Workshop of
Classical Moduli Theory
and its Related Topics

 Invited Speakers

Kiryong Chung (Kyungpook National University)

Changho Han (Korea University)

Yeonjae Hong (Sungkyunkwan University)

Sukmoon Huh (Sungkyunkwan University)

Min-Gyo Jeong (Sungkyunkwan University)

Jaehyun Kim (Ewha Womans University)

Jeong-Seop Kim (KIAS)

Minseong Kwon (Morningside Center of Mathematics, AMSS)

Hyenho Lho (Chungnam National University)

Dongsun Lim (Sungkyunkwan University)

Jeongseok Oh (Seoul National University)

Abstract

November 20 (Thu)

14:00-14:40 Hyeonho Lho
Integrality of Tevelev degrees of local and formal geometry

This talk studies integrality phenomena for Tevelev degrees in local and formal geometry, and for their difference, the effective Tevelev degree, for smooth complete intersections in projective space. For local models given by split negative bundles, I use torus localization, asymptotic analysis of the I-function, and a detailed description of the quantum Euler class and hyperplane relation in quantum cohomology to prove a strong integrality and divisibility theorem. I then construct a formal quantum cohomology theory for complete intersections, establish integrality for the corresponding formal Tevelev degrees, and compare the local and formal theories to obtain integrality of effective degrees in some stable range.

15:00-15:40 Changho Han
Compact moduli of K3 surfaces via trigonal maps

Classifications of K3 surfaces with finite automorphisms have been worked out for several decades by many people, including Artebani, Sarti, and Taki. K3 surfaces with order 3 automorphisms have interesting geometric structures. For example, Kondo showed that certain K3 surfaces can be recovered from canonical curves of genus 4. Also, Taki showed that many such K3 surfaces admit a generically isotrivial elliptic fibration with a trisection, whose general fibers have j-invariant 0; this induces a trigonal map from this trisection to a projective line. Using this idea, I will explain, based on the joint works with Valery Alexeev, Anand Deopurkar, and Philip Engel, how compactifying moduli of trigonal maps give new insight on classifying semistable degenerations of K3 surfaces (as boundary members of appropriate compactifications of moduli of K3 surfaces with nonsymplectic automorphisms).

16:00-16:40 Minseong Kwon
Sphericality of spaces of conics on rational homogeneous spaces

Rational curves play an important role in the study of Fano varieties. Among them, lines on rational homogeneous spaces can be described via Tits geometry, and it is well known that for a rational homogeneous space of Picard rank 1, the space of lines is homogeneous if and only if it is associated to a long simple root. In this talk, I will discuss conics on rational homogeneous spaces. Namely, I will show that given a rational homogeneous space of Picard rank 1, the Hilbert scheme of conics is spherical if and only if it is associated to a long simple root.  This talk is based on a joint work with Nicolas Perrin.

November 21 (Fri)

10:00-10:40 Jeong-Seop Kim
Moduli of rational curves on Fano threefolds

Fano varieties contain many rational curves. There has been a series of questions concerning the geometry of these families, such as the irreducibility or smoothness of the moduli space of rational curves of fixed degree. In this talk, I will begin with a brief survey of previous results and then introduce the study of the geometry of the compact moduli spaces of rational curves of low degree. This talk is based on joint work with Kiryong Chung and Jaehyun Kim.

11:00-11:40 Jaehyun Kim
Relative Cones of Du Val del Pezzo Surfaces of Degree 1 with Picard Rank 2

The minimal resolutions of Du Val del Pezzo surfaces admit various P1-fibrations depending on the singularity type. These fibrations yield natural A1-bundle structures on the surfaces with an ample boundary. In this talk, we focus on Du Val del Pezzo surfaces of degree 1 with Picard rank 2 that arise as compactifications of an A1-bundle over some affine curve. This is joint work with Dae-Won Lee and Masatomo Sawahara.

14:00-14:40 Min-Gyo Jeong
Loci of generalized logarithmic sheaves in moduli spaces of semistable torsion-free sheaves on smooth complete intersections

A subspace of the moduli space of semistable torsion-free sheaves on a smooth complete intersection with a fixed Chern classes can be realized as a locus of logarithmic sheaves associated with an arrangement of divisors on the complete intersection. In this talk, I will introduce a generalized logarithmic tangent sheaf on a complete intersection, which we call the net logarithmic tangent sheaf, and provide explicit descriptions of the closure of the locus of net logarithmic sheaves in the corresponding moduli space on a smooth quadric and a cubic surface in projective 3-space. This talk is based on joint work with Sukmoon Huh.

15:00-15:40 Sukmoon Huh
The resolution of two rational maps from the space of plane cubic curves

The assignment to a plane cubic curve its Hessian cubic is a 3-1 map. The Hessian cubic comes with a non-trivial 2-torsion divisor class, and it allows one to represent the Hessian as the symmetric determinant of a net of cubics. This defines a birational map from the projective space to the Grassmannian G(3,6). The assignment to a plane cubic curve its Caylean cubic is also a 3-1 map. We resolve the indeterminacy loci of the two rational maps from the space of plane cubic curves. The main ingredient is to consider the moduli space of stable sheaves of rank two on the projective plane with the Chern classes (0,3) and interpret its general point as a logarithmic tangent sheaf associated to the plane cubic curve. This is a joint work with Simone Marchesi and Joan Pons-Llopis.

16:00-16:40 Jeongseok Oh
Introduction to virtual cycles I

A typical way of producing enumerative invariants or counting geometric objects is to construct a virtual cycle which is a homology cycle on the moduli space (of the objects). Usually we count the objects satisfying certain conditions that can be described as cohomology classes on the moduli space. Then capping them gives a number. However, the construction of virtual cycle has nothing to do with moduli space -- it is constructed on an arbitrary space (with some structure). I will explain the motivation (from moduli space), the construction (on arbitrary space) of virtual cycles. If time allows, I will present some examples to show how to use it.

November 22 (Sat)

10:00-10:20 Dongsun Lim
Virtual Poincaré polynomial of moduli space of semistable sheaves of rank two on reducible curves

We give an explicit description of the moduli space of rank-two semistable sheaves on a stable curve C obtained by gluing two smooth curves at a point. We show that this moduli space is irreducible and birational to a projective bundle over the product of the moduli spaces of stable vector bundles on each component curve. This result holds for any choice of polarization. As an application, we compute the virtual Poincaré polynomial of this moduli space and describe how it relates to the corresponding polynomial in the smooth case.

10:20-10:40 Yeonjae Hong
Counting aCM Toric Bundles of Rank Two on the Veronese Surface

Torus-equivariant vector bundles on a toric variety can be described as collections of filtrations of vector spaces satisfying a certain compatibility condition. Invariants such as cohomology can also be described directly in terms of these filtrations. In this talk, I will discuss the classification of rank-2 aCM toric bundles on the Veronese surface and compute the number of underlying bundles, counted up to twist and up to isomorphism.

11:00-11:40 Jeongseok Oh
Introduction to virtual cycles II

A typical way of producing enumerative invariants or counting geometric objects is to construct a virtual cycle which is a homology cycle on the moduli space (of the objects). Usually we count the objects satisfying certain conditions that can be described as cohomology classes on the moduli space. Then capping them gives a number. However, the construction of virtual cycle has nothing to do with moduli space -- it is constructed on an arbitrary space (with some structure). I will explain the motivation (from moduli space), the construction (on arbitrary space) of virtual cycles. If time allows, I will present some examples to show how to use it.

Organizers

Kiryong Chung (Kyungpook National University)

Sukmoon Huh (Sungkyunkwan University)

Jeong-Seop Kim (KIAS)