Symposium in Algebraic Geometry Busan, Haeundae Rivera Hotel 2014. 12. 29 (Mon) ~ 12. 30 (Tue) |
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12/29
최인송
Title: Number of maximal subbundles in view of secant varieties
Abstract: Given a vector bundle (or G-bundle more generally), the subbundles of maximal degree with fixed rank are called the maximal subbundles. The dimension of the space of maximal subbundles can be computed using relevant cohomology groups. When the dimension is zero, it is an interesting problem to compute the number of maximal subbundles. In this talk, I would like to review the known results, including the relation to the secant varieties. Finally, I will suggest a few open problems in this direction.
이경용
Title: Positivity for cluster algebras
Abstract: Cluster algebras are fundamental objects in mathematics and physics. All important algebraic/combinatorial/geometric/topological/physical objects are conjectured to have cluster algebra structures. We introduce cluster algebras and their remarkable properties. Positivity is a central theme in this field. In joint work with Schiffler, we prove the positivity conjecture. This implies that the Euler characteristics of certain quiver Grassmannians are positive.
We also explain a very recent work of Gross, Hacking, Keel and Kontsevich along this line.
정대웅
Title: On Conjecture O of Galkin, Golyshev and Iritani for G/P.
Abstract: I will present a proof of Conjecture O of Galkin, Golyshev and Iritani for G/P which underlies so-called Gamma conjectures I, II of them. Roughly, Conjecture O states that for Fano manifolds X, the quantum multiplication operator of the first Chern class of X has a positive real eigenvalue modulus with multiplicity one, and all eigenvalues of the modulus R can be obtained from the principal eigenvalue R by rotating by an angle depending on the Fano index of X. For some special homogeneous spaces, all eigenvalues and eigenvectors of the operator are computed.
박진형
Title: A bound for Castelnuovo-Mumford regularity by double point divisors
Abstract: One way to study a projective variety is to take general projections. The non-isomorphic locus of a general projection gives an effective divisor, which is called a double point divisor. After a quick review of basic results on double point divisors, I present applications to Castelnuovo-Mumford regularity of a smooth projective variety. I first show a sharp bound for Castelnuovo-Mumford regularity of a structure sheaf, and I classify the extremal and the next to extremal cases. By generalizing Mumford's method, I give a Castelnuovo type bound for normality of a smooth projective variety. This is joint work with Sijong Kwak.
이완석
Title: On syzygies of divisors on rational normal scrolls
Abstract: I will talk about the minimal free resolutions of divisors of rational normal scrolls. With Euisung Park, we recently got an interesting progress which provides several improvements for former results. In particular, I will briefly show our results about the syzygies of rational curves of almost minimal degree, elliptic normal curves of low degree and curves of maximal regularity which are contained in a rational normal surface scroll.
12/30
김정훈
Title: Deformations of Poisson invertible sheaves
Abstract: In this talk, I will explain deformations of a algebraic Poisson scheme, which extends flat deformation theory of algebraic schemes. I will identity infinitesimal deformations and obstructions for smooth Poisson algebraic varieties. Given a Poisson variety, a Poisson line bundle is a line bundle which is equipped with a flat Poisson connection. I will define simultaneous deformations of a Poisson variety and a Poisson invertible sheaf on it, which extends a simultaneous deformation theory of a variety and a invertible sheaf on it. I will identify infinitesimal deformations and obstructions for smooth projective Poisson varieties.
황동선
Title: Log del Pezzo surfaces of rank one
Abstract: I will discuss the classification problem of log del Pezzo surfaces of rank one. After a brief review on the previous attempts toward this goal, I will present a way to enumerate all log del Pezzo surfaces of rank one by completing the classification program initiated by De-Qi Zhang.
최인송
Title: Number of maximal subbundles in view of secant varieties
Abstract: Given a vector bundle (or G-bundle more generally), the subbundles of maximal degree with fixed rank are called the maximal subbundles. The dimension of the space of maximal subbundles can be computed using relevant cohomology groups. When the dimension is zero, it is an interesting problem to compute the number of maximal subbundles. In this talk, I would like to review the known results, including the relation to the secant varieties. Finally, I will suggest a few open problems in this direction.
이경용
Title: Positivity for cluster algebras
Abstract: Cluster algebras are fundamental objects in mathematics and physics. All important algebraic/combinatorial/geometric/topological/physical objects are conjectured to have cluster algebra structures. We introduce cluster algebras and their remarkable properties. Positivity is a central theme in this field. In joint work with Schiffler, we prove the positivity conjecture. This implies that the Euler characteristics of certain quiver Grassmannians are positive.
We also explain a very recent work of Gross, Hacking, Keel and Kontsevich along this line.
정대웅
Title: On Conjecture O of Galkin, Golyshev and Iritani for G/P.
Abstract: I will present a proof of Conjecture O of Galkin, Golyshev and Iritani for G/P which underlies so-called Gamma conjectures I, II of them. Roughly, Conjecture O states that for Fano manifolds X, the quantum multiplication operator of the first Chern class of X has a positive real eigenvalue modulus with multiplicity one, and all eigenvalues of the modulus R can be obtained from the principal eigenvalue R by rotating by an angle depending on the Fano index of X. For some special homogeneous spaces, all eigenvalues and eigenvectors of the operator are computed.
박진형
Title: A bound for Castelnuovo-Mumford regularity by double point divisors
Abstract: One way to study a projective variety is to take general projections. The non-isomorphic locus of a general projection gives an effective divisor, which is called a double point divisor. After a quick review of basic results on double point divisors, I present applications to Castelnuovo-Mumford regularity of a smooth projective variety. I first show a sharp bound for Castelnuovo-Mumford regularity of a structure sheaf, and I classify the extremal and the next to extremal cases. By generalizing Mumford's method, I give a Castelnuovo type bound for normality of a smooth projective variety. This is joint work with Sijong Kwak.
이완석
Title: On syzygies of divisors on rational normal scrolls
Abstract: I will talk about the minimal free resolutions of divisors of rational normal scrolls. With Euisung Park, we recently got an interesting progress which provides several improvements for former results. In particular, I will briefly show our results about the syzygies of rational curves of almost minimal degree, elliptic normal curves of low degree and curves of maximal regularity which are contained in a rational normal surface scroll.
12/30
김정훈
Title: Deformations of Poisson invertible sheaves
Abstract: In this talk, I will explain deformations of a algebraic Poisson scheme, which extends flat deformation theory of algebraic schemes. I will identity infinitesimal deformations and obstructions for smooth Poisson algebraic varieties. Given a Poisson variety, a Poisson line bundle is a line bundle which is equipped with a flat Poisson connection. I will define simultaneous deformations of a Poisson variety and a Poisson invertible sheaf on it, which extends a simultaneous deformation theory of a variety and a invertible sheaf on it. I will identify infinitesimal deformations and obstructions for smooth projective Poisson varieties.
황동선
Title: Log del Pezzo surfaces of rank one
Abstract: I will discuss the classification problem of log del Pezzo surfaces of rank one. After a brief review on the previous attempts toward this goal, I will present a way to enumerate all log del Pezzo surfaces of rank one by completing the classification program initiated by De-Qi Zhang.