Conference on Derived Categories and
Singularities in Algebraic Geometry

Organizers

• 김영락 (부산대학교)

• 김인균 (연세대학교)

• 정기룡 (경북대학교)

• 원준영 (이화여자대학교)

• 박진형 (KAIST)

Main Lecturers

• 이경석 (University of Miami)
  Title : Vector bundles on elliptic surfaces
  
  Abstract : Studying vector bundles (and more generally coherent sheaves) on elliptic surfaces has been one of the central topics in many branches of mathematics, for example, algebraic geometry, differential geometry, mathematical physics, and topology to name a few. In this lecture series, we will give a gentle introduction to the theory of vector bundles (and coherent sheaves) on elliptic surfaces and its applications. We will also discuss how the theory of derived category is related to the study of coherent sheaves on elliptic surfaces.
  
  Lecture Note (08/16) Lecture Note (08/17) Lecture Note (08/18)
  
  

• 정승조 (전북대학교)
  Title : Singular tour to hypersurface singularities
  
  Abstract : A hypersurface singularity is a basic object in singularity theory. It is not only simple enough to understand but also interesting to have many invariants, e.g. Bernstein-Sato polynomials, Steenbrink spectra, log canonical thresholds, etc. The main goal of this series is to introduce these invariants and to discuss the relation between them with simple examples. No prior knowledge required.
  
  Roughly, the lecture series goes like:
  Lecture 1. Topological aspect of hypersurface singularities.
  Lecture 2. Algebraic aspect of hypersurface singularities.
  Lecture 3. Geometric aspect of hypersurface singularities.
  
  Lecture Note (08/16) Lecture Note (08/17) Lecture Note (08/18)

Assistant Lecturers

• 김정섭 (KIAS)
  Title : Introduction to vector bundles
  
  Abstract : We will introduce some basic notions, including complex manifolds, vector bundles, coherent sheaves, and stability. Also, we will provide examples and non-examples of these concepts along with explicit computations. For instance, we will explore an example that explains the differences in the moduli of vector bundles on surfaces compared to curves.
  
  Lecture Note (08/15) Lecture Note (08/16)
  
  

• 이대원 (이화여자대학교)
  Title : Introduction to singularities in algebraic geometry
  
  Abstract : In this lecture series, we will present various singularities that appear in algebraic geometry and related birational transformations, such as blow-up and log resolution. In particular, we will explore singularities of the minimal model program, rational singularities, and others, and also discuss their relationships. Additionally, we will introduce the notion of log canonical thresholds, which play a crucial role in birational geometry.
  
  Lecture Note (08/15) Lecture Note (08/16)

Poster Talks

• 김민훈 (경북대학교)
  Title : Ribbon concordance is a partial order
  
  Abstract : We discuss Ian Angol's result that   show that ribbon concordance forms a partial ordering on the set of knots, answering a question of Gordon. The proof makes use of representation varieties of the knot groups to SO(N) and relations between them induced by a ribbon concordance.
  
  

• 김영락 (부산대학교)
  Title : Remark on Ulrich complexity of small cubics
  
  Abstract : For a nonzero homogeneous form F of degree d in N variables, the Ulrich complexity uc(F) is defined as the minimal exponent r such that there is a linear matrix factorization (A, B) of F^r. In the case, A is a dr × dr matrix composed of homogeneous linear forms whose determinant is a constant multiple of F^r. For a fixed N and d=2, the Ulrich complexity decreases when F becomes more singular since F will have smaller quadric ranks. On the other hand, when N=4 and d=3 (the case of cubic surfaces), the Ulrich complexity of a general F is 1 but it may jump into 2 when F has an E_6-singularity. We discuss some cases when N=5 and d=3 and their applications.
  
  

• 김재현 (이화여자대학교)
  Title : Cylinders in del Pezzo surfaces
  
  Abstract : Cylinder is an A¹-ruled Zariski open subset in a normal projective variety over some affine variety. If the boundary of cylinder is defined by an effective member in numerical class of given divisor, then the cylinder is called polar for the divisor. The existence of such a structure has deep connections to certain group actions on the corresponding affine cone. From this point of view, the ample polar cylindricity of del Pezzo surfaces of degree at least 3 has been extensively studied in many areas. In this talk, we focus on the ample polar cylinders in smooth del Pezzo surfaces of lower degrees and explain some method to characterize them.
  
  

• 유상범 (공주교육대학교)
  Title : Towards intersection cohomology of the moduli of representations of the 3-Kronecker quiver with dimension vector (3,3)
  
  Abstract : The purpose of this work is to compute the intersection cohomology of the moduli M of representations of the 3-Kronecker quiver with dimension vector (3,3). It is known that M is singular. In this talk, we introduce Kirwan's algorithm which is a method to compute the intersection cohomology of a GIT quotient and then how to apply the method to the moduli M. This is a joint work with Kiryong Chung.
  
  

• 윤영호 (충북대학교)
  Title : Hodge ideals and spectrum of isolated hypersurface singularities
  
  Abstract : Algebraic geometry uses rings to understand geometry. Since the singularity is local, we study ideals of a local ring to understand the geometry of the singularity. Spectrum encodes the geometrical information of hypersurface singularities. We illustrate how to recover the information from (higher) multiplier ideals and Hodge ideals.
  
  

• 정기룡 (경북대학교)
  Title : Construction of quintic del Pezzo and Mukai variety
  
  Abstact : In this presentation, we attempt to construct quintic del-Pezzo and Mukai varieties from the following two perspectives: Sublocus of Hilbert scheme of points/curves and $\mathrm{SL}_n$-invariant space. We then explain the relationship between these perspectives from a polarity viewpoint. We may also present the generation of Macaulay2 code for relevant calculations if necessary. Almost all results are well known to experts.
  
  

• 좌동욱 (KIAS)
  Title : Graded matrix factorization categories and its Hochschild homology
  
  Abstract : As its name suggests, graded matrix factorization categories is an enhancement of ordinary matrix factorization. This extra information - grading - is quite essential in the study of "gauged sigma models" and it corresponds to the choice of characters appeared in GIT quotients construction. In the joint work with B. Sreedhar, we calculate Hochschild invariants of graded matrix factorizations when the structure morphism is of Deligne-Mumford type.
  
  

• 최성락 (연세대학교)
  Title : Introduction to potential triple
  
  Abstract : Birkar-Zhang's generalized pairs have been playing important roles in the recent development of birational geometry. They are ubiquitous and many unsolved problems can be stated in terms of generalized pairs. We introduce a notion which is even more general than the generalized pairs and state some results.
  
  

• 최준호 (KIAS)
  Title : Gonality sequence and higher secant varieties
  
  Abstact : Let C be a smooth curve, L be a very ample line bundle on C, and S(C) be the homogeneous coordinate ring of C for the embedding by L. According to the gonality conjecture solved by Ein-Lazarsfeld and Rathmann, the gonality of C determines the shape of the minimal free resolution of S(C) when L has sufficiently large degree. In this presentation we generalize the gonality conjecture for the gonality sequence of C and higher secant varieties to C. This is a joint work with Prof. Sijong Kwak and Prof. Jinhyung Park.