권민성 (Morningside Center of Mathematics)

 

Classification of equivariantly embedded homogeneous Legendrian subvarieties

of nilpotent orbits

Minseong Kwon

Postdoctoral Researcher, Morningside Center of Mathematics, AMSS, CAS

 

 

Abstract
Adjoint varieties are rational homogeneous spaces associated to simple Lie algebras, and it is well known that they are the only homogeneous Fano contact manifolds. Moreover, for adjoint varieties of type A or C, a classification of equivariantly embedded homogeneous Legendrian subvarieties is classically well known. In this talk, I will explain a classification of equivariantly embedded homogeneous Legendrian subvarieties of nilpotent orbits, which are homogeneous contact manifolds including adjoint varieties.

 

 

 

 

 

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김동현 (연세대학교)

 

Existence of Quasi-Monomial Minimizers and the Anti-Canonical MMP

Donghyeon Kim

Postdoctoral Researcher, Yonsei University

 

 

Abstract
I define the log canonical threshold of a pseudo-effective $Q$-divisor $D$ using Nakayama’s asymptotic order, and prove the existence of a quasi-monomial minimizer—an analogue of the Jonsson–Mustaţă conjecture (proved by Chenyang Xu). I also introduce the notion of pklt for a pair $(X,Delta)$, which generalizes klt log Calabi–Yau type pairs, and show that any pklt pair $(X,Delta)$ admits a $-(K_X+Delta)$-MMP with scaling by an ample divisor. 

This is a joint work with Sung Rak Choi, Sungwook Jang and Dae-Won Lee.

 

 

 

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김신영 (강원대학교)

 

 

Deformation rigidity of some quasi-homogeneous varieties with Picard number one

Shin-young Kim

Assistant professor, Kangwon National University 

 

 

Abstract
We investigate the global deformation rigidity problem of rational homogeneous manifolds of Picard number one which were developed by Hwang, Mok and others. In particular, we focus on the role of varieties of minimal rational tangents. Starting with similar ideas, we introduce some recent global deformation rigidity results of some quasi-homogeneous varieties, symmetric varieties and horospherical varieties, with Picard number one.

 

 

 

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문현석 (성신여자대학교)

 

 

On the rank index of projective curves of almost minimal degree

Hyunsuk Moon

Assistant Professor, Sungshin Women`s University

 

 

Abstract
In this talk, we investigate the rank index of projective curves C in P^r of degree r+1 when C is a projection of the standard rational normal curve in P^{r+1} from a point p. Here, the rank index is defined to be the least integer k such that the homogeneous ideal can be generated by quadratic polynomials of rank less than or equal to k. Our results show that the rank index of C is at most 4, and it is exactly equal to 3 when the projection center p is a coordinate point of P^{r+1}. And we will investigate the case when p is on the third secant of C.

 

 

 

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이동협 (연세대학교)

 

 

Castelnuovo-Mumford Regularity of Zero-Dimensional Schemes

Dong Hyeop Lee

Postdoctoral Researcher, Yonsei University

 

 

Abstract
For zero-dimensional schemes, the Castelnuovo–Mumford regularity provides a measure of their algebraic complexity. While it is known that zero-dimensional schemes whose regularity is close to a well-known upper bound exhibit specific geometric configurations, the geometric configurations of schemes with low regularity are not yet fully understood. In this talk, I describe the geometric configurations of zero-dimensional schemes within a specific regularity range. Based on this configuration, I compute the Betti table of the scheme and explain how its syzygy shows the underlying geometric structure. This is a joint work with Euisung Park.
 

 

 

 

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이상현 (아주대학교)

 

 

Blow-up formula of Gromov-Witten invariants using Master space

Sanghyeon Lee

Assistant Professor, Ajou University

 

 

Abstract

Blow-up formula relates Gromov-Witten invariants of the base space with Gromov-Witten invariants on blow-up space. 

Some of them are proved, and some of them still remain conjectural. For example, Gathman studied the case when genus=0 and the blow-up locus is finite points, and Lai studied the case when genus=0 and the normal bundle of the blow-up locus is convex. Recently, Chen-Du studied the case when the normal bundle is positive.

Their method is based on torus localization on master space relating stable map space of the base and blow-up space.

Basically I will use Chen-Du's argument and prove some additional results when the curve class on blow-up space contains line classes in the exceptional divisor.

 

 

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정민교 (성균관대학교)

 

 

Torelli problem for logarithmic sheaves and related topics

Min-gyo Jeong

Postdoctoral Researcher, Sungkyunkwan University

 

 

 

Abstract
A Torelli-type theorem implies that a subspace of the moduli space of semistable vector bundles on a projective variety can be realized as a locus of logarithmic sheaves associated with arrangements of divisors on the variety. In this talk, I will introduce the Torelli problem for logarithmic sheaves associated with an arrangement of divisors on a smooth projective variety. After reviewing some previous results, I will introduce a generalized logarithmic sheaf termed the net logarithmic sheaf. This concept provides a method to describe the moduli space of semistable sheaves, specifically allowing us to describe non-generic boundary points that are not realized as the logarithmic sheaves in the classical sense.

 

 

 

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최도영 (KIAS)

 

 

Singularities and Syzygies of secant varieties of smooth projective varieties

Doyoung Choi

Research Fellow,  Korea Institute for Advanced Study (KIAS)

 

 

 

Abstract
We study the higher secant varieties of a smooth projective variety embedded in projective space. We prove that when the variety is a surface and the embedding line bundle is sufficiently positive, these varieties are normal with Du Bois singularities and the syzygies of their defining ideals are linear to the expected order. We show that the cohomology of the structure sheaf of the surface completely determines whether the singularities of its secant varieties are Cohen--Macaulay or rational. We also prove analogous results when the dimension of the original variety is higher and the secant order is low, and by contrast we prove a result that strongly implies these statements do not generalize to higher dimensional varieties when the secant order is high. Finally, we deduce a complementary result characterizing the ideal of secant varieties of a surface in terms of the symbolic powers of the ideal of the surface itself, and we include a theorem concerning the weight one syzygies of an embedded surface -- analogous to the gonality conjecture for curves -- which we discovered as a natural application of our techniques.

 

 

 

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최영욱 (영남대학교)

 

Tschirnhausen Module of a Covering of a Smooth Curve and Its Applications

Youngook Choi

Professor, Yeungnam University

 

 

 

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한종인 (KIAS)

 

 

Castelnuovo theory of syzygies

Jong In Han

Research Fellow,  Korea Institute for Advanced Study (KIAS)

 

 

Abstract
The Castelnuovo theory developed by G. Castelnuovo, G. Fano, D. Eisenbud, and J. Harris induces several upper bounds on the number of quadrics defining projective varieties. By Han-Kwak and Han-Kwak-Park, it is known that some of the bounds can be extended to upper bounds on the number of syzygies. Their results give a hint that there might exist a hierarchy structure on graded Betti numbers in the quadratic strand. We show there indeed exists a hierarchy structure.