Program

Junho Choe (KIAS)

Lecture 1: Equations, relations and syzygies: from free resolution to Koszul cohomology
Projective algebraic geometry studies interactions between geometry of projective varieties and algebraic properties of their homogeneous equations. In 1890 Hilbert proved a fundamental theorem on equations, relations and syzygies, namely Hilbert’s syzygy theorem. It says that any finitely generated module over a polynomial ring has free resolution of length less than or equal to the number of variables. And in 1984 Green introduced and analyzed Koszul cohomology to determine the shape of (minimal) free resolutions. In this talk we review basic notions from free resolution to Koszul cohomology.
Lecture 2: Vanishing and nonvanishing of Koszul cohomology groups
In this talk we discuss vanishing and nonvanishing theorems on Koszul cohomology groups for smooth curves. One is Green’s (2g+1+p)-theorem, and another is the gonality conjecture that was raised by Green and Lazarsfeld in 1986 and proved by Ein and Lazarsfeld in 2015. To this end we compare Koszul cohomology with sheaf cohomology.
Lecture 3: Computations by Macaulay2
Macaulay2 is a computer algebra system which is useful to observe the situation and find examples when studying algebraic geometry. In this talk we see how to use Macaulay2 with several applications in research. Subjects to be addressed are higher secant varieties, the Eisenbud-Goto regularity conjecture, Gaussian graphical models and so forth.

Hakho Choi (QSMS)
An introduction to the topology of the normal surface singularities
In this series of talks, we will explore the topological aspects of isolated singularities, which serve as a bridge between two branches of mathematics: algebraic geometry and topology. Our discussions will begin by introducing the fundamental concepts of isolated singularities. We will then shift our focus to normal surface singularities, which play a crucial role not only in algebraic surfaces but also in 4-dimensional symplectic/smooth topology.

Lecture 1: Topology of isolated singularities
We will start by examining the algebraic properties of an isolated singularity X and explore the basic topological characteristics of its link L.
Lecture 2: Symplectic aspects of normal surface singularities
This session will provide an introduction to the 'Milnor fillable' contact structure on the link of a normal surface singularity, offering an overview from both the singularity and symplectic viewpoints.
Lecture 3: Deformations of singularities: Milnor fibers vs. Symplectic fillings
We will compare two smooth objects associated with a normal surface singularity: 'Milnor fibers' derived from deformation theory and 'minimal symplectic fillings' derived from symplectic topology.

Minseong Kwon (KAIST)
Rational Curves in Adjoint Varieties
Complex geometry is naturally related with both algebraic geometry and Riemannian geometry. Adjoint varieties are one of the keys linking these three areas. They are rational homogeneous spaces defined by the adjoint representations of simple complex Lie groups, which are equipped with complex-contact structures. Moreover, they are in bijective correspondence with compact Wolf spaces, i.e. simply connected symmetric quaternionic-Kähler manifolds, which appear in Riemannian geometry. In this talk, I will focus on the moduli spaces of rational curves of low degree contained in adjoint varieties. The main topic is their varieties of minimal rational tangents (called VMRTs) and a well-known relation between the Wolf spaces and the moduli spaces of conics in adjoint varieties.

Jong In Han (KAIST)
Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties
Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question "which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?", we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases.
The extremal varieties of dimension n, codimension e, and degree d are exactly characterized by the following two types:
(i) Varieties with d=e+2, depth X=n, and Green-Lazarsfeld index a(X)=0,
(ii) Arithmetically Cohen-Macaulay varieties with d=e+3.
This is a generalization of G. Castelnuovo, G. Fano, and E. Park’s results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak.
In addition, we show that every variety X that belongs to (i) or (ii) is always contained in a unique rational normal scroll Y as a divisor. Also, we describe the divisor class of X in Y. This is a joint work with S. Kwak and E. Park.

Doyoung Choi (KAIST)
Degree of the 3-secant variety
In this paper, we present a formula for the degree of the 3-secant variety of a nonsingular projective variety embedded by a 5-very ample line bundle. The formula is provided in terms of Segre classes of the tangent bundle of a given variety. We use the generalized version of double point formula to reduce the calculation into the case of the 2-secant variety. Due to the singularity of the 2-secant variety, we use secant bundle as a nonsingular birational model and compute multiplications of desired algebraic cycles.

Myoungyoun Kim (Ewha Womans Univ.)
Weyl actions on Picard group of Rational surfaces
In this talk, we discuss the Picard group of del Pezzo surfaces from the view of Root lattices and Dual lattices along the Weyl groups. We consider the special divisor classes of Picard group which are lattices points called lines, ruling and exceptional systems. Based on the correspondence with Weyl actions between these special divisor classes of del Pezzo surfaces and the geometry of Gosset polytopes of type (r - 4)₂₁, we study certain Er-type root lattices embedded within the standard Lorentzian lattice Zr+1 (3 ≤ r ≤ 8). We explain the hierarchy of periodicity of affine lattice planes as the roots lattices and their duals. This is a joint work with Jae-hyouk Lee.

Ju A Lee (Seoul National Univ.)
Signature of surface bundles, Kodaira fibrations, and Lefschetz fibrations
Surface bundles over surfaces, Lefschetz fibrations, and Kodaira fibrations constitute an interesting family of examples of symplectic 4-manifolds and complex surfaces. I’d like to discuss about the old and new results on the signature of those manifolds with fibrations both on algebraic geometric and topological sides.

Hyungkee Yoo (IMS)
2-Connected indecomposable Laplacian Integral Graphs
For some positive integer k, if the finite cyclic group Zₖ can freely act on a graph G, then we say that G is k-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this talk, we investigate a class of 2-connected k-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of k-symmetric graphs in which all Laplacian eigenvalues are integers.

Jaewoo Jung (IBS-CCG)
A decomposition for bounds on regularity of monomial ideals
We study the Castelnuovo-Mumford regularity of monomial ideals through their associated simplicial complexes. Our main result is a decomposition of the complex that bounds the regularity of corresponded ideals. We improve known bounds on the regularity of monomial ideals by using our main theorem with results in structural graph theory.

Jaekwan Jeon (Chungnam National Univ.)
A strategy for proving Kollár conjecture
In 1991, J. Kolllár conjectured that every smoothing of a rational surface singularity is induced from a partial resolution of the singularity, which is known a P-modification. A few examples are known that the conjecture holds. Recently, H. Park and D. Shin propose a strategy for proving the conjecture. I will give a sketch of the strategy. And as an example, I will demonstrate how the strategy can be applied to weighted homogeneous surface singularities.

Min-Gyo Jeong (Sungkyunkwan Univ.)
Logarithmic sheaves and the Torelli problem on the blown-up surface
Let 𝓓={D₁, ..., Dₘ} be an arrangement of reduced and effective divisors on a smooth (projective) variety X. Then one can define the sheaf Ωₓ¹(log 𝓓) of differential 1-forms with logarithmic poles along 𝓓. We investigate some properties of the logarithmic sheaf associated to the given 𝓓 on the blow-up of a surface at finitely many points and then study the one of interesting problems related to it, which is called the Torelli problem.

Sungwook Jang (Yonsei Univ.)
ACC of plc threshold
The potential log discrepancy is first introduced by S. R. Choi and J. Park. These invariants are preserved along the MMP for anticanonical divisor. Also the notion of pklt, defined by potential log discrepancy, characterizes Fano type variety. We define the potential log canonical threshold and prove that the set of these thresholds satisfies ascending chain condition (ACC).

Jeong-Hoon Ju (Pusan National Univ.)
A new CP decomposition of the determinant tensor
The determinant tensor detₙ for n×n matrices is a classical object in linear and multilinear algebra, and its tensor rank measures the complexity of the determinant as a function. However, its tensor rank is widely unknown, in particular, for n ≥ 4. So it is meaningful to improve the upper bound of its tensor rank. A direct method to improve the upper bound is to find a more efficient CP decomposition of detₙ. In this poster, we present a new CP decomposition of det₄ which is derived by Least Absolute Shrinkage and Selection Operator (LASSO) technique. In addition, considering some symmetries in that decomposition, we present a generalized formula for det₂ₖ so that rank(detₙ) ≤ n!/(2⌊(n−2)/2⌋) when the base field is not of characteristic 2.