arXiv subjects

Sijong Kwak

Publications and source records attributed to Sijong Kwak.

At least 19 recordsLinked to original sources

arXiv2025

Hierarchical structure of graded Betti numbers in the quadratic strand

The classical results, initiated by Castelnuovo and Fano and later refined by Eisenbud and Harris, provide several upper bounds on the number of quadrics defining a nondegenerate projective variety. Recently, it has been revealed that these bounds extend naturally to certain linear syzygies, suggesting the presence of a hierarchical structure governing the quadratic strand of graded Betti numbers. In this article, we establish such a hierarchy in full generality. We first prove sharp upper bounds for $β_{p,1}(X)$ depending on the degree of a projective variety $X$, extending the classical quadratic bounds to all linear syzygies and identifying the extremal varieties in each range. We then introduce geometric conditions that describe how containment of $X$ in low-degree varieties influences syzygies, and we show that these conditions stratify the quadratic strand into a finite sequence of hierarchies. This leads to a complete description of all possible extremal behavior. We also prove a generalized $K_{p,1}$-theorem, demonstrating that the vanishing of $β_{p,1}(X)$ detects containment in a variety of minimal degree at each hierarchy.

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arXiv2025

Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question "which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?", we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases. The extremal varieties of dimension $n$, codimension $e$, and degree $d$ are exactly characterized by the following two types: (i) varieties with $d = e+2$, $\operatorname{depth} X =n$, and Green-Lazarsfeld index $a(X)=0$, (ii) arithmetically Cohen-Macaulay varieties with $d = e+3$. This is a generalization of G. Castelnuovo, G. Fano, and E. Park's results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak. In addition, we show that every variety $X$ that belongs to (i) or (ii) is always contained in a unique rational normal scroll $Y$ as a divisor. Also, we describe the divisor class of $X$ in $Y$.

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arXiv2025

Surface counterexamples to the Eisenbud-Goto conjecture

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in $\mathbb{P}^5$, and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in $\mathbb{P}^4$ and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces.

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arXiv2023

On the first non-trivial strand of syzygies of projective schemes and Condition ${\mathrm ND}(l)$

Let $X\subset\mathbb{P}^{n+e}$ be any $n$-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property $\mathbf{N}_{d,p}~(d\ge 2, ~p\geq 1)$, which means that $X$ is $d$-regular up to $p$-th step in the minimal free resolution and the other is a new notion $\mathrm{ND}(\ell)$ which generalizes the classical "being nondegenerate" to the condition that requires a general finite linear section not to be contained in any hypersurface of degree $\ell$. First, we introduce condition $\mathrm{ND}(\ell)$ and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first non-trivial strand of syzygies, which generalize results in the quadratic case to higher degree case, and provide characterizations for the extremal cases. Further, after regarding some consequences of property $\mathbf{N}_{d,p}$, we characterize the resolution of $X$ to be $d$-linear arithmetically Cohen-Macaulay as having property $\mathbf{N}_{d,e}$ and condition $\mathrm{ND}(d-1)$ at the same time. From this result, we obtain a syzygetic rigidity theorem which suggests a natural generalization of syzygetic rigidity on $2$-regularity due to Eisenbud-Green-Hulek-Popescu to a general $d$-regularity.

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arXiv2023

Syzygies of secant varieties of smooth projective curves and gonality sequences

The purpose of this paper is to prove that one can read off the gonality sequence of a smooth projective curve from syzygies of secant varieties of the curve embedded by a line bundle of sufficiently large degree. More precisely, together with Ein-Niu-Park's theorem, our main result shows that the gonality sequence of a smooth projective curve completely determines the shape of the minimal free resolutions of secant varieties of the curve of sufficiently large degree. This is a natural generalization of the gonality conjecture on syzygies of smooth projective curves established by Ein-Lazarsfeld and Rathmann to the secant varieties.

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arXiv2022

Determinantal characterization of higher secant varieties of minimal degree

A variety of minimal degree is one of the basic objects in projective algebraic geometry and has been classified and characterized in many aspects. On the other hand, there are also minimal objects in the category of higher secant varieties, and their algebraic and geometric structures seem to share many similarities with those of varieties of minimal degree. We prove in this paper that higher secant varieties of minimal degree have determinantal presentation of two types, i.e. scroll type and Veronese type. Our result generalizes the del Pezzo-Bertini classification for varieties of minimal degree. Also, as a consequence, we show that for any smooth projective variety having higher secant variety of minimal degree, the embedding line bundle admits a special decomposition into two line bundles as so do those of the well-known examples: varieties of minimal degree, smooth del Pezzo varieties, Segre varieties and 2-Veronese varieties.

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arXiv2021

A matryoshka structure of higher secant varieties and the generalized Bronowski's conjecture

In projective algebraic geometry, there are classical and fundamental results that describe the structure of geometry and syzygies, and many of them characterize varieties of minimal degree and del Pezzo varieties. In this paper, we consider analogous objects in the category of higher secant varieties. Our main theorems say that there is a matryoshka structure among those basic objects including a generalized $K_{p,1}$ theorem, syzygetic and geometric characterizations of higher secant varieties of minimal degree and del Pezzo higher secant varieties, defined in this paper. For our purpose, we prove a weak form of the generalized Bronowski's conjecture raised by C. Ciliberto and F. Russo that relates the identifiability for higher secant varieties to the geometry of tangential projections.

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arXiv2021

Componentwise linearity of projective varieties with almost maximal degree

The degree of a projective subscheme has an upper bound in term of the codimension and the reduction number. If a projective variety has an almost maximal degree, that is, the degree equals to the upper bound minus one, then its Betti table has been described explicitly. We build on this work by showing that for most of such varieties, the defining ideals are componentwise linear and in particular the componentwise linearity is suitable for classifying the Betti tables of such varieties. As an application, we compute the Betti table of all varieties with almost maximal degree and componentwise linear resolution.

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arXiv2019

The reduction number and degree bound of projective subschemes

In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with almost maximal degree. We also give explicit Betti tables for almost maximal cases. Interesting examples are provided to understand our main results.

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arXiv2018

A bound for Castelnuovo-Mumford regularity by double point divisors

Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\text{reg} (\mathcal{O}_X)$ is not even bounded above by any polynomial function of $d$ when $X$ is not smooth. For a normality bound in the smooth case, we establish that $\text{reg}(X) \leq n(d-2)+1$, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that $\text{reg}(X) \leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\text{reg}(X)$ under suitable assumptions.

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arXiv2018

Classification and syzygies of smooth projective varieties with 2-regular structure sheaf

The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo-Mumford regularity.

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arXiv2015

Geometric properties of projective manifolds of small degree

The aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb P^r$ of degree $d \leq r+2$, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalization of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb P^r$ of degree $d \leq r$ with counterexamples for $d=r+1, r+2$. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb P^r$ of dimension $n$ and degree $d \leq n(r-n)+2$ is Calabi-Yau, and give an example that shows this bound is also sharp.

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arXiv2014

On Syzygies, degree, and geometric properties of projective schemes with property $\textbf{N}_{3,p}$

For an algebraic set $X$ (union of varieties) embedded in projective space, we say that $X$ satisfies property $\textbf{N}_{d,p}$, $(d\ge 2)$ if the $i$-th syzygies of the homogeneous coordinate ring are generated by elements of degree $< d+i$ for $0\le i\le p$ (see \cite{EGHP2} for details). Much attention has been paid to linear syzygies of quadratic schemes $(d=2)$ and their geometric interpretations (cf. \cite{AK},\cite{EGHP1},\cite{HK},\cite{GL2},\cite{KP}). However, not very much is actually known about the case satisfying property $\textbf{N}_{3,p}$. In this paper, we give a sharp upper bound on the maximal length of a zero-dimensional linear section of $X$ in terms of graded Betti numbers (Theorem 1.2 (a)) when $X$ satisfies property $\textbf{N}_{3,p}$. In particular, if $p$ is the codimension $e$ of $X$ then the degree of $X$ is less than or equal to $\binom{e+2}{2}$, and equality holds if and only if $X$ is arithmetically Cohen-Maucalay with $3$-linear resolution (Theorem 1.2 (b)). This is a generalization of the results of Eisenbud et al. (\cite{EGHP1,EGHP2}) to the case of $\textbf{N}_{3,p}$, $(p\leq e)$.

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arXiv2013

Sharp bounds for higher linear syzygies and classifications of projective varieties

In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections. Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebraic set $X$ and those of its inner projection $X_q$. Using these results, we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and next-to-extremal case are. This is a generalization of Castelnuovo and Fano's results on the number of quadrics containing a given variety and another characterization of varieties of minimal degree and del Pezzo varieties from the viewpoint of `syzygies'. Note that our method could be also applied to get similar results for more general categories (e.g. connected in codimension one algebraic sets).

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arXiv2012

Analysis on some infinite modules, inner projection, and applications

A projective scheme $X$ is called `quadratic' if $X$ is scheme-theoretically cut out by homogeneous equations of degree 2. Furthermore, we say $X$ satisfies `property $\textbf{N}_{2,p}$' if it is quadratic and the quadratic ideal has only linear syzygies up to first $p$-th steps. In the present paper, we compare the linear syzygies of the inner projections with those of $X$ and obtain a theorem on `embedded linear syzygies' as one of our main results. This is the natural projection-analogue of `restricting linear syzygies' in the linear section case, \cite{EGHP1}. As an immediate corollary, we show that the inner projections of $X$ satisfy property $\textbf{N}_{2,p-1}$ for any reduced scheme $X$ with property $\textbf{N}_{2,p}$. Moreover, we also obtain the neccessary lower bound $(\codim X)\cdot p -\frac{p(p-1)}{2}$, which is sharp, on the number of quadrics vanishing on $X$ in order to satisfy $\textbf{N}_{2,p}$ and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme $X$. These results admit an interesting `syzygetic' rigidity theorem on property $\textbf{N}_{2,p}$ which leads the classifications of extremal and next to extremal cases. For these results we develope the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by M. Green. This new method allows us to treat a wider class of projective schemes which can not be covered by the Koszul cohomology techniques, because these are not projectively normal in general.

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arXiv2011

Generic Initial ideals of Singular Curves in Graded Lexicographic Order

In this paper, we are interested in the generic initial ideals of \textit{singular} projective curves with respect to the graded lexicographic order. Let $C$ be a \textit{singular} irreducible projective curve of degree $d\geq 5$ with the arithmetic genus $ρ_a(C)$ in $\p^r$ where $r\ge 3$. If $M(I_C)$ is the regularity of the lexicographic generic initial ideal of $I_C$ in a polynomial ring $k[x_0,..., x_r]$ then we prove that $M(I_C)$ is $1+\binom{d-1}{2}-ρ_a(C)$ which is obtained from the monomial $$ x_{r-3} x_{r-1}\,^{\binom{d-1}{2}-ρ_a(C)}, $$ provided that $\dim\Tan_p(C)=2$ for every singular point $p \in C$. This number is equal to one plus the number of non-isomorphic points under a generic projection of $C$ into $\p^2$. %if $°(C)=3,4$ then $M(I_C)= °(C)$ by the direct computation. Our result generalizes the work of J. Ahn for \textit{smooth} projective curves and that of A. Conca and J. Sidman \cite{CS} for \textit{smooth} complete intersection curves in $\p^3$. The case of singular curves was motivated by \cite[Example 4.3]{CS} due to A. Conca and J. Sidman. We also provide some illuminating examples of our results via calculations done with {\it Macaulay 2} and \texttt {Singular} \cite{DGPS, GS}.

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arXiv2011

The Degree Complexity of Smooth Surfaces of codimension 2

D.Bayer and D.Mumford introduced the degree complexity of a projective scheme for the given term order as the maximal degree of the reduced Gröbner basis. It is well-known that the degree complexity with respect to the graded reverse lexicographic order is equal to the Castelnuovo-Mumford regularity (\cite{BS}). However, little is known about the degree complexity with respect to the graded lexicographic order (\cite{A}, \cite{CS}). In this paper, we study the degree complexity of a smooth irreducible surface in $\p^4$ with respect to the graded lexicographic order and its geometric meaning. Interestingly, this complexity is closely related to the invariants of the double curve of a surface under the generic projection. As results, we prove that except a few cases, the degree complexity of a smooth surface $S$ of degree $d$ with $h^0(\mathcal I_S(2))\neq 0$ in $\p^4$ is given by $2+\binom{°Y_1(S)-1}{2}-ρ_{a}(Y_{1}(S))$, where $Y_1(S)$ is a double curve of degree $\binom{d-1}{2}-ρ_{a}(S \cap H)$ under a generic projection of $S$ (Theorem \ref{mainthm2}). Exceptional cases are either a rational normal scroll or a complete intersection surface of $(2,2)$-type or a Castelnuovo surface of degree 5 in $\p^4$ whose degree complexities are in fact equal to their degrees. This complexity can also be expressed only in terms of the maximal degree of defining equations of $I_S$ (Corollary \ref{cor:01} and \ref{cor:02}). We also provide some illuminating examples of our results via calculations done with {\it Macaulay 2} (Example \ref{Exam:01}).

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arXiv2009

Graded mapping cone theorem, multisecants and syzygies

Let $X$ be a reduced closed subscheme in $\mathbb P^n$. As a slight generalization of property $\textbf{N}_p$ due to Green-Lazarsfeld, we can say that $X$ satisfies property $\textbf{N}_{2,p}$ scheme-theoretically if there is an ideal $I$ generating the ideal sheaf $\mathcal I_{X/¶^n}$ such that $I$ is generated by quadrics and there are only linear syzygies up to $p$-th step (cf. \cite{EGHP1}, \cite{EGHP2}, \cite{V}). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property $\textbf{N}_{2,p}$(cf. \cite{CKP}, \cite{EGHP1}, \cite{EGHP2} \cite {KP}). In this case, the Castelnuovo regularity and normality can be obtained by the blowing-up method as $\reg(X)\le e+1$ where $e$ is the codimension of a smooth variety $X$ (cf. \cite{BEL}). On the other hand, projection methods have been very useful and powerful in bounding Castelnuovo regularity, normality and other classical invariants in geometry(cf. \cite{BE}, \cite{K}, \cite{KP}, \cite{L} \cite {R}). In this paper, we first prove the graded mapping cone theorem on partial eliminations as a general algebraic tools and give some applications. Then, we bound the length of zero dimensional intersection of $X$ and a linear space $L$ in terms of graded Betti numbers and deduce a relation between $X$ and its projections with respect to the geometry and syzygies in the case of projective schemes satisfying property $\textbf{N}_{2,p}$ scheme-theoretically. In addition, we give not only interesting information on the regularity of fibers and multiple loci for the case of $\textbf{N}_{d,p}, d\ge 2$ but also geometric structures for projections according to moving the center.

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