YeungNam Workshop on
Algebraic Geometry XVII

Invited Speakers

Jaekwan Jeon (IBS-CGP)

Jaewoo Jung (DGIST BRL-AGSTA)

Jihun Park (IBS-CGP & POSTECH)

YongJoo Shin

Title and Abstract 

Jaekwan Jeon (IBS-CGP)
Title: Picture deformations and incidence matrices for weighted homogeneous surface singularities
Abstract: In the deformation theory of sandwiched singularities developed by de Jong and van Straten, one-parameter smoothings of sandwiched singularities can be described by so-called picture deformations of suitable curve singularities. Each picture deformation naturally determines an incidence matrix encoding the incidence relations among the curve components. A key question in using these combinatorial objects to study the deformation space is whether distinct picture deformations give rise to distinct incidence matrices. In this talk, we address this question in the case of weighted homogeneous surface singularities. This is joint work with D. Shin and H. Park. 

Jaewoo Jung (DGIST BRL-AGSTA)
Title: Sums of Squares on Rational Curves of Almost Minimal Degree
Abstract: On a rational normal curve, every nonnegative quadratic form is a sum of squares of linear forms, but this need not hold for a general curve. How is this distinction reflected by the geometry of the variety? In this talk, we study this question for rational curves of almost minimal degree, viewed as projections of rational normal curves. For projections from a point outside the third secant variety, we show that the Hankel index is determined by the almost real rank of the binary form corresponding to the center of projection. This connects the geometry of the sums-of-squares cone with real apolarity and decompositions of binary forms. We then discuss our recent results on rational curves, focusing on how their geometry is reflected in associated Hankel matrices and their ranks. We explain how these results suggest viewing the Hankel index of a rational curve as a measure of the complexity of the linear recurrences governing its parametrization and of the secant geometry of its projections. We conclude by outlining several questions about how the Hankel index behaves under projection, degeneration, and other natural operations on rational curves. 

The talk is based on joint work with Grigoriy Blekherman and Justin Chen, and ongoing joint work with Maciej Dawid Gałązka and Hyunsuk Moon.

Jihun Park (IBS-CGP & POSTECH)
Title: Fano varieties and Sasaki-Einstein connected sums $(S^2\times S^3)^{\# n}$ and $(S^3\times S^4)^{\# n}$
Abstract: We review the construction of Sasaki-Einstein metrics arising from Fano varieties. In particular, we explain Kollár's construction of such metrics on every connected sum of $S^2\times S^3$. Recent advances in K-stability and the topology of highly connected manifolds have made it possible to investigate the 7-dimensional analogue of Kollár's result, namely the construction of Sasaki-Einstein metrics on connected sums of $S^3\times S^4$. In this talk, we present the 7-dimensional analogue of Kollár's result. This is joint work in progress with D. Jeong, I. Kim, and J. Won.

YongJoo Shin
Title: Optimal Bound of $(-2)$-Curves on a Numerical Godeaux Surface
Abstract: Let $Y$ be a numerical Godeaux surface, that is, a smooth minimal surface of general type with $p_g(Y)=0$ and $K_Y^2=1$. We study the maximal number $\mu(Y)$ of pairwise disjoint $(-2)$-curves on $Y$. Miyaoka's inequality gives the general bound $\mu(Y) \le 7$, and the existence of a numerical Godeaux surface with seven disjoint $(-2)$-curves had remained open in the work of Keum on projective surfaces with many nodes.

In this talk, we prove the optimal bound $\mu(Y) \le 6$. The bound is sharp, as shown by examples arising from singular Godeaux surfaces constructed by Rito, Gleissner, and Ruhland.

The proof combines lattice theory, binary codes associated with configurations of disjoint $(-2)$-curves, and covering constructions. Assuming the existence of seven such curves, the orthogonal complement of the canonical class is identified with $E_8(-1)$. The resulting binary code yields compatible 2-divisibility relations, which lead to a bidouble cover. A final discriminant computation in the unimodular numerical lattice gives a contradiction.

The original motivation comes from the study of involutions on minimal surfaces of general type with $p_g=0$ and $K^2=7$. The optimal bound above excludes cases (2), (3)(a), and (3)(b) in the classification table of Chen-Shin-Zhang, thereby refining the possible quotient configurations of such involutions. 

This is joint work with Yifan Chen.

Participants (Registrants to Date)

금종해, 김영락, 김인균, 박경동, 박지훈, 신용주, 원준영, 이완석, 전재관, 정기룡, 정재우, 조용화, 최영욱, 홍연재, 김상필.

Organizers

Yonghwa Cho (Gyeongsang National University)

Kyeong-Dong Park (Gyeongsang National University)

Previous Workshops

• YeungNam Workshop on Algebraic Geometry XVI (2026년 5월, 포항 IBS POSTECH Campus) https://cgp.ibs.re.kr/conferences/2026YWAG

• YeungNam Workshop on Algebraic Geometry XV (2025년 11월, 부경대) https://sites.google.com/view/yeungnam-workshop-on-ag-15

• YeungNam Workshop on Algebraic Geometry XIV (2025년 5월, DGIST) https://sites.google.com/view/ynseminarag2025spring

• YeungNam Workshop on Algebraic Geometry XIII (2024년 10월, 창원대) https://sites.google.com/view/ywag2024

• YeungNam Workshop on Algebraic Geometry XII (2024년 4월, 영남대) https://sites.google.com/yu.ac.kr/2024-04-yeungnam-workshop

• YeungNam Workshop on Algebraic Geometry XI (2023년 10월, 포항 IBS POSTECH Campus) https://cgp.ibs.re.kr/conferences/the_11th_yeongnam_workshop_on_algebraic_geometry

• YeungNam Seminar on Algebraic Geometry X (2023년 4월, 부산 라발스호텔) https://sites.google.com/view/ywag2023s

• YeungNam Seminar on Algebraic Geometry IX (2022년 10월, 진주) https://sites.google.com/view/ynseminarag2022autumn

• 영남대수기하학회 VIII (2022년 5월, 울산대) https://sites.google.com/view/ynseminarag2022

• 영남 대수기하 워크샵 (2021년 10월, 대구) https://sites.google.com/view/21fall-yn-ag-workshop

• YeungNam Seminar on Algebraic Geometry VI (2019년 10월, 경북대) https://sites.google.com/site/ynseminarag/

• YeungNam Seminar on Algebraic Geometry V (2019년 5월, 부경대) 

• 4th YeungNam Workshop on Alegebraic Geometry (2018년 11월, DGIST) https://sites.google.com/view/ywag4th/

• The 3rd Yeongnam workshop on algebraic geometry (2017년 12월, 포항 IBS-CGP) https://cgp.ibs.re.kr/conferences/the_3rd_yeongnam_workshop_on_algebraic_geometry

• YeungNam Seminar on Algebraic Geometry II (2017년 4월, 영남대) 

• YeungNam Seminar on Algebraic Geometry I (2016년 11월, 영남대) 

후원: 한국연구재단(NRF)