Aim: 한국에서 대수기하관련 공부를 하는 분들이 연구결과를 공유합니다. 

Participants:

1. 김민훈 (이화여대)

2. 김연수 (전남대)

3. 김인균 (KIAS)

4. 노현호 (충남대)

5. 박경동 (경상국립대)

6. 박진형 (KAIST)

7. 오정석 (서울대)

8. 원준영 (이화여대)

9. 이경석 (Postech)

10. 임우남 (연세대)

11. 정기룡 (경북대)

12. 정대웅 (충북대)

13. 정승조 (전북대)

14. 조용화 (경상국립대)

15. 최성락 (연세대)

16. 허석문 (성균관대)

17. 홍규식 (전주대)

18. 황동선 (IBSCCG)

Organisers: 김영훈(KIAS), 김인균(KIAS), 오정석(서울대), 정승조 (전북대), 홍규식 (전주대)

Schedule: 2025년 9월 17일 

12:30-13:30 점심

13:30-14:30 김연수

Title: Aubert duals of certain unitary representations for metaplectic groups

Abstract: The study of unitary representations of reductive groups plays a central role in the Langlands program; full classification of unitary representation is called the unitary dual problem. Among the tools developed to solve this problem, the Aubert involution provides a nontrivial construction of unitary representation and may serve as a step toward resolving the unitary dual problem.

In this talk, I will introduce the unitary dual problem and briefly review some related prior work. I will then briefly explain the Aubert dual and discuss ongoing joint work with Gyujin Oh on constructing Aubert duals of certain unitary representations of metaplectic groups.

14:30-15:30 

16:00-17:00 정기룡 

Title: Quartic curves in the quintic del Pezzo threefold

Abstract: Let X be the quintic del Pezzo threefold. By adjunction formula, the general intersection X with linear subspace H of codimension two is an elliptic quintic curve E_5. If we choose the linear subspace H containing a line lying in X, then E_5 is the union of that line and rational quartic curve meeting at two points. In this talk, we prove that each rational quartic curve takes arise in this way even though the curve may not be reducible. This is a parallel study with that of arXiv:2412.17721 and is a joint work with Jaehyun Kim and Jeong-Seop Kim.

17:00-18:00 홍규식

Title: Hypersurfaces in P^4 with positive defect

Abstract: Let V be a hypersurface in P^4 with ordinary double and triple points as the only singularities, and let μ_2 and μ_3 be the number of ordinary double points (triple points, respectively) of V. Then we prove that if μ_2 + 11μ_3 ≥ (11d^4-50d^3+85d^2-70d+48)/24, then the defect of V is positive. This is based on a joint work with Seung-Jo Jung.

18:30-20:00 저녁

20:00-23:00 토론

2025년 9월 18일

09:00-10:00 박진형

Title: Rank 3 quadratic equations for smooth projective varieties embedded by sufficiently positive line bundles 

Abstract: Han-Lee-Moon-Park proved that the defining ideals of Veronese embeddings are generated by rank 3 quadrics, and they conjectured that the same is true for smooth projective varieties embedded by sufficiently positive line bundles. In this talk, we confirm their conjecture using the geometry of Hilbert schemes of points. This is joint work with Daniele Agostini.

10:00-11:00 오정석

Title: A pullback between sheaves on log Calabi-Yau 4-folds 

Abstract: Given a section of a bundle with quadratic form, we construct a specialisation map between K-groups of matrix factorisations of the quadratic function from the space to the normal cone of the zero locus. When the section is isotropic, the quadratic function becomes zero and the construction recovers usual specialisation map in Fulton-MacPherson's intersection theory. 

When a log Calabi-Yau 4-fold (X,D) is given, we apply the construction to define a pullback from sheaves on D to ones on X after assuming the space of sheaves on D is a critical locus globally. For a Calabi-Yau 4-fold X, we artificially define "the space of sheaves on D" to be a point so that its structure sheaf pulls back to the virtual structure sheaf of the space of sheaves on X.

This is a joint work in progress with Dongwook Choa and Richard Thomas.

11:30-12:30 김인균

Title : K-stability of blow-ups of the weighted projective planes

Abstract: The K-stability of smooth del Pezzo surfaces is well understood. For instance, the projective plane is K-polystable, but it becomes K-unstable after one or two blow-ups, regains K-stability after three blow-ups, and retains this stability until the Fano condition is violated after eight blow-ups.

In this talk, we investigate the K-stability of blow-ups of weighted projective planes with weights (1, 1, n) and demonstrate that singular del Pezzo surfaces arising from n+4 blow-ups are K-stable. This result highlights a striking parallel between the stability patterns of the projective plane and the weighted projective plane, offering new insights into the broader relationship between blow-ups and K-stability in algebraic geometry.

12:30-13:30 점심

13:30-14:30 정승조

Title: Singularities and Topology of Projective Hypersurfaces

Abstract: This talk explores how singularities dictate the topology of projective hypersurfaces. In contrast to smooth cases, where the topology is determined by degree, singularities introduce a complexity that classical theorems by Lefschetz and Kato only partially explain, leaving a critical intermediate range of dimensions. We show how the modern theory of vanishing cycles analyzes this gap, revealing that the entire topological difference between a singular hypersurface and its smooth version is concentrated within this range. If time permits, I will present joint work with Prof. Hong along these lines.

14:30-15:30 원준영

Title : Birational geometry of Fano 3-fold WCIs

Abstract : Let W be a smooth projective variety. If K_W is not pseudo-effective, then by, the Minimal Model Program produces a birational model V of W which admits a very special fibre structure V → S called a Mori fibre space. We study rigidity and solidity in weighted complete intersections.

16:00-17:00 임우남

Title: Chern filtration of moduli spaces of sheaves

Abstract: The cohomology ring of moduli spaces of sheaves has been a central topic in enumerative geometry due to its rich additional structures, such as tautological generators and relations, Hall algebra, perverse filtration, etc. In this talk, I will introduce a relatively new structure, called the Chern filtration, and explain why it is interesting and how it interacts with existing structures. The main focus will be on moduli spaces of one-dimensional sheaves on surfaces and moduli spaces of bundles on curves. This is based on joint works with Y. Kononov, M. Moreira, W. Pi.

17:00-18:00 노현호

Title : Integrality of Tevelev degrees of local and formal CY geometry

Abstract : We study the integrality of Tevelev degrees of local and formal geometries of projective spaces. Combined with the result of virtual Tevelev degrees by Buch and Pandharipande, we study the integrality of the effective degrees of complete intersections in projective spaces.

18:30-20:00 저녁

2025년 9월 19일

09:00-10:00 이경석

Title: Intersection forms of surfaces isogenous to a product with p_g=0 of unmixed type

Abstract: Intersection forms play key roles in the theory of 4-manifolds, including algebraic surfaces. In their very interesting recent paper "Mori dream spaces and Q-homology quadrics", Cascini, Catanese, Chen and Keum asked if one can determine the intersection forms of surfaces isogenous to a product with p_q=0 of unmixed type. In this talk, I will discuss intersection forms of these surfaces. This talk is based on a joint work in progress with JongHae Keum and Do Geon Kim.

10:00-11:00 조용화

Title: Counting h^0 on primary Burniat surfaces

Abstract: A (primary) Burniat surface is a surface of general type with p_g=q=0 and K^2=6, which can be obtained by a smooth bidouble cover of a smooth del Pezzo surface of degree 6. The Picard group of it is an abelian group of rank 4 with torsions of order 64. Alexeev discovered that the semigroup of effective divisors is spanned by the ramification curves. In this talk, I will discuss an extension of Alexeev's result, where we developed an algorithm to compute h^0(D) for an arbitrary divisor D. As an application, we provide a remark on Ulrich bundles on primary Burniat surfaces.

11:30-12:30 최성락

Title: Birational properties of potential triples and a valuative approach toward -K-MMP.

Abstract: We introduce a new birational framework which can be used to study a variant of minimal model program.

We first define triples that are more general than the generalized pairs and present their basic properties.

As a main result, we show that the log canonical thresholds of klt triples can be computed by quasimonomial valuations. The results in this talk are based on the joint work with S.Jang, D.Kim, and D.Lee.

12:30-13:30 점심

13:30-18:00 토론

18:30-20:00 저녁

2025년 9월 20일

09:00-10:00 허석문

Title : On a rational self-map of the projective space of plane cubic curves. 

Abstract : The family of plane cubic curves is parametrized by the projective space of dimension nine. The assignment to a plane cubic curve its Hessian cubic is a 3-1 map. The Hessian cubic comes with a non-trivial 2-torsion divisor class, and it allows one to represent the Hessian as the symmetric determinant of a net of cubics. This defines a birational map from the projective space to the Grassmannian G(3,6). We describe this picture, using the notion of net logarithmic sheaves. This is a joint work with Simone Marchesi and Joan Pons-Llopis. 

10:00-11:00 박경동

Title: Complete intersection hyperkaehler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties 

Abstract: We classify fourfolds with trivial canonical bundle which are zero loci of general global sections of completely reducible equivariant vector bundles over exceptional homogeneous varieties of Picard number one. By computing their Hodge numbers, we see that there exist no hyperkaehler fourfolds among them. From the previous results, this implies that a hyperkaehler fourfold represented as the zero locus of a general global section of a completely reducible equivariant vector bundle over a rational homogeneous variety of Picard number one is one of the two cases described by Beauville-Donagi and Debarre-Voisin.