Program & Abstract

Jun-Muk Hwang (Institute for Basic Science)
Varieties of minimal rational tangents on cubic threefolds
  The tangent directions of lines through a general point x on a smooth cubic threefold X determine a collection of six points on a conic, namely, a point p(x) in M0,6 (up to ordering). As we vary x over X, does the point p(x) also vary in the maximal possible way over M0,6? An affirmative answer for a general cubic threefold is given by G. Jennings in 1987 using the method of moving frames in differential geometry. We explain an algebro-geometric proof using the Segre cubic primal and discuss it's generalization to higher dimensions.

Meng Chen (Fudan University)
Several moduli spaces of canonical threefolds with small invariants
  We present an application of MMP to determine several moduli spaces of canonical 3-folds with relatively small invariants. As a byproduct, the Noether inequality for 3-folds with pg=5 is proved. This is a joint work with Yong Hu and Chen Jiang.

Yujiro Kawamata (University of Tokyo)
On twisted non-commutative deformations of algebraic varieties
  Lowen and Van den Bergh developed a deformation theory of abelian categories. I would like to explain a geometric approach to the twisted non-commutative deformations of algebraic varieties. I will also talk about the problem of extending tilting bundles under such deformations.

Young-Hoon Kiem (Korea Institute for Advanced Study)
Gromov-Witten invariants for branched covers
  A fundamental idea in computing Gromov-Witten invariants is to push the computation to simpler spaces like projective spaces. When the target manifold X is a complete intersection in a projective space P, the virtual fundamental class of the moduli space M(X) of stable maps to X coincides with the cosection localized virtual fundamental class of the moduli space of stable maps to P with an additional field. Hence we can enumerate curves in X by studying certain decorated moduli spaces of curves in P. In this talk, I will extend this idea to the case where the target manifolds are branched covers of simpler spaces. Based on a joint work with Hyeonjun Park.

De-Qi Zhang (National University of Singapore)
Wild automorphisms of projective varieties (maps having no invariant proper subsets), and Motivations from Gelfant-Kirillov dimension
  An automorphism g on a projective variety X is a wild automorphism if it does not fix any Zariski closed proper subset of X. It has been conjectured by Reichstein, Rogalski and J J Zhang 18 years ago that such an X is an abelian variety, and g is of zero entropy. We report the progress on it, and also the motivation of it from the Gelfand-Kirillov dimension. This is based on the joint paper with Oguiso and early work with HY Lin and TC Dinh.

Keiji Oguiso (University of Tokyo)
Automorphisms with Zariski dense orbits of Calabi-Yau threefolds with birational c2-contraction
  We work over C or Q̅. After explaining some background, motivation of the problem, I would like to describe the full automorphism groups of Calabi-Yau threefolds with birational c2-contraction (two special Calabi-Yau threefolds) and then determine which automorphism admits Zariski dense orbit in terms of (number theoretic property of) the first dynamical degree and/or the non-existence of an invariant non-constant rational function, with explicit examples.

Jongil Park (Seoul National University)
A study on ℚ-homology projective planes with quotient singularities
  A normal projective surface with the same Betti numbers of the projective plane ℂℙ2 is called a rational homology projective plane (briefly a ℚ-homology ℂℙ2). People working in algebraic geometry and topology have long studied a ℚ-homology ℂℙ2 with possibly quotient singularities. It is now known that it has at most five such singular points, but it is still mysterious so that there are many unsolved problems left. In this talk, I'll review some known results and open problems in this field which might be solved and might not be solved in near future. In particular, I'd like to review some recent progress on the classification of ℚ-homology ℂℙ2s with quotient singularities. This is a joint work with Woohyeok Jo and Kyungbae Park.

Francesco Polizzi (Università degli Studi di Napoli Federico II)
Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants
  We investigate some finite, non-abelian quotients G of the pure braid group on two strands P2(Σb), where Σb is a closed Riemann surface of genus b. Building on our previous work on some special systems of generators on finite groups that we called "diagonal double Kodaira structures", we prove that, if G has not order 32, then |G| ≥ 64, and we completely classify the cases where equality holds. In the last section, as a geometric application of our algebraic results, we construct two 3-dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental groups. This is a joint work with Pietro Sabatino.

Giancarlo Urzúa (Pontificia Universidad Católica de Chile)
Classification of Horikawa surfaces with T-singularities
  Horikawa surfaces are nonsingular minimal complex surfaces of general type with K2=2pg-4. They were classified by Eiji Horikawa in the 70s. An interest in degenerations of Horikawa surfaces arises because of a famous problem proposed by Horikawa in 1976 about their diffeomorphism type when K2=16t. Particularly interesting for this problem is to know about degenerations with only T-singularities. Until now there was only one example of such degenerations due to Yongnam Lee and Jongil Park 2011 (found by Fintushel-Stern 1996 as smooth 4-manifold), we call them Lee-Park examples. Together with Jaime Negrete and Vicente Monreal, in this recent pre-print https://arxiv.org/pdf/2410.02943 we classify all Horikawa surfaces with T-singularities, and we prove that for pg≥10 the only smoothable surfaces are Lee-Park examples (and for just one component when there are two). In this classification we introduce a new family of surfaces, which we call small surfaces. Horikawa T-surfaces have 8 families of small surfaces, and so it proposes new diffeomorphism questions via rational blowdown. The novel techniques in our pre-print allow us to classify T-surfaces with K2=2pg-3, e.g. quintics and I-surfaces. My talk will be about the various ideas behind the proofs.

Jungkai Chen (National Taiwan University)
On irregular threefolds of general type
  There are several recent advances in the study of threefolds of general type. Among threefolds of general type, irregular threefolds are considered to be more well-behaved. In this talk, we are going to introduce some recent works in progrss along the investigation of irregular threefolds. In particular, we are going to show the lowever bound of canonical volume, Noether-Severi boundary, and some new examples. The talk is based on a joint work in progress with Y. Hu, Z. Jiang and T. Zhang.

Gian Pietro Pirola (Università degli Studi di Pavia)
Some results and problems on the sub-fields of the rational field of algebraic varieties
  Let X be a complex algebraic variety of dimension n. A consequence of the fundamental work of Birkar, Cascini, Hacon and Mc Kernan is that the set M(X) of generically finite rational dominant maps f: X⇢Y, where Y is of general type (up to birational transformation of X and Y) is finite. Let m(X) be its cardinality of M(X).
  We discuss the following related results and problems:
(1) Give upper bounds of m(X) (de Franchis Problem);
(2) Compute m(X) when [X] is the general point of some moduli space (joint work with Yongnam Lee).
If there is time we will also discuss a problem (of Zariski) on the possible Galois groups of the splitting fields associated dominant maps f: X⇢Pn.

Donghoon Hyeon (Seoul National University)
Moduli space of languages

Yongnam Lee (KAIST & Institute for Basic Science)
Looking back on my past while researching algebraic geometry

Jihun Park (POSTECH & Institute for Basic Science)
Simply connected positive Sasakian 5-manifolds and log del Pezzo surfaces
  Sasakian geometry is a vibrant field at the intersection of differential geometry, topology, complex geometry, and algebraic geometry, with applications ranging from theoretical physics to geometric analysis. In this talk, we explore closed simply connected 5-manifolds capable of hosting positive Sasakian structures.

Fabrizio Catanese (Universität Bayreuth)
Automorphisms acting trivially on rational/integral cohomology and the case of properly elliptic surfaces
  For a cKM X the group Autℚ(X) of numerically trivial automorphisms is the subgroup acting trivially on rational cohomology, and the group Autℤ(X) of cohomologically trivial automorphisms is the subgroup acting trivially on integral cohomology. If Aut0(X) is the connected component of the identity in the Lie group Aut(X), then we have a sequence of inclusions of normal subgroups Aut0(X) < Autℤ(X) < Autℚ(X), and Lieberman and Fujiki in the 70’s showed that Aut0(X) has finite index in Autℚ(X). The main question which I started to investigate with Wenfei Liu is that of giving bounds for the respective indices, after we realized that many results in the literature were incorrect. Already for algebraic surfaces the situation is not clear, except for surfaces of Kodaira dimension 0 or negative which was clarified together with W. Liu. In joint work with him and Matthias Schuett we studied the case of S minimal of Kodaira dimension 1 and χ>0. Here we show that the group Autℚ(X) of numerically trivial automorphisms can be arbitrarily large, and essentially described: moreover there upper bounds depending on the bigenus P2(S) and on the irregularity q(S). More difficult is the study of Autℤ(S). Part I, which is joint work also with Davide Frapporti and Christian Gleissner is devoted to the case where χ=0, and S is isogenous to an elliptic product. If Aut0(S) is infinite (i.e., S is pseudo-elliptic), the index of Aut0(S) inside Autℤ(X) is at most 2, and we classify exactly the cases where the index is 2. If S is not pseudo elliptic, but with χ=0, we show that Autℤ(S) can be only ℤ/2, ℤ/3, (ℤ/2)2, and that the first two cases do effectively occur. Also for χ>0, we have  examples where Autℤ(S) has 2 or 3 elements. It is an interesting question to find examples of minimal surfaces with Kodaira dimension 1 or 2 having |Autℤ(X)| ≥ 4.

JongHae Keum (Korea Institute for Advanced Study)
Cox rings of regular surfaces
  The Cox ring of a variety is the total coordinate ring, i.e., the direct sum of all spaces of global sections of all divisors. When this ring is finitely generated, the variety is called Mori dream (MD). A necessary condition for being MD is the finite generatedness of the divisor class group, i.e., the vanishing of the irregularity. Smooth rational surfaces with big anticanonical divisor are MD. So are all del Pezzo surfaces. Rational elliptic surfaces are not MD, if they are general. A K3 surface or an Enriques surface is MD if and only if its automorphism group is finite. In this talk I will consider the case of surfaces of general type with geometric genus pg=0, and provide several examples that are MD. I will also provide examples that are not MD. This is a joint work with Kyoung-Seog Lee.