Reference: (Lukas Braun: arXiv:2004.00522) The local fundamental group of a Kawamata log terminal singularity is finite
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
Mini-Courses on Computational Algebraic Geometry
Date: 2023.05.24-06.14 (Wednesdays) 03:30PM-05:30PM
Venue: Rm. 515, NCTS (Cosmology Bldg., NTU)
Speaker: Christopher Peterson (Colorado State University)
Give a broad overview of selected topics in computational and applied aspects of Algebra, Geometry, and Topology. This includes the fundamental computational tools of Gröbner bases, Homotopy Continuation, Persistent Homology. We will also discuss basic algorithms/computations on Grassmann, Flag, and Stiefel manifolds.
Course Outline & Descriptions
Theme 1: Grobner Bases and Symbolic Computation: Theory, algorithms, and applications.
Theme 2: Homotopy Continuation and Numeric Computation: Theory, algorithms, and applications.
Theme 3: Real Algebraic Geometry and Probabilistic Computation: Theory, algorithms, and applications.
Theme 4: Grassmann, Flag, and Steifel Manifolds: Theory, algorithms, and applications.
Theme 5: Persistent Homology: Theory, algorithms, and applications.
Registration: https://forms.gle/YKK5XiCT8HPKqvReA
2023.06.09 (Friday)
03:30PM-05:00PM
Speaker: 張志煥 (video)
Reference: (arXiv: 2108.01506) Moduli Spaces of Quasi-Trivial Sheaves
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
Student Seminar at CCU
An Invitation to Algebraic Geometry
An Invitation to Algebraic Geometry
The purpose of this seminar is to provide an intuitive introduction to modern algebraic geometry.
Lecturer: 陳榮凱 Jungkai Chen
Venue: CCU
Week 1 (2/20) 03:00PM-05:00PM
Lecture 1: Projective space, projective varieties and elliptic curves (video)
Ref: [1, section 4.1, 4.2, 5.1]
Week 2(2/27) 03:00PM-05:00PM
National holiday
Week 3 (3/6) 03:00PM-05:00PM
Lecture 2: Bezout’s theorem and intersection theory (video)
Ref: [1, section 3.1, 3.2, 3.3, 5.3]
Week 4, 5
NCTS Higher Dimensional Algebraic Geometry
3/13-3/17 mini-courses
3/20-3/24 workshop
Week 6 (3/27) 03:00PM-05:00PM
Lecture 3: Singularities and blowups (video)
Ref: [1, chapter 7]
Week 7 (4/3)
Spring break
Week 8 (4/10) 03:00PM-05:00PM
Midterm week 4/10-4/14
Week 9 (4/17)
TA session (video)
Week 10 (4/24) 03:00PM-05:00PM
Lecture 4: Canonical divisors and Riemann-Roch theorem for curves (video)
Ref: [1, chapter 8]
Week 11 (5/1) 03:00PM-05:00PM
Lecture 5: Riemann-Roch theorem for surfaces (video)
Ref: [2, section V.1]
Week 12 (5/8)
TA session (video)
Week 13 (5/15) 03:00PM-05:00PM
Lecture 6: Projective morphisms (video)
Week 14 (5/22)
TA session (video)
Week 15 (5/29) 03:00PM-05:00PM
Lecture 7: Vanishing theorem and extension theorem (video)
Week 16 (6/5)
Presentation
References:
[1] W. Fulton, Algebraic Curves
[2] R. Hartshorne, Algebraic Geometry
It is recommended to read sections 1.1-1.7 and 2.1-2.5 of [1] by yourselves in advance.
2023.05.26 (Friday)
10:30AM-11:30AM
Speaker: Shih-Hsin Wang (University of Utah) (video)
Title: Families of Jets on Higher Du Val Singularities
Abstract: We will reveal a natural correspondence between families of arcs originally studied by Nash and families of jets through the singular locus. Then we focus on normal locally complete intersection varieties with rational singularities. In particular, we introduce the notion of higher Du Val singularities, a higher dimensional version of Du Val singularities that preserve many of their features, and more generally, the notion of higher compound Du Val singularities, whose definition parallels that of compound Du Val singularities. For such singularities, we prove a one-to-one correspondence exists between families of arcs and families of jets through the singularities. As an application, we give a solution to the Nash problem for higher Du Val singularities.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2023.05.11 (Thursday)
03:30PM-05:30PM
Lecture 5. Derived Functors in algebraic geometry, I ([1, Sec 3.3])
Speaker: 黃建順 (video)
Lecture 6. Derived Functors in algebraic geometry, II ([1, Sec 3.3])
Speaker: 陳毅鴻 (video)
Reference: [1] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
2023.05.04 (Thursday)
03:30PM-05:30PM
Lecture 3. Derived Functors ([1, Sec 2.2])
Speaker: 張志煥 (video) (5/11 viedo)
Lecture 4. Derived Categories of coherent sheaves ([1, Sec 3.1])
Speaker: 吳宏宜 (video)
Reference: [1] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
2023.04.20 (Thursday)
03:30PM-05:30PM
Lecture 1. Triangulated categories and exact functors ([1, Sec 1.2, 1.3])
Speaker: 林俊廷 (video)
Lecture 2. Derived Categories of abelian categories ([1, Sec 2.1])
Speaker: 施政邦 (video) (4/27 video)
Reference: [1] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
2023.03.31 (Friday) NCTS Seminar in Algebraic Geometry
04:45PM-05:45PM
Speaker: Marta Benozzo (video)
Title: On the canonical bundle formula in positive characteristic
Abstract: An important problem in birational geometry is trying to relate in a meaningful way the canonical bundles of the source and the base of a fibration. Recently, the problem has been approached with techniques from the minimal model program. These methods can be used to prove a canonical bundle formula result in positive characteristic.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2023.03.31 (Friday) NCTS Seminar in Algebraic Geometry
03:30PM-04:30PM
Speaker: Yujiro Kawamata (video)
Title: NC deformations, semi-orthogonal decompositions and Q-Gorenstein smoothing
Abstract: I will explain how NC deformations of coherent sheaves can sometimes be used to produce locally free sheaves.
I will also explain some consequences on semi-orthogonal decompositions of derived categories and singularity categories.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2023 NCTS Higher Dimensional Algebraic Geometry, Mini-courses and Workshop
Date: 2023.03.13-03.24
Venue: Rm. 515, NCTS (Cosmology Bldg., NTU)
Invited Speakers:
Week 1: 3/13-3/17, 2023 (mini-courses) (video playlist)
Paolo Cascini (Imperial College London)
Alexander Kuznetsov (Steklov Mathematical Institute of Russian Academy of Sciences)
Joaquin Moraga (University of California, Los Angeles)
Yukinobu Toda (Kavli Institute for the Physics and Mathematics of the Universe)
Week 2: 3/20-3/24, 2023 (workshop) (video playlist)
Iacopo Brivio (National Center for Theoretical Sciences)
Nero Budur (Katholieke Universiteit Leuven)
Paolo Cascini (Imperial College London)
Hsin-Ku Chen (Korea Institute For Advanced Study)
Alessio Corti (Imperial College London)
Tien Cuong Dinh (National University of Singapore)
Jun-Muk Hwang (Institute for Basic Science)
Alexander Kuznetsov (Steklov Mathematical Institute of Russian Academy of Sciences)
Kuan-Wen Lai (University of Bonn)
Yongnam Lee (Institute for Basic Science)
Hsueh-Yung Lin (National Taiwan University)
James McKernan (University of California, San Diego)
Joaquin Moraga (University of California, Los Angeles)
Kieran O'Grady (Sapienza Università di Roma)
Shinnosuke Okawa (Osaka University)
Mihai Paun (University of Bayreuth)
Mihnea Popa (Harvard University)
Yuri Prokhorov (Steklov Mathematical Institute)
Burt Totaro (University of California, Los Angeles)
Chenyang Xu (Princeton University)
De-Qi Zhang (National University of Singapore)
For more information, please click here.
Preprint Seminar
The purpose of this seminar is to present the most up-to-date results that appeared in arXiv in recent two years.
The purpose of this seminar is to present the most up-to-date results that appeared in arXiv in recent two years.
2023.02.24 (Friday) 03:30PM-05:30PM (video)
Yen-An Chen:
Infinitesimal structure of log canonical thresholds.,
(Liu, Meng, Xie 2209.11369)
2023.03.03 (Friday) 03:30PM-05:30PM (video)
Jheng-Jie Chen:
On explicit birational geometry for minimal $n$-folds of canonical dimension $n-1$., (Chen, Esser and Wang, arXiv: 2201.08966)
2023.03.10 (Friday) 03:30PM-05:30PM (video)
Bin Nguyen:
On the components of the Main Stream of the moduli space of surfaces of general type with $p_g=q=2$.
(Alessandro and Catanese, arXiv:2212.14872)
2023.03.09 (Thursday)
03:30PM-05:30PM
Lecture 11. Log-canonical Surface Singularities (Section 4.1)
Speaker: 張繼剛 (video)
Lecture 12. Du Val Singularities (Section 4.2)
Speaker: 吳宏宜 (video)
Reference: [1] J. Kollár, S. Mori, Birational Geometry of Algebraic Varieties
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
2023.03.02 (Thursday)
03:30PM-05:30PM
Lecture 9. Negativity Lemma (Lemma 3.38-3.41)
Speaker: Hajime Nakahashi (video)
Lecture 10. Running the MMP (Section 3.7)
Speaker: Jheng-Jie Chen (video)
Reference: [1] J. Kollár, S. Mori, Birational Geometry of Algebraic Varieties
Location: Rm. 440, NCTS (Astro-Math Bldg., NTU)
2023.02.16 (Thursday)
02:30PM-05:30PM
Lecture 4. Existence of the Minimal Model Program (section 2.2 of [1])
Speaker: 張宏彬 (video)
Lecture 5. Discrepancies (section 2.3 of [1], up to subsection 2.33)
Speaker: 陳毅鴻 (video)
Lecture 6. Singularities in MMP (section 2.3 of [1], after subsection 2.34)
Speaker: 蘇品丞 (video)
Reference: [1] J. Kollár, S. Mori, Birational Geometry of Algebraic Varieties
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
2023.02.10 (Friday)
10:00AM-11:00AM
Speaker: Keita Goto (Kyoto University) (video)
Title: On affine structures which come from Berkovich Geometry
Abstract: In the SYZ mirror symmetry context, SYZ fibrations are often studied. In particular, it is well known that they induce affine structures on their base spaces. On the other hand, Kontsevich and Soibelman introduce a non-Archimedean analog of SYZ fibrations, which is called non-Archimedean SYZ fibrations later. Non-Archimedean SYZ fibrations induce affine structures as well as SYZ fibrations. Moreover, they predict a certain equivalence between SYZ fibrations and non-Archimedean SYZ fibrations for maximally degenerating family of polarized Calabi-Yau varieties. In particular, two affine structures coming from these different origins are expected to coincide. We proved the equivalence for K-trivial finite quotients of polarized abelian varieties by introducing what we call hybrid SYZ fibrations. In this talk, we introduce the above theorem. This talk is based on a joint work (https://arxiv.org/abs/2206.14474) with Yuji Odaka.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2023.02.09 (Thursday)
03:30PM-05:30PM
Lecture 2. Ampleness Criterion (1.5 of [1])
Speaker: 施政邦 (viedo)
Lecture 3. Minimal Model of Surfaces and an Overview of Higher Dimensional Varieties (1.4 of [1] and Prop. 2.5, Example 2.7, subsection 2.14 in section 2.1 of [1])
Speaker: 鄭容濤 (video)
Reference: [1] J. Kollár, S. Mori, Birational Geometry of Algebraic Varieties
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
2023.02.06 (Monday)
04:30PM-05:30PM
Speaker: Federico Fallucca (University of Trento) (video)
Title: On the degree of the canonical map of surfaces of general type
Abstract:
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2023.02.06 (Monday)
03:00PM-04:00PM
Speaker: Pedro Núñez (University of Freiburg) (video)
Title: Adapted differentials on Campana orbifolds
Abstract: Adapted differentials play an important role in Campana's birational classification program. In this talk we will discuss two results concerning them. These differential forms are usually only defined over suitably ramified covers, and the first result shows that they can be seen as a qfh-sheaf on the whole category of schemes over the orbifold. The second result ensures that we can extend (reflexive) adapted differential 1-forms on such a suitably ramified cover to a resolution of singularities of the cover, provided that the orbifold is klt. As an application, we construct the Albanese of a Campana orbifold with quotient singularities.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.12.30 (Friday)
04:00PM-0.5:00PM
Speaker: Geoffrey Mboya (University of Oxford) (video)
Title: Singular threefold families of K3 surfaces
Abstract: This talk will be on the classification problem of projective fibrations f:X->P^1 over the projective line, where the general fibre of f is one of the “famous 95” list of Fletcher–Reid K3 hypersurfaces in weighted projective spaces; and the fibration f : X → P^1 embeds into a weighted projective bundle as an anticanonical hypersurface with at most Ordinary Double Point singularities. I will demonstrate how to construct twisting data on F, leading to finite lists of mildly singular CY3s. I will also comment on a projective version of deformations of these CY3s that revisit earlier works of Gross and Ruan. This is part of my submitted DPhil thesis and joint work arXiv:2210.10559 with Balazs Szendroi.
Location: Rm. 505, NCTS (Cosmology Bldg., NTU)
2022.12.29 (Thursday)
03:30PM-04:30PM
Speaker: 蘇品丞 (video)
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
2022.12.19 (Monday)
04:00PM-0.5:00PM
Speaker: Chuyu Zhou (EPFL) (video)
Title: On multiplicities of fibers of Fano fibrations.
Abstract: In this talk, we will focus on a conjecture on multiplicities of fibers of Fano fibrations, and present that several important conjectures in birational geometry (including Mckernan-Shokurov conjecture and boundedness conjecture for RC Calabi-Yau varieties) could be reduced to this multiplicity conjecture.
Location: Rm. 505, NCTS (Cosmology Bldg., NTU)
2022.12.09 (Friday)
09:00AM-10:00AM
Speaker: Mads Villadsen (Stony Brook University) (video)
Title: Generic Vanishing and Chen-Jiang Decompositions
Abstract: The Generic Vanishing theorem of Green and Lazarsfeld describes the behaviour of the cohomology of direct images of canonical bundles to abelian varieties when twisted by line bundles of degree zero. I will discuss Chen-Jiang decompositions of these direct images, first introduced by J. Chen and Z. Jiang for generically finite morphisms to abelian varieties, which explain their generic vanishing behaviour and certain positivity properties in detail. In particular, I will discuss how to prove the existence of these decompositions using classical variational Hodge theory.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
Workshop - Younger Generation in Moduli Spaces and Related Topics
2022.12.02 - 2022.12.04
Organizers:
Jungkai Chen (NTU), Ching-Jui Lai (NCKU)
Agenda:
2022/12/02 (Friday) Venue: 3173, Mathematics Bldg.
16:30-16:40 Welcome address
16:45-17:45 Talk 1: Bin Nguyen (NCTS) Some new algebraic surfaces with maximal Picard number
2022/12/03 (Saturday) Venue: 36106, Science Bldg.
09:00-09:45 Presentation A-1
09:50-10:35 Presentation A-2
10:50-11:35 Presentation B-1
11:40-12:25 Presentation B-2
13:30-14:30 Talk 2: Ryo Yamagishi (NTU) McKay correspondence and moduli of G-constellations
14:50-15:35 Presentation C-1
15:40-16:25 Presentation C-2
2022/12/04 (Sunday) Venue: 36106, Science Bldg.
09:00-09:45 Presentation D-1
09:50-10:35 Presentation D-2
10:40-10:50 Closing address
Titles and speakers:
A. C. Sabbah, Introduction to the theory of D-modules lecture notes (Nankai 2011) (beginning part)
陳毅鴻、黃建順
B. S. Mukai. On the moduli space of bundles on K3 surfaces. I. Vector bundles on algebraic varieties (Bombay, 1984), 341–413,
Tata Inst. Fund. Res. Stud. Math., 11, Tata Inst. Fund. Res., Bombay, 1987
鄭容濤、張志煥
C. G. Brown, Flips arising as quotients of hypersurfaces. Math. Proc. Cambridge Phil. Soc.127, (1999), 13-31
張宏彬、蘇品丞
D. T. Bridgeland, Flops and derived categories, Invent. Math. 147, (2002), 613-632
施政邦、中橋一萌
Location: National Cheng-Kung University
2022.09.15 (Thursday)
Part A. 03:30PM-04:30PM
Speaker: Hong-Bin Chang (張宏彬) (video)
Title: Examples of invariants and moduli
Ref. Mukai's book Chapter 1
Part B. 04:30PM-05:30PM
Speaker: Yi-Hong Chen (陳毅鴻) (video)
Title: Classical Hodge theory
Ref. Steenbrink's book, 1.1-1.2
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
2022.05.03
May 3 (Tuesday) 04:00PM-05:00PM
Speaker: Marta Benozzo (University College London)
Title: Iitaka conjecture for anticanonical divisors in positive characteristics
Abstract: Given a fibration f: X -> Z, a natural question is how we can relate the Kodaira dimensions of X, Z, and the fibres. The Iitaka conjecture addresses this problem. In characteristic 0, it is proven in many cases, while it is known to not hold in general in positive characteristics. Recently, a similar statement for the anticanonical divisors was proven in characteristic 0. The same results can be extended also to positive characteristics in low dimensions. At the end, we will see also some counterexamples constructed from Tango-Raynaud surfaces.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.04.29
Apr. 29 (Friday) 09:00AM-10:00AM
Speaker: Jayan Mukherjee (Brown University)
Title: Deformations and moduli of irregular canonical covers with $K^2=4p_g-8$
Abstract: In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $K^2 = 4p_g-8$, for any even integer $p_g \geq 4$. These surfaces also have unbounded irregularity $q$. We carry out our study by investigating the deformations of the canonical morphism $\varphi: X \to \mathbb{P}^N$, where $\varphi$ is a quadruple Galois cover of a smooth surface of minimal degree. As a result, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality $p_g > 2q-4$, with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. The existence of jumping subloci is a contrast with the moduli of surfaces with $K^2 = 2p_g- 4$, studied by Horikawa. These irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus $g\geq 3$, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.04.28
Apr. 28 (Thursday) 09:00AM-10:30AM
Speaker: Renee Bell (U Penn)
Title: Monodromy of Tamely Ramified Covers of Curves
Abstract: The étale fundamental group \pi_1^{et} in algebraic geometry formalizes an analogy between Galois theory and topology, extending our intuition to spaces in which loops, as defined traditionally, do not yield meaningful information. For a curve X over an algebraically closed field of characteristic 0, finite quotients of $\pi_1^{et}$ can be described solely in topological terms, but in characteristic p, dramatic differences and new phenomena have inspired many conjectures, including Abhyankar's conjectures. Let k be an algebraically closed field of characteristic p and let X be the projective line over k with three points removed. In joint work with Booher, Chen, and Liu, we show that for each prime p ≥ 5, there are families of tamely ramified covers with monodromy the symmetric group S_n or alternating group A_n for infinitely many n, producing these covers from moduli spaces of elliptic curves, and relating the fiber of these covers to the Markoff surface.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.04.15
Apr. 15 (Friday) 03:00PM-04:00PM
Speaker: Meng Chen (Fudan University)
Title: On explicit birational geometry of higher dimensional varieties
Abstract: In this seminar, I will talk about some new progress on estimating the lower bound of canonical volumes and the upper bound of canonical stability indices for higher dimensional varieties of general type. I will mainly introduce the construction of examples with small volumes and a general machinery of studying the canonical stability index.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.04.01
Apr. 1 (Friday) 10:00AM-11:00AM
Speaker: Zhi Jiang (Fudan University)
Title: Cohomological support loci and pluricanonical systems of irregular varieties
Abstract: We will discuss a generalization of a result of Chen and Hacon saying that the cohomological support loci of a variety of general type and of maximal Albanese dimension generates the Picard variety. We then discuss various applications of this kind of results.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.03.25
Mar. 25 (Friday) 09:00AM-10:00AM
Speaker: Shihoko Ishii (The University of Tokyo)
Title: A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds
Abstract: In the talk I will show that the minimal log discrepancy of every pair consisting of a smooth 3-fold and a "general" real ideal is computed by the divisor obtained by at most two weighted blow ups.
Our proof suggests the following conjecture: Every pair consisting of a smooth N-fold and a "general” real ideal is computed by a divisor obtained by at most N-1 weighted blow ups.
This is regarded as a weighted blow up version of Mustata-Nakamura’s conjecture.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
Workshop - Invitation to a Singular Promenade
2022.03.18 - 2022.03.20
Organizers:
Jungkai Chen (NTU), Ching-Jui Lai (NCKU), Hsueh-Yung Lin (NTU)
Aim & Scope:
The purpose of this workshop is to understand singularities from various aspects of mathematics, for students as well as active researchers. In addition to advanced research talks, we will have some students’ presentations, following some chapters of Ghys’ beautiful book “A Singular mathematical promenade”.
Agenda:
2022/03/18 (Friday)
19:00-19:50 林學庸 (NTU) Where are we? Where do we go?
20:00-20:20 陳毅鴻 (NTU) Intersecting polynomials: Maxim Kontsevich
20:20-20:40 梁珮欣 (NTHU) Separable permutations
2022/03/19 (Saturday)
09:30-09:50 黃建順 (NTU) Moebius and his band
09:50-10:10 蘇品丞 (NTU) Moebius necklaces
10:30-11:20 陳正傑 (NCU) Accumulation points of threefold canonical thresholds
13:00-13:20 施政邦 (NTU)
13:20-13:40 張志煥 (NTU) Some formal algebra
13:40-14:00 姚皓勻 (NTU) Convergence Le calcul des limites de Cauchy
14:20-14:40 范谷瑜 (NTU) Resolution of singularities
14:40-15:00 公奕 (Kyoto Univ.) Victor Puiseux, at last!
15:10-15:30 蔡以心 (NTU) Analytic chord diagrams I: an algorithm
15:30-15:50 張宏彬 (NTU) Analytic chord diagrams II: interlace graphs
15:50-16:10 鄭容濤 (NTU)
2022/03/20 (Sunday)
09:30-10:20 陳延安 (NCTS) Foliated MLD and LCT
10:40-11:30 賴青瑞 (NCKU) From RDP's to klt singularities
11:30-11:45 陳榮凱 (NTU) Closing Remark
Location: National Dong-Hua University, Meilun Campus, Hua-Lien
2022.03.17
Mar. 17 (Thursday) 3:50PM-5:10PM
Speaker: Vicente Lorenzo (Universidade de Lisboa)
Title: Group Actions on Horikawa Surfaces
Abstract:
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.03.11 (rescheduled)
Mar. 11 (Friday) 4:00PM-5:30PM
Speaker: Alexei Latyntsev (University of Oxford)
Title: Quantum Vertex Algebras and Cohomological Hall Algebras
Abstract: There is an extremely rich history of interaction between string theory and the mathematics of moduli spaces, for instance cohomological Hall algebras/algebras of BPS states, or vertex/chiral algebras. In this talk, I will explain a link between two of these: Joyce's vertex algebras attached to the moduli stack of objects in an abelian category, and one dimensional CoHAs. This is based on my recent paper 2110.14356, whose main result says that the cohomologies of such stacks are ``quantum vertex algebras": the factorisation/vertex analogues of quasitriangular bialgebras. The main technical tool is a "bivariant" Euler class which makes torus localisation work in this context. I will discuss applications of these techniques to CoHAs of coherent sheaves on a curve and future directions.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.02.25
Feb. 25 (Friday) 10:00AM-11:00AM
Speaker: Yongnam Lee (Korea Advanced Institute of Science and Technology)
Title: Total dual VMRT and its Application to the Bigness of the Tangent Bundle of Fano Threefolds
Abstract: Recently, the bigness of the tangent bundle of Fano threefolds with Picard number 1 were determined by Höring, Liu, and Shao by using the total dual VMRT. In this talk, I will explain the theory of the total dual VMRT and its application to study the bigness of the tangent bundle of a Fano threefold or the blow-up of nondegenerate smooth curve in P^3. Through an explicit description of irreducible effective divisors in P[T_x] constructed from the total dual VMRT of the family of lines and conics, we determine successfully the bigness of the tangent bundle of whole 36 deformation types of Fano threefolds with Picard number 2. This is a joint work with Hosung Kim and Jeong-Seop Kim.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.01.21
Jan. 21 (Friday) 10:00AM-11:00AM
Speaker: Rahul Singh (University of Pittsburgh)
Title: Counting Parabolic Principle G-bundles with Nilpotent Sections over P^1
Abstract: A Higgs bundle over an algebraic curve is a vector bundle with a twisted endomorphism. An important question is to calculate the volume of the groupoid of Higgs bundles over nite elds. In 2014, Olivier Schiffmann succeeded in finding the corresponding generating function and together with Mozvogoy reduced the problem to counting pairs of a vector bundle and a nilpotent endomorphism. It was generalized recently by Anton Mellit to the case of Higgs bundles with regular singularities. An important step in Mellit's calculations is the case of P^1 and 2 marked points, which allows him to relate the corresponding generating function with the Macdonald polynomials.
It is a natural question to generalize Mellit's calculations to arbitrary reductive groups. In my work, I consider the case of P^1 with 2 points for an arbitrary split connected reductive group G over Fq . Firstly, I give an explicit formula for the number of Fq-rational points of generalized Steinberg varieties of G. Secondly, for each principal G-bundle over , I give an explicit formula counting the number of triples consisting of parabolic structures at 0 and infinity and compatible nilpotent sections of the associated adjoint bundle.
Jan. 21 (Friday) 4:00PM-5:00PM
Speaker: Jeroen Hekking (KTH Royal Institute of technology)
Title: Derived Blow-ups Using Rees Algebras and Virtual Cartier Divisors
Abstract: The blow-up B of a scheme X in a closed subscheme Z enjoys the universal property that for any scheme X' over X such that the pullback of Z to X' is an effective Cartier divisor, there is a unique morphism of X' into B over X. It is well-known that the blow-up commutes along flat base change.
In this talk, I will discuss a derived enhancement B' of B, namely the derived blow-up, which enjoys a universal property against all schemes over X, satisfies arbitrary (derived) base-change, and contains B as a closed subscheme. To this end, we will need some elements from derived algebraic geometry, which I will review along the way. This will allow us to construct the derived blow-up as the projective spectrum of the derived Rees algebra, and state its functor of points in terms of virtual Cartier divisors, using Weil restrictions.
This is based on ongoing joint work with Adeel Khan and David Rydh.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2022.01.07
Jan. 7 (Friday) 4:00PM-5:00PM
Speaker: Marco Andreatta (Università di Trento, Italy)
Title: Lifting from An ample section the case of weighted blow-ups
Abstract: A classical method to study a projective variety is to consider its hyperplane section and ”lift” the properties of the section to the variety. This is sometime called Apollonius method and it works well since in general a variety is at least as special as any of its hyperplane sections. For example a weighted projective space can be an hyperplane section only of a weighted projective space (S. Mori 1975). We extend this result in a ”relative situation”, namely we consider f : X → Z to be a local, projective, divisorial contraction between normal varieties of dimension n with Q-factorial singularities and Y ⊂ X to be a f-ample Cartier divisor. If f|Y : Y → W has a structure of a weighted blow-up then f : X → Z, as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f : X → Z from a variety X with terminal Q-factorial singularities, which contracts a prime divisor E to an isolated Q-factorial singularity P ∈ Z, such that −(KX + (n − 3)L) is f-ample, for a f-ample Cartier divisor L on X. Using the above result, the existence of a ”good” general section of L and the existing results in dimension 3, we prove that (Z, P) is a hyperquotient singularity and f is a weighted blow-up.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.12.17
Dec. 17 (Friday) 10:00AM-11:00AM
Speaker: Kento Fujita (Osaka University)
Title: The Calabi problem for Fano threefolds
Abstract: There are 105 irreducible families of smooth Fano threefolds, which have been classified by Iskovskikh, Mori and Mukai. For each family, we determine whether its general member admits a Kaehler-Einstein metric or not.
This is a joint work with Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Suess and Nivedita Viswanathan.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.12.10
Dec. 10 (Friday) 3:00PM-4:00PM
Speaker: Yuri Prokhorov (Steklov Mathematical Institute, Russia)
Title: Towards a classification of Q-Fano threefolds
Abstract: We survey recent progress in classification of singular Fano threefolds with special emphasis on birational transformations between them.
Location: Rm. 509, NCTS (Cosmology Bldg., NTU)
2021.12.03
Dec. 03 (Friday) 10:00AM-11:00AM
Speaker: Priyankur Chaudhuri (Maryland)
Title: Nef and abundant divisors, generalized abundance and canonical bundle formula.
Abstract: In the 1980's, Kawamata showed that if the canonical divisor K_X of a normal projective variety X with possibly mild singularities is nef and abundant, then it is semiample (i.e. some multiple of it moves without base points). From the point of view of the abundance conjecture, this is considered to be an important result. In this talk, using canonical bundle formulas, I will discuss some generalizations of this well known theorem. The question we consider is this: if L is a nef line bundle on X such that K_X+L is nef and abundant, then is K_X+L semiample? We will answer some special cases of this question, in particular the cases when either K_X is effective or when the stable base locus of L has codimension atleast 2. This essentially extends Kawamata's theorem to the setting of the generalized abundance conjecture of Lazić and Peternell. I will also discuss an inductive approach to this conjecture involving nef dimension.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.11.26
Nov. 26 (Friday) 10:00AM-12:00AM
Speaker: Chi-Wei Chang (NCTS)
Title: Seshadri Constants and Characterization of Projective Spaces
Abstract: In this talk, I will discuss the main theorem of “Characterization of projective spaces by Seshadri contants” by Yuchen Liu and Ziquan Zhuang, saying that any Q-Fano variety X such that the Seshadri constant of the anti-canonical divisor is strictly greater than dim X at one smooth point must be the projective space. I will also discuss its relation to another conjectural characterization of projective spaces via Seshadri constants of tangent bundles, which was proposed in “Seshadri Constants for Vector Bundles” by Mihai Fulger and Takumi Murayama.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.11.19
Nov.19 (Friday) 10:00AM-12:00AM
Speaker: Jheng-Jie Chen (NCU)
Title: Study on an effective upper bound for anti-canonical volumes of canonical Q-Fano threefolds
Abstract: Chen Jiang discussed his recent works on anti-canonical volume of three dimensional projective varieties with only terminal singularities in our lab on 9/9. In this talk, I would like to go into the details of his paper (joint work with Yu Zou) "an effective upper bound for anti-canonical volumes of canonical Q-Fano threefolds". More precisely, the details of the arguments of Theorem 3.3, Propositions 4.1, 4.2 will be included.
Reference: Chen Jiang and Yu Zou, "An effective upper bound for anti-canonical volumes of canonical Q-Fano threefolds", arXiv: 2107.01037v1
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
NCTS Workshop on Arithmetic and Algebraic Geometry
2021.11.12
Nov.12 (Friday) 10:00AM-15:30PM
Organizers:
Jungkai Chen (NTU) Ching-Jui Lai (NCKU) Hsueh-Yung Lin (NTU)
Aim & Scope:
Many arithmetic problems can be understood in terms of algebraic geometry, which helps to develop many useful tools and theory. The purpose of this workshop is to introduce some of the problems and current research in this active topic.
Agenda:
10:00-10:50 Keiji Oguiso (Univ. of Tokyo)
11:10-12:00 Hang Fu (NTU)
13:30-14:20 Liang-Chung Hsia (NTNU)
14:40-15:30 Ting-Yu Lee (NTU)
Location: R101, Provincial Government Building, Tunghai University
2021.11.11
Nov.11 (Thursday) 4:00PM-5:00PM
Speaker: Konstantin Shramov (Higher School of Economics)
Title: Biratonal Automorphism Groups of Algebraic Varieties
Abstract: I will survey various results on the structure of finite groups acting by automorphisms and birational automorphisms of algebraic varieties in characteristic zero. The main focus will be on the boundedness properties of such groups, including the so-called Jordan property. If time permits, I will explain how (some of) these results generalize to the case of varieties over fields of positive characteristic.
Location: broadcast in Room 440, Astro-Math Bldg., Online seminar
Join Zoom Meeting
https://us02web.zoom.us/j/87908681099?pwd=OVVDVUpveXZxK1BER1VHMnBZVmcxUT09
Meeting ID: 879 0868 1099
Passcode: LabBirGeom
2021.11.05
Nov.05 (Friday) 10:00AM-11:00AM
Speaker: Tong Zhang (East China Normal University)
Title: Noether-Severi inequality and equality for irregular threefolds of general type
Abstract: In this talk, I will introduce the optimal Noether-Severi inequality for all smooth and irregular threefolds of general type. It answers in dimension three an open question of Z. Jiang. I will also present a complete description of canonical models of smooth and irregular threefolds of general type attaining the Noether-Severi equality. This is a joint work with Yong Hu.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.10.22
Oct.22(Friday) 10:00AM-12:00AM
Speaker: Jheng-Jie Chen (NCU)
Title: Computing 3-fold canonical thresholds
Abstract: In this talk, we will briefly introduce the canonical thresholds, Sarkisov Program and the classification of divisorial contractions that contract divisors to points. Then, I show that the set T^{can}_{3} of 3-fold canonical thresholds satisfies the ascending chain condition (ACC). Besides, we give the computation T^{can}_{3} in the half open interval (1/2,1).
Location: Rm 101, Astro-Math Bldg.
2021.10.15
Oct.15(Thursday) 10:00AM-12:00AM
Speaker: Yen-An Chen (NCTS)
Title: Semistability and foliations
Abstract: This is a survey talk on semistability and foliations. The topics we will cover are 1) the maximal destabilizing subsheaf of the tangent sheaf is a foliation; 2) algebraic integrability of the foliations; and 3) generic semipositivity of the tangent sheaf if the anticanonical class is nef.
Location: Rm 101, Astro-Math Bldg.
2021.10.07
Oct.07 (Thursday) 10:00AM-11:00AM
Speaker: Justin Lacini (Kansas University)
Title: On pluricanonical maps of varieties of general type.
Abstract: Hacon and McKernan have proved that there exist integers rn such that if X is a smooth variety of general type and dimension n, then the pluricanonical maps |rKX| are birational for r ≥ rn. These values are typically very large: for example r3 ≥ 27 and r4 ≥ 94. In this talk we will show that the r-th canonical maps of smooth threefolds and fourfolds of general type have birationally bounded fibers for r ≥ 2 and r ≥ 4 respectively. Furthermore, we will generalize these results to higher dimensions in terms of the constants rn and we will discuss recent progress on a conjecture of Chen and Jiang.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.09.30
Sep.30 (Thursday) 4:00PM-5:00PM
Speaker: Ivan Cheltsov (University of Edinburgh)
Title: K-stable Fano threefolds
Abstract: A smooth Fano manifold X admits a Kahler-Einstein metric if and only if X is K-polystable (K-stable if the automorphism group of X is finite). For smooth del Pezzo surfaces (two-dimensional Fano manifolds), Tian proved that the K-polystability is equivalent to the reductivity of the automorphism group. In this talk, I will explain which smooth Fano threefolds (three-dimensional Fano manifolds) are K-polystable and which are not K-polystable. This is a joint project with Carolina Araujo (IMPA), Ana-Maria Castravet (Versailles), Kento Fujita (Osaka), Anne-Sophie Kaloghiros (London), Jesus Martinez-Garcia (Essex), Constantin Shramov (Moscow), Hendrik Suess (Jena) and Nivedita Viswanathan (Loughborough).
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.09.23
Sep.23 (Thursday) 10:00AM-12:00AM
Speaker: lacopo Brivio (NCTS)
Title: Frobeniuns stable pluricaonical system
Ref. arXiv 2010.08897
Location: Rm. 515, NCTS (Cosmology Bldg., NTU) CTS(Cosmology Bldg.,NTU)
2021.09.16
Sep.16 (Thursday) 10:00AM-12:00AM
Speaker: Chi-kang Chang (NTU)
Title: Kawamata's boubdedness of Q-Fano 3-folds
Ref. Y . Kawamata, "Boundedness of Q-Fano threefolds, " in Proc. Int. Conf. on Algebra, Novosibirsk, 1989(Am. Math. Soc., Providence, Rl, 1992), Part3, Contemp. Math. 131, pp. 439-445.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.09.09
Sep.09 (Thursday) 10:00AM-11:00AM
Speaker: Chen Jiang (Fudan)
Title: Explicit boundedness of canonical Fano 3-folds: known results and open problems
Abstract: Motivated by the classification of canonical Fano 3-folds, we are interested in boundedness results on diffrent kinds of canonical Fano 3-folds, such as anticanonical systems, indices, degrees, and so on. I will summarize known results with some progress (based on joint works with Meng Chen and Yu Zou) and open problems in this area.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.09.02
Sep. 02 (Thursday) 3:00PM-4:00PM
Speaker: Paolo Cascini (Imperial)
Title: On the minimal model for algebraically integrable foliations
Abstract: Every fibration, or more in general, every dominant rational map between normal varieties, defines a natural foliation, which is called algebraically integrable. The canonical sheaf of such a foliation behaves, in many aspects, as the canonical sheaf of a normal variety. I will describe some recent results in this direction, such as a cone theorem, and some applications on the canonical bundle formula. In particular, this provides a proof of a conjecture by Shokurov.
This is joint work with Ambro, Shokurov and Spicer.
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.08.26
Aug. 26 (Thursday) 10:00AM-12:00AM
Speaker: Ching-Jui Lai
Title: On anti-canonical volume of Fano threefolds
Location: Rm. 515, NCTS (Cosmology Bldg., NTU)
2021.08.19
Aug. 19 (Thursday) 10:00AM-12:00AM
Speaker: Jungkai Chen (NTU)
Title: Some open problems on varieties of general type
Location: Rm. 440, Astro-Math Blag.
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Hajime Nakahashi
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Yi-Hong Chen
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Phín-sîng Soo
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郭倍誠
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張志煥
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黃建順
臺灣大學數學系
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臺灣大學數學系
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Annie Wang
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Roufang Chou
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