Program

Abstract

We give a bound on the dimension of the linear automorphism group of a projective variety in terms of its fundamental forms at a general point. Moreover, we show that the bound is achieved precisely when the projective variety is projectively equivalent to an Euler-symmetric variety. As a by-product, we determine the Lie algebra of infinitesimal automorphisms of an Euler-symmetric variety. This is a joint work with Qifeng Li.

Abstract

A normal projective surface with the same Betti numbers of the projective plane $\mathbb{CP}^2$ is called a rational homology projective plane or a $\mathbb{Q}$-homology $\mathbb{CP}^2$. People working in algebraic geometry and topology have long studied a $\mathbb{Q}$-homology $\mathbb{CP}^2$ with possibly quotient singularities.

It is well-known that it has at most five such singular points, but it is still mysterious what is the optimal upper bound for the number of quotient singular points. This is the algebraic Montgomery--Yang problem, which is one of the most famous conjectures in this field.

Recently, we announced a complete resolution of the algebraic Montgomery--Yang problem. The goal of this talk is to give a hidden idea how we approached to resolve it.

This is a joint work with Woohyeok Jo and Kyungbae Park.

Abstract

Koszul modules and resonance loci originated in geometric group theory and made a relatively recent appearance in algebraic geometry. We plan to discuss several applications to illustrate their usefulness. Based on joint work with G. Farkas, C. Raicu, A. Suciu.

Abstract

The Segre locus, the locus of points from which a projective variety is projected non-birationally onto its image, naturally appears when we consider inner projections of projective varieties. In this talk, we present work in progress aimed at computing the defining equations of the Segre locus from given defining equations of a projective variety. Due to computational constraints, our approach assumes that the variety is defined over a field of large positive characteristic. After theoretical considerations, we present explicit calculations for some typical examples of projective varieties with nonempty outer or inner Segre loci. 

Abstract

My talk will begin with the Castelnuovo--Mumford regularity of Castelnuovo-type, coming from a study of cohomologies of locally Cohen--Macaulay modules, which is also related with some geometry of successive generic hyperplane sections. After giving a survey on the regularity, I will talk on a study of the structure of vector bundles on the projective space, especially quasi-Buchsbaum bundles based on a study of the behavior of intermediate cohomologies, and will give a classification of vector bundles on the projective space with small cohomologies in terms of standard system of parameters, including a characterization of null-correlation bundles with corresponding skew-symmetric matrices.

Abstract

In this talk I present results about the behavior of mixed multiplicities and multiplicity sequence under hyperplane sections (in a joint work with C. Polini and B. Ulrich).  We are motivated by the $\mu^\ast$-invariants of an isolated hypersurface singularity, which are the Milnor numbers of the general linear sections of the singularity.  When the singularity is non-isolated, we must replace the Milnor number by the polar multiplicities and Segre numbers, which are variants of mixed multiplicities and multiplicity sequences. Our results show that in most cases, these multiplicities up to an index not greater than the dimension of the base ring minus $1$ remain unchanged by a generic hyperplane section.

Abstract

A monomial curve $C$ is defined by a sequence of coprime integers $0= a_0 < a_1 < \cdots < a_k =: d$. A gap of this sequence is $a_{i+1} - a_i - 1$. Gruson--Lazarsfeld--Peskine bound (1983) says that $\operatorname {reg}  (C) \le d - k +2$, which is equal to the sum of all gaps plus $2$. Lvovsky (1996) showed that it is enough to take the sum of two largest gaps plus $2$. In this paper, under some specific conditions,  we give several new bounds which are better than Lvovsky's bound. Our method relies on the study of Apery sets and Frobenius numbers. From this we can give new criteria to check the (arithmetically) Cohen--Macaulay and Buchsbaum properties of $C$. We also give an application to studying the structure of sumsets.

This is a joint work with Doan Quang Tien.

Abstract

Let $X^n\subset\mathbb P^N$ be a smooth projective variety covered by a maximal irreducible family of lines, parametrized by a subvariety $F^{n-1}\subset G(1,N)$. The universal family of lines $Y=Y_F=\{([l],x)\in F\times X\mid x\in l\}$ has natural projections: $q:Y\to F$, making $Y$ into a line bundle over $F$, and $p:Y\to X$. We denote by $\mu=\deg{p}$ the number of lines through a general point, counted with multiplicity. The pair $(X,F)$ is called $F$-connected if two general points of $X$ can be joined by a chain of lines from $F$, with consecutive lines meeting; in this case $X$ is called a \textit{multiruled variety}. It can be shown that, under mild conditions, the ramification divisor $R$ of $p$ has the form $R=q^{-1}(D)$, where $D\subset F$ is the divisor formed by \textit{special lines\/}.

A natural way to understanding the geometry of $X$ involves studying the concurrent lines which translates into questions about special finite subschemes in $F$. Surprisingly, this viewpoint reveals a connection with higher order abnormality of $F$. Let $H_F$ be the (Pl\"ucker) hyperplane class in $F$. The variety $F$ is called \textit{$r$-normal} if every divisor in the linear system $|F(rH)|$ is cut by a degree-$r$ hypersurface in the ambient Pl\"ucker space; otherwise $F$ is called $r$-abnormal. It turns out that in many cases when $D\sim rH_F$ and $r\in\mathbb Z$, particularly for multiruled complete intersections, $F$ fails to be $r$-normal, $D$ is not cut by a hypersurface of degree $r$, and the $r$-th Veronese embedding $v_r(F)$ is the (isomorphic) projection of the variety $\tilde F_r$, the image of $F$ under the map defined by the complete linear system $|\mathcal O_F(r)|$, from the point $u$ corresponding to $D$. The center of projection $u$ is a highly singular point of the $(\mu-1)$-st secant variety $S^{\mu-1}\tilde F_r$, the locus of linear subspaces spanned by general collections of $\mu$ points of $\tilde F_r$.

The point $u\in S^{\mu-1}\tilde F_r$ is so singular that each point $[\tilde l]\in\tilde F_r$ is contained in its entry locus, and if $v_l=<[l],u>\cap S^{\mu-2}\tilde F_r$, then the entry locus of $v_l$ is the curve in $\tilde F_r$ parametrizing the lines $m\subset X$ such that $m\cap l\ne\varnothing$. For a point $x\in l$, the point $v_r([l])$ and the $\mu-1$ points $v_r([m_1]),\dots,[v_r(m_{\mu-1}])$ corresponding to the residual lines passing through $x$ (counting multiplicities) span a $(\mu-2)$-dimensional linear space obtained by projecting from $u$ the corresponding $(\mu-1)$-dimensional space in $<\tilde F_r>$ containing $u$; this is a kind of the Cayley-Bacharach property. Thus the complicated data involving concurrent lines at various points of $X$ have common origin at a single highly singular point of a suitable secant variety of $\tilde F_r$. This approach is new even in the most classical and well-studied threefold cases, when $X$ is a cubic $V_3$ or the intersection of two quadrics $V_4$ ($r=2$ in the first case and $r=1$ in the second one). For example, for $V_3$ the local $2$-abnormality is due to the fact that the six lines through a general point lie on a conic of directions cut out on the tangent quadric cone; what is important is that these local abnormalities have a global origin.

The study of lines also yields abnormal varieties in a different setting. Let $\Sigma\subset G(1,N)$ be a linear congruence of lines, that is the section of $G(1,N)$ by a (not necessarily general) linear subspace such that the projection $p:Y_\Sigma\to\mathbb P^N$ is birational, let $X\subset\mathbb P^N$, $\dim{X}=n$ be the jump locus of $p$, and assume $X$ to be smooth. It can be shown that the lines of $\Sigma$ are then the $r$-secant lines of $X$ and that $X$ is $(r-2)$-abnormal. The simplest example is the projected Veronese surface in $\mathbb P^4$, where $n=2$ and $r=3$; this congruence was thoroughly studied by Castelnuovo. A similar example with $N=5$, $n=3$ and $r=4$ is the Palatini scroll, and a more interesting example is the Peskine variety for which $N=9$, $n=6$ and $r=4$. It would be nice to have a unified view of the abnormality arising in these instances and the consequences it has for the corresponding theories.

Abstract

Let $C$ be a general $\nu$-gonal curve of genus $g$. We study Brill--Noether loci of stable rank-two vector bundles on $C$, focusing on bundles of speciality three. We construct and describe irreducible components of these loci via extensions of line bundles, and determine the degree ranges for which such components exist. These components are generically smooth and uniruled, and can be either regular or superabundant. Our approach combines the study of degeneracy loci in extension spaces, the Lange--Narasimhan criterion for stability, and the rank-two Petri map. We also discuss the corresponding loci with fixed determinant. This is joint work with Professor Flaminio Flamini (University of Rome Tor Vergata) and Professor Seonja Kim (Chungwoon University).

Abstract

I first met Professor Sijong Kwak in 2000 at Seoul National University, where we began studying commutative algebra together. Over the years, we had the opportunity to work on several problems in algebraic geometry and commutative algebra.

In this talk, I will share some memories of our mathematical journey, from our early seminars to our joint work on projections of algebraic varieties, partial elimination ideals, syzygies, and generic initial ideals. I will also briefly discuss our last joint paper with Kangjin Han, published in 2023. Finally, I will describe some recent work on affine fat point schemes, Gröbner bases, and Hermite–Vandermonde matrices, revisiting ideas from our earlier studies.

Rather than focusing on technical details, I hope to reflect on the mathematics we explored together and the memories associated with it.

Abstract

The Eisenbud--Goto regularity conjecture predicts a sharp bound for the Castelnuovo--Mumford regularity of projective varieties embedded in projective space. Two methods have been developed to attack this conjecture for projective varieties with mild singularities. The first one is to control the complexity of fibers of general projections, and the second one exploits the positivity of double point divisors. Sijong Kwak has made significant contributions to both approaches. In this talk, I will survey the history and recent progress on the Eisenbud--Goto conjecture with a focus on Sijong Kwak's achievements.

Abstract

In this talk, I recall that some problems and results on syzygy behaviors under projection mappings, which were initiated by guidance of Sijong Kwak. The main technical tool for this study is Partial Elimination Ideals (PEIs). I'll present some key results and underlying ideas in this journey and also propose some future directions to pursue.

Abstract

Higher secant varieties lie at the center of additive decomposition problems, and their study is ongoing with active contributions from many algebro-geometers. One of the core questions in this area is how many distinguished additive decompositions a given vector admits, including whether such a decomposition is unique. The latter is known as the identifiability problem. Interestingly, despite its geometric nature, identifiability has unexpected connections with algebraic aspects of higher secant varieties.

In this talk, we present algebraic structures of higher secant varieties, especially those of minimal degree in the sense of Ciliberto and Russo, focusing on their minimal free resolutions. These results were obtained jointly with Professor Sijong Kwak and other collaborators.

Abstract

Subspace arrangements provide a natural meeting point of projective geometry, commutative algebra, and combinatorics. Given a finite collection of linear subspaces of a projective space, one may study their union through its defining ideal and ask how its geometric configuration is reflected in the algebraic invariants of that ideal. Two fundamental invariants in this context are the Castelnuovo-Mumford regularity and the Hilbert function and polynomial. Despite the simple geometric description of these varieties, understanding their defining ideals can be difficult. A classical result in this area is due to Hartshorne and Hirschowitz who settled in the early ’80 the case of generic lines. 

This talk presents recent developments on these questions, focusing on generic and transversal subspace arrangements. The first part concerns bounds on regularity. The second part develops a homological approach to the Hilbert polynomial and local cohomology, based on a complex naturally associated with a family of ideals.

Abstract

In this talk, we establish effective vanishing criteria for the Koszul cohomology of tautological line bundles on symmetric products of smooth projective curves. As an application, we obtain explicit degree bounds for these line bundles to satisfy property $N_p$. Our results generalize classical theorems of Green on syzygies of curves and projective spaces. The main ingredient is a vanishing theorem for weight-one syzygies, proved by induction on the dimension of the symmetric product using a filtration of the kernel bundle. Our results also provide an evidence for a folklore conjecture concerning syzygies of adjoint line bundles on higher-dimensional varieties.

This is joint work with Peter Yi Wei.

Abstract

In 1944 Beniamino Segre proved that the Fermat quartic surface in the $3$-dimensional projective space has infinitely many discrete automorphisms. It was the first example of an algebraic surface with infinite discrete automorphism group. Since then, it has long been an open problem to find generators of the automorphism group. On the other hand, other Fermat hypersurfaces of dimension $>1$ and degree $>2$ admit only projectively linear automorphisms, hence their automorphism groups are finite.

With Keiji Oguiso and Xun Yu, we solve this problem, providing 16 explicit generators, all of finite order. I will talk on this result, along with history of computation of automorphism groups of other K3 surfaces.

We will discuss promising research directions and explore new perspectives on existing problems, with the aim of identifying unexplored questions and potential avenues for further study.

We will discuss promising research directions and explore new perspectives on existing problems, with the aim of identifying unexplored questions and potential avenues for further study.