KU Lectures in Algebraic Geometry 2026:
Surfaces, Syzygies, and Sheaves 

This lecture series presents a sample of areas of algebraic geometry actively pursued in Korea. It is aimed at students beginning to specialize — senior undergraduates, master's students, and PhD students in their early years.

Registration: Link

Speakers:

• Yonghwa Cho (Gyeongsang National University)

• Yeongrak Kim (Pusan National University)

• Sukmoon Huh (Sungkyunkwan University)

Titles and Abstracts:

• Yonghwa Cho (Gyeongsang National University): Geometry of nodal surfaces

  1. Lecture 1: Cubic and quartic surfaces with nodal singularities
     Abstract: In this series of lectures, we consider nodal surfaces, which are complex projective surfaces in \mathbb{P}^3 whose singularities are all nodes. In the first lecture, we introduce basic definitions and collect some elementary results, primarily focusing on nodal cubic and quartic surfaces.

  2. Lecture 2: (Half-)even sets of nodes and binary codes
     Abstract: The notion of (half-)even sets of nodes originates in the (16,6)-configuration of Kummer quartic surfaces, and has turned out to be a powerful machinery for the study of nodal surfaces. The symmetric difference of two such sets is again a (half-)even set, which naturally leads to the definition of binary codes associated with nodal surfaces. We briefly review how these structures are used to obtain important results on nodal surfaces of low degree.

  3. Lecture 3: Nodal-Severi varieties
     Abstract: In the projective space parametrizing the surfaces of degree d in \mathbb{P}^3, let F(d,n) be the locally closed subset parametrizing nodal surfaces with precisely n nodes. A classical question of Kummer asks, for each fixed d, what is the largest number of n such that F(d,n) is nonempty. Understanding the geometry of F(d,n) is an essential problem in the classification of nodal surfaces. In the final talk, I will discuss recent results on maximally nodal surfaces of degree d<=6. Part of this talk is based on joint work with F. Catanese, S. Coughlan, D. Frapporti, M. Kiermaier, and S. Kurz.

• Yeongrak Kim (Pusan National University): A quick overview on syzygies of algebraic curves
  Abstract: The word 'syzygy' is often used in astronomy that refers to three or more celestial bodies lying on a straight line. In mathematics, the term was brought by Sylvester in 1850 to represent linear relations between certain functions. We can find some applications of syzygies to classical invariant theory from several works of Cayley and of Sylvester. After the celebrating Hilbert's syzygy theorem, syzygies and higher syzygies become a bridge between the geometry of algebraic varieties and homological algebra. The study of syzygies in algebraic geometry becomes exploding after Green introduced Koszul cohomology groups in 1984. This lecture series aims to skim through these wonderful development in a quick way, particularly focuses on syzygies of algebraic curves.
  
  I managed to divide this lecture series into three parts. The first lecture will be consisted of recalling basic notions for algebraic curves, and a quick introduction on the study of special linear series and Brill-Noether theory. The second lecture will contain a brief introduction on syzygies and Koszul cohomology groups. The last lecture will be devoted to syzygies of algebraic curves by addressing a flagship problem so-called Green's conjecture, and also a few concrete examples.

• Sukmoon Huh (Sungkyunkwan University): Introduction to the theory on the moduli space of stable sheaves

  1. Lecture 1: Stability and moduli functor / moduli space of stable bundles on curves
     Abstract: The construction of moduli spaces is one of the central themes in modern algebraic geometry. In this lecture, we introduce the notion of stability for vector bundles on smooth projective curves and explain why stability provides the appropriate framework for constructing meaningful moduli spaces. Beginning with the concepts of slope and (semi)stability, we discuss Jordan–Hölder filtrations and S-equivalence, which naturally lead to the classification of semistable bundles. We then formulate the moduli problem in terms of the moduli functor and explain how it is represented by the moduli space of stable bundles and corepresented by the moduli space of semistable bundles. Time permitting, we will briefly discuss the geometric properties of these moduli spaces, including their dimension, irreducibility, and projectivity.

  2. Lecture 2: Moduli space of stable sheaves on surfaces
     Abstract: The theory of stable sheaves extends the study of vector bundles on curves to higher-dimensional varieties and has become a fundamental tool in algebraic geometry. In this lecture, we focus on smooth projective surfaces and introduce Gieseker stability and slope stability with respect to a chosen polarization. We explain how the moduli functor of semistable sheaves is constructed and how Geometric Invariant Theory leads to projective moduli spaces. We also discuss several important examples on the projective plane and K3 surfaces, illustrating the rich geometry of these moduli spaces and their dependence on the choice of polarization. If time permits, we will briefly mention wall-crossing phenomena and some recent developments connecting moduli spaces of sheaves with derived categories and birational geometry.

Venue:

Hana Science Hall B214 (하나과학관 지하 214호), Korea University

Nearest station: Anam Station (안암역), Seoul Metro Line 6 (6호선), Exit 4

[View on Naver Map] · [Kakao Map]

Organizer:

• Changho Han (Korea University)

• Euisung Park (Korea University)

Sponsors:

• 한국연구재단 (National Research Foundation(NRF) of Korea)

• 4차 BK21 고려대학교 수리과학 미래인재 교육연구팀