Algebraic Geometry International Conference




June 3-7, 2019                            KIAS 1503, Seoul, Korea

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Masayuki Kawakita (Kyoto University)

- Title: On boundedness of divisors computing the minimal log discrepancy on a smooth threefold

- Abstract: The minimal log discrepancy is an invariant of singularities which is closely related to termination of flips. When one works on a smooth variety, the ACC for minimal log discrepancies is equivalent to a certain boundedness of divisors computing the minimal log discrepancy. I will introduce an approach towards this boundedness in dimension three.


Yongnam Lee (KAIST)

- Title: The moduli space of smooth ample hypersurfaces in projective spaces or in abelian varieties,

- Abstract: In this talk, we give a structure theorem for projective manifolds $W_0$ with the property of admitting a 1-parameter deformation where $W_t$ is a hypersurface in a projective smooth manifold $Z_t$. Their structure is the one of special iterated univariate coverings which we call of normal type, which essentially means that the line bundles where the univariate coverings live are tensor powers of the normal bundle to the image $X$ of $W_0$. We give an application to describe an irreducible connected component of moduli space of smooth ample hypersurfaces in projective spaces or in abelian varieties. This is a joint work with Fabrizio Catanese.


Joonyeong Won (KIAS CMC)

- Title: Simply connected Sasaki-Einstein rational homology 5-sphere

- Abstract: We study existence of Sasaki-Einstein metric on 5-dimensional Samale manifold. In particular we completely determine which connected rational homology 5-sphere admit Sasaki-Einstein metrics.


Yoshinori Gongyo (University of Tokyo)

- Title: A generalization of Batyrev's cone conjecture
- Abstract: We discuss a generalization of Batyrev's cone conjecture for curves moving in co dimension $l$. Thus we propose the cone conjecture integrating Mori's cone theorem and the Batyrev's cone conjecture. Moreover we prove a weak version of this conjecture by proving some rationality theorem. This is a work in progress with Sung Rak Choi.


Takuzo Okada (Saga University)

- Title: Birationally superrigid Fano 3-folds of codimension 4

- Abstract: We have satisfactory results on both classification and birational (super)rigidity for (terminal) Fano 3-folds of index 1 embedded in a weighted projective space as a codimension at most 3 subvariety. I will talk about birational superrigidityof Fano 3-folds of index 1 embedded in a wps as a codimension 4 subvariety.


Stefan Schreieder (Ludwig Maximilian University of Munich)

- Title: Variation of stable birational types

- Abstract: We explain how to use decompositions of the diagonal to study the variation of the stable birational types of Fano hypersurfaces over fields of arbitrary characteristic. This had been initiated by Shinder, whose method works in characteristic zero.


Yuri Prokhorov (Steklov Institute & Higher School of Economics)

- Title: Rationality of Fano 3-folds over nonclosed fields

- Abstract: I discuss rationality problem for smooth Fano threefoldsof Picard rank one over nonclosed fields. The talk is based on joint work with Alexander Kuznetsov (inprogress).


Seung-Jo Jung (Seoul National University)

- Title: Hodge ideals and spectrum of isolated hypersurface singularities

- Abstract: Recently, Mustaţă-Popa introduced Hodge ideals for Q-divisors on a smooth variety. This talk gives a brief introduction to their theory and discusses a connection between the Hodge ideals and the Hodge spectrum. This is based on joint work with I.-K. Kim, Y. Yoon and M. Saito.


Costya Shramov (Steklov Institute & Higher School of Economics)

- Title: Automorphisms of Severi-Brauer surfaces

- Abstract: I will discuss finite groups acting by automorphisms and birational automorphisms of Severi-Brauer surfaces. We will see that they are bounded provided that the base field is a function field, and also make some observations on their structure in the general case. Also, we will see that in certain cases this applies to automorphism groups of higher-dimensional Severi-Brauer varieties.


Keiji Oguiso (University of Tokyo)

- Title: Complex dynamics of inertia groups on surfaces
- Abstract: We study the inertia groups of some smooth rational curves on 2-elementary K3 surfaces and singular K3 surfaces from the view of topological entropy, with an application to a long standing open question of Coble on the inertia group of a generic Coble surface. This is a joint work with Professor Xun Yu. 


Kyoung-Seog Lee (IBS-CGP)

- Title : Mori dream surfaces of general type with p_g=q=0

- Abstract : After their discovery, surfaces of general type with p_g=q=0 have been intensively studied and investigated. Especially, there have been lots of new developments in the theory of minimal surfaces of general type with p_g=q=0 in these days. In the first part of this talk, I will briefly review these new developments. Then I will discuss effective, nef, semiample cones and Cox rings of surfaces of general type with p_g=q=0. This talk is based on joint works (in progress) with Davide Frapporti and JongHae Keum.


Taro Sano (Kobe University)

- Title: Construction of non-Kähler Calabi-Yau 3-folds by smoothing normal crossing varieties

- Abstract: It is an open problem whether there are finitely many topological types of Calabi-Yau 3-folds. Kawamata--Namikawa developed log deformation theory of normal crossing varieties. By using this, we can construct examples of Calabi-Yau varieties from normal crossing varieties with uniruled irreducible components. I will explain construction of some non-Kähler Calabi-Yau 3-folds by this method. If time permits, I will also explain an example of involutions on a family of K3 surfaces which do not lift biregularly over the total space. This is based on joint work with Kenji Hashimoto.


Young-Hoon Kiem (Seoul National University)

- Title: Virtual cobordism classes

- Abstract: Many important homological invariants in algebraic geometry like ordinary Borel-Moore homology, Chow groups and K-groups share properties like projective pushforward, smooth pullback, A1-homotopy and Chern classes, which are the fundamental tools in intersection theory. These properties are often codified as the notion of oriented Borel-Moore homology theory. Recently Levine and Morel constructed a universal oriented Borel-Moore homology theory, called algebraic cobordism. The universality implies that once we prove a result on algebraic cobordism, the same should hold for all oriented Borel-Moore homology theories. In this talk, I will show how we can generalize key notions and techniques in virtual intersection theory, such as virtual fundamental class, virtual pullback, torus localization and cosection localization, to algebraic cobordism. Based on joint work with Hyeonjun Park.


De-Qi Zhang (National University of Singapore)

- Title: Equivariant Minimal Model Program, with a view towards algebraic and arithmetic dynamics

- Abstract: We elaborate the notion of “int-amplified” endomorphism f of a normal projective variety X, a property weaker than “polarised” yet preserved by products. We show that the existence of such a single f guarantees that every Minimal Model Program (MMP) is equivariant w.r.t. a finite-index submonoid of the whole monoid SEnd(X) of all surjective endomorphisms of X. Applications of the equivariant MMP are discussed: Kawaguchi-Silverman conjecture on the equivalence of arithmetic and dynamic degrees; Characterisation of a sub variety with Zariski dense periodic points. Some parts are based on joint work with Meng.


Jinhyung Park (Sogang University)

- Title: Singularities and syzygies of secant varieties of nonsingular projective curves

- Abstract: In recent years, singularities, defining equations, and syzygies of secant varieties have attracted considerable attention. In this talk, we consider the k-th secant variety of a nonsingular projective curve of genus g embedded by a line bundle of degree greater than 2g+2k+p. I show that the k-th secant variety has normal Du Bois singularities, is arithmetically Cohen-Macaulay, and satisfies the property N_{k+2,p}. This result is a generalization of Green's (2g+1+p)-theorem, and it settles conjectures of Sidman-Vermeire and Ullery. This is joint work with Lawrence Ein and Wenbo Niu.


Ivan Cheltsov (University of Edinburgh)

- Title: G-birationally super-rigid Fano varieties with small alpha-invariants.

- Abstarct: Recently, Stibitz and Zhuang proved K-stability of birationally super-rigid Fano varieties whose alpha-invariants are greater than 1/2. They asked whether alpha-invariants of birationally super-rigid Fano varieties are always greater than 1/2 or not. In this talk, we answer negatively an equivariant version of Stibitz-Zhuang question. Namely, we present series of examples of smooth G-birationally super-rigid Fano varieties with (arbitrary) small G-invariant alpha-invariants. This is a joint work with Ziquan Zhuang (Princeton).


Yong Hu (KIAS)

- Title: Fibered varieties over curves with low slope and sharp bounds in dimension three.

- Abstract: We will first construct varieties of any dimension $n>2$ fibered over curves with low slopes. These examples violate the conjectural slope inequality of Barja and Stoppino. Then we will forcus on finding the lowest possible slope when $n=3$. This is a joint work with Tong Zhang.