In 2021-2022, I organized a seminar at KIAS(usually at online) on algebraic geometry and related topics. The followings are the list of talks had been held in the seminar.

9) 2022.03.30(Wed), Lee, Donggun(Seoul National University) 
Title : Representations on the cohomology of the moduli space of rational curves  
Abstract : The moduli space of rational curves has a natural action of the symmetric group which permutes the marked points. In this talk, we present a formula for the character of the representation of the symmetric group of the cohomology of the moduli of rational curves. This formula reduces the computation to a purely combinatorial problem. As a corollary, we compute an explicit formula for cohomologies of degree at most 6 and check that they are permutation representations. We also show that the sum of cohomologies of degrees 2p and 2p-2 is a permutation representation for each p. A key ingredient is the wall crossing of delta-stability on moduli of quasi-maps. This is a joint work with Jinwon Choi and Young-Hoon Kiem.   

8) 2022.01.26(Wed),  Nam, Sungwoo(University of Illinois at Urbana-Champaign)
Title : Gromov-Witten invariants of local reducible surfaces
Abstract : Gromov-Witten invariants of some noncompact CY 3-folds have been intensively studied both by physicists and mathematicians. Especially, for a smooth projective Del Pezzo surface S, its local Gromov-Witten invariants can be thought of as contributions to the three-fold invariant containing S. In this talk, motivated from the studies on 5d SCFT by physicists, we will discuss how to generalize this problem to certain reducible surfaces and how to compute some low-degree invariants. These are all toric surfaces glued along torus invariant divisor, but with the conjectural existence of a CY 3-folds, can be applied to other examples. This is a work in progress.

7) 2022.01.19(Wed), Lim, Woonam(ETH Zurich)
Title : Virasoro constraints for stable pairs on Calabi-Yau 4-folds
Abstract : We conjecture Virasoro constraints for stable pair theories on Calabi-Yau 4-folds. This provides one more example of ubiquitous Virasoro constraints on sheaf-theoretic moduli spaces in various dimensions.  As supporting evidence for the conjecture, we  prove that Virasoro constraints for a Fano 3-fold are compatible with Virasoro constraints for a corresponding elliptic CY 4-fold. We also conjecture Virasoro constraints for moduli of torsion sheaves and study their relationship to stable pair Virasoro constraints. This is a joint work in progress with  Younghan Bae and Miguel Moreira.

6) 2021. 09. 24(Fri),  Dhruv Ranganathan(University of Cambridge)
Title :  Gromov-Witten theory and invariants of matroids
Abstract : Matroids are combinatorial generalizations of hyperplane arrangements, and are intimately tied to the geometry of Grassmannians. In the last several years, a number of remarkable geometric structures have been found in this combinatorial world. I will discuss the Gromov-Witten theory and quantum cohomology for these combinatorial objects, based on the geometry of de Concini and Procesi’s theory of wonderful compactifications. The upshot is that the results point to a well-defined (logarithmic) Gromov-Witten theory for matroids. The direction is essentially fully of open questions, and I will discuss some of them. This is joint work with Jeremy Usatine at Brown.

5) 2021. 07. 02, Park, Jun-Yong (IBS-CGP)
Title: Enumerating fibrations of curves over P^1
Abstract : Through arithmetic invariants of Hom stacks parameterizing rational curves on moduli stacks of curves, we enumerate algebraic curves and abelian varieties with precise lower order terms ordered by bounded discriminant height over Fq(t) which renders new heuristics over Q through the global fields analogy. This is joint work with Changho Han and Hunter Spink.

4) 2021. 06. 04, Kyoung-Seog Lee(University of Miami)
Title : Derived categories and motives of moduli spaces of vector bundles on curves
Abstract : Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. On the other hand, moduli spaces of vector bundles on curves provide attractive and important examples of algebraic varieties and there have been intensive works studying them. In this talk, I will discuss derived categories and motives of moduli spaces of vector bundles on curves. This talk is based on several joint works with I. Biswas, T. Gomez, H.-B. Moon and M. S. Narasimhan.

3) 2021. 04. 02, Jingchen Niu(University of Arizona)
Title: Towards desingularizations of the moduli spaces of stable maps to projective spaces
Abstract: The moduli spaces of stable maps to projective spaces play a prominent role in algebraic and symplectic geometry. Following the fundamental work [VakilZinger], desingularizations of such spaces for genus 1 and 2 have been achieved from different perspectives: [RanganathanSantos-ParkerWise] and [BattistellaCarocci] via log geometry, and [HuLi] and [HuLiNiu] via constructive blowing-ups. Towards possible generalizations to higher genera, we provide certain interpretations of [HL] and [HLN] in [HuNiu1] and [HuNiu2] using "twisted fields". In this talk, I will explain some motivation and examples for [HN1,2].

2) 2021.03.12, Hyunsuk Moon(KAIST)
Title : Rank 3 quadratic generators of Veronese Embeddings
Abstract : Let L be a very ample line bundle on a projective scheme X defined over an algebraically closed field. We say that (X, L) satisfies property QR(k) if the homogeneous ideal of the linearly normal embedding X ⊂ PH^0 (X, L) can be generated by quadrics of rank less than or equal to k. Many classical varieties such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree satisfy property QR(4).

In this talk, we will introduce the idea of showing that the Veronese Embedding satisfies property QR(3) for all dimensions and degrees. Also, we can apply this result to show some asymptotic behavior of any projective scheme. Namely, (i) if X is m-regular, then (X,L^d) satisfies QR(3) for all d>=m and (ii) if A is an ample line bundle, then (X,A^d) satisfies QR(3) for sufficiently big composition number d. We expect that the QR(3) holds for all suffieiciently ample line bundles on X as in the case of N_p condition and determinantal presentation.

1) 2021. 02. 19, Yoon-Joo Kim(Stony Brook University)
Title: Cohomology decomposition of compact hyper-Kähler manifolds
Abstract: Compact hyper-Kähler (HK) manifolds are higher dimensional generalizations of K3 surfaces. Looijenga, Lunts and Verbitsky showed the cohomology of HK manifolds admits a nontrivial algebraic group action. It is the most rigid structure on the cohomology of HK that we have so far. In this talk, I will explain how this structure can be used to study cohomology of HK manifolds, and compute them explicitly for all currently known deformation types of HK. This is joint work with M. Green, R. Laza and C. Robles.