arXiv subjects

Euisung Park

Publications and source records attributed to Euisung Park.

At least 19 recordsLinked to original sources

arXiv2026

Secant rank and syzygies of projections of elliptic normal curves

We study the syzygies of projections of elliptic normal curves. Let $C \subset \mathbb{P}^{d-1}$ be an elliptic normal curve of degree $d \ge 5$, and let $C_q$ denote the projection of $C$ from a point $q$. We obtain sharp bounds for the Green--Lazarsfeld index of $C_q$ in terms of the secant rank of $q$. More precisely, if $q \in C^s \setminus C^2$, where $C^s$ is the $s$-th secant variety of $C$, then $\mathrm{index}(C_q) \le s-3$, and equality holds for a general point $q$ of $C^s$. In particular, $\mathrm{index}(C_q) = \lceil \frac{d}{2} \rceil - 3$ for a general point $q$ in $\mathbb{P}^{d-1}$. The proof realizes projected elliptic curves as hyperplane sections of elliptic ruled surface scrolls and exploits the known syzygetic properties of these scrolls.

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arXiv2026

On the rank index of projective curves of almost minimal degree

In this article, we investigate the rank index of projective curves $\mathscr{C} \subset \mathbb{P}^r$ of degree $r+1$ when $\mathscr{C} = π_p (\tilde{\mathscr{C}})$ for the standard rational normal curve $\tilde{\mathscr{C}} \subset \mathbb{P}^{r+1}$ and a point $p \in \mathbb{P}^{r+1} \setminus \tilde{\mathscr{C}}^3$. Here, the rank index of a closed subscheme $X \subset \mathbb{P}^r$ is defined to be the least integer $k$ such that its homogeneous ideal can be generated by quadratic polynomials of rank $\leq k$. Our results show that the rank index of $\mathscr{C}$ is at most $4$, and it is exactly equal to $3$ when the projection center $p$ is a coordinate point of $\mathbb{P}^{r+1}$. We also investigate the case where $p \in \tilde{\mathscr{C}}^3 \setminus \tilde{\mathscr{C}}^2$.

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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

Rank 3 Quadratic Generators of Veronese Embeddings: The Characteristic 3 Case

This paper investigates property QR(3) for Veronese embeddings over an algebraically closed field of characteristic $3$. We determine the rank index of $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (d))$ for all $n \geq 2$, $d \geq 3$, proving that it equals $3$ in these cases. Our approach adapts the inductive framework of [HLMP 2021], re-proving key lemmas for characteristic $3$ to establish quadratic generation by rank $3$ forms. We further compute the codimension of the span of rank $3$ quadrics in the space of quadratic equations of the second Veronese embedding, showing it grows as ${n+1 \choose 4}$. This provides a clear explanation of the exceptional behavior exhibited by the second Veronese embedding in characteristic $3$. Additionally, we show that for a general complete intersection of quadrics $X \subset \mathbb{P}^r$ of dimension at least $3$, the rank index of $(X,\mathcal{O}_X (2))$ is $4$, thereby confirming the optimality of our main bound. These results complete the classification of the rank index for Veronese embeddings when ${\rm char}(\mathbb{K}) \ne 2$.

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arXiv2025

On quadratic persistence and Pythagoras numbers of totally real projective varieties

In this paper, we study the relationship between quadratic persistence and the Pythagoras number of totally real projective varieties. Building upon the foundational work of Blekherman et al. in arXiv:1902.02754, we extend their characterizations of arithmetically Cohen-Macaulay varieties with next-to-maximal quadratic persistence to arbitrary case. Our main result classifies totally real non-aCM varieties of codimension $c$ and degree $d$ that exhibit next-to-maximal quadratic persistence in the cases where $c=3$ and $d \geq 6$ or $c \geq 4$ and $d \geq 2c+3$. We further investigate the quadratic persistence and Pythagoras number in the context of curves of maximal regularity and linearly normal smooth curves of genus 3.

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arXiv2024

On rank 3 quadratic equations of Veronese varieties

This paper studies the geometric structure of the locus $Φ_3 (X)$ of rank $3$ quadratic equations of the Veronese variety $X = ν_d (\mathbb{P}^n)$. Specifically, we investigate the minimal irreducible decomposition of $Φ_3 (X)$ of rank $3$ quadratic equations and analyze the geometric properties of the irreducible components of $Φ_3 (X)$ such as their desingularizations. Additionally, we explore the non-singularity and singularity of these irreducible components of $Φ_3 (X)$.

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arXiv2024

Castelnuovo-Mumford regularity of finite schemes

Let $Γ\subset \mathbb{P}^n$ be a nondegenerate finite subscheme of degree $d$. Then the Castelnuovo-Mumford regularity ${\rm reg} (Γ)$ of $Γ$ is at most $\left\lceil \frac{d-n-1}{t(Γ)} \right\rceil +2$ where $t(Γ)$ is the smallest integer such that $Γ$ admits a $(t+2)$-secant $t$-plane. In this paper, we show that ${\rm reg} (Γ)$ is close to this upper bound if and only if there exists a unique rational normal curve $C$ of degree $t(Γ)$ such that ${\rm reg} (Γ\cap C) = {\rm reg} (Γ)$.

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arXiv2024

Some remarks on the $\mathcal{K}_{p,1}$ Theorem

Let $X$ be a non-degenerate projective irreducible variety of dimension $n \ge 1$, degree $d$, and codimension $e \ge 2$ over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Let $β_{p,q} (X)$ be the $(p,q)$-th graded Betti number of $X$. M. Green proved the celebrating $\mathcal K_{p,1}$-theorem about the vanishing of $β_{p,1} (X)$ for high values for $p$ and potential examples of nonvanishing graded Betti numbers. Later, Nagel-Pitteloud and Brodmann-Schenzel classified varieties with nonvanishing $β_{e-1,1}(X)$. It is clear that $β_{e-1,1}(X) \neq 0$ when there is an $(n+1)$-dimensional variety of minimal degree containing $X$, however, this is not always the case as seen in the example of the triple Veronese surface in $\mathbb{P}^9$. In this paper, we completely classify varieties $X$ with nonvanishing $β_{e-1,1}(X) \neq 0$ such that $X$ does not lie on an $(n+1)$-dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties whose Picard number is $\le n-1$.

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arXiv2023

On rank 3 quadratic equations of projective varieties

Let $X \subset ¶^r$ be a linearly normal variety defined by a very ample line bundle $L$ on a projective variety $X$. Recently it is shown in \cite{HLMP} that there are many cases where $(X,L)$ satisfies property $\textsf{QR} (3)$ in the sense that the homogeneous ideal $I(X,L)$ of $X$ is generated by quadratic polynomials of rank $3$. The locus $Φ_3 (X,L)$ of rank $3$ quadratic equations of $X$ in $¶\left( I(X,L)_2 \right)$ is a projective algebraic set, and property $\textsf{QR} (3)$ of $(X,L)$ is equivalent to that $Φ_3 (X)$ is nondegenerate in $¶\left( I(X)_2 \right)$. In this paper we study geometric structures of $Φ_3 (X,L)$ such as its minimal irreducible decomposition. Let \begin{equation*} Σ(X,L) = \{ (A,B) ~|~ A,B \in {\rm Pic}(X),~L = A^2 \otimes B,~h^0 (X,A) \geq 2,~h^0 (X,B) \geq 1 \}. \end{equation*} We first construct a projective subvariety $W(A,B) \subset Φ_3 (X,L)$ for each $(A,B)$ in $Σ(X,L)$. Then we prove that the equality \begin{equation*} Φ_3 (X,L) ~=~ \bigcup_{(A,B) \in Σ(X,L)} W(A,B) \end{equation*} holds when $X$ is locally factorial. Thus this is an irreducible decomposition of $Φ_3 (X,L)$ when ${\rm Pic} (X)$ is finitely generated and hence $Σ(X,L)$ is a finite set. Also we find a condition that the above irreducible decomposition is minimal. For example, it is a minimal irreducible decomposition of $Φ_3 (X,L)$ if ${\rm Pic}(X)$ is generated by a very ample line bundle.

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arXiv2023

On the rank index of some quadratic varieties

Regarding the generating structure of the homogeneous ideal of a projective variety $X \subset \mathbb{P}^r$, we define the rank index of $X$ to be the smallest integer $k$ such that $I(X)$ can be generated by quadratic polynomials of rank at most $k$. Recently it is shown that every Veronese embedding has rank index $3$ if the base field has characteristic $\ne 2, 3$. In this paper, we introduce some basic ways of how to calculate the rank index and find its values when $X$ is some other classical projective varieties such as rational normal scrolls, del Pezzo varieties, Segre varieties and the Plücker embedding of the Grassmannian of lines.

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arXiv2022

On the rank of quadratic equations for curves of high degree

Let $\mathcal{C} \subset \mathbb{P}^r$ be a linearly normal curve of arithmetic genus $g$ and degree $d$. In \cite{SD}, B. Saint-Donat proved that the homogeneous ideal $I(\mathcal{C})$ of $\mathcal{C}$ is generated by quadratic equations of rank at most $4$ whenever $d \geq 2g+2$. Also, in \cite{EKS} Eisenbud, Koh and Stillman proved that $I(\mathcal{C})$ admits a determinantal presentation if $d \geq 4g+2$. In this paper, we will show that $I(\mathcal{C})$ can be generated by quadratic equations of rank $3$ if either $g=0,1$ and $d \geq 2g+2$ or else $g \geq 2$ and $d \geq 4g+4$.

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arXiv2021

Rank 3 Quadratic Generators of Veronese Embeddings

Let $L$ be a very ample line bundle on a projective scheme $X$ defined over an algebraically closed field $\Bbbk$ with ${\rm char}~\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf{QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset \mathbb{P}H^0 (X,L)$ can be generated by quadrics of rank $\leq k$. Many classical varieties such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree satisfy property $\mathsf{QR}(4)$. In this paper, we first prove that if ${\rm char}~\Bbbk \neq 3$ then $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (d))$ satisfies property $\mathsf{QR}(3)$ for all $n \geq 1$ and $d \geq 2$. We also investigate an asymptotic behavior of property $\mathsf{QR}(3)$ for any projective scheme. Namely, we prove that $(i)$ if $X \subset \mathbb{P} H^0 (X,L)$ is $m$-regular then $(X,L^d )$ satisfies property $\mathsf{QR}(3)$ for all $d \geq m$ and $(ii)$ if $A$ is an ample line bundle on $X$ then $(X,A^d )$ satisfies property $\mathsf{QR}(3)$ for all sufficiently large even number $d$. These results provide an affirmative evidence for the expectation that property $\mathsf{QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green-Lazarsfeld's condition $\mathrm{N}_p$ and Eisenbud-Koh-Stillman's determininantal presentation in [EKS88]. Finally, when ${\rm char}~\Bbbk = 3$ we prove that $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (2))$ fails to satisfy property $\mathsf{QR}(3)$ for all $n \geq 3$.

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arXiv2020

On completely decomposable defining equations of points in general position in $\mathbb{P}^n$

The study of the defining equations of a finite set $Γ\subset \mathbb{P}^n$ in linearly general position has been actively attracted since it plays a significant role in understanding the defining equations of arithmetically Cohen-Macaulay varieties. In \cite{T}, R. Treger proved that $I(Γ)$ is generated by forms of degree $\leq \lceil \frac{|Γ|}{n}\rceil$. Since then, Treger's result have been extended and improved in several papers. The aim of this paper is to reprove and improve the above Treger's result from a new perspective. Our main result in this paper shows that $I(Γ)$ is generated by the union of $I(Γ)_{\leq \lceil \frac{|Γ|}{n}\rceil -1}$ and the set of all completely decomposable forms of degree $\lceil \frac{|Γ|}{n}\rceil$ in $I(Γ)$. In particular, it holds that if $d \leq 2n$ then $I(Γ)$ is generated by quadratic equations of rank $2$. This reproves Saint-Donat's results in \cite{SD1} and \cite{SD2}.

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arXiv2019

On hyperquadrics containing projective varieties

Classical Castelnuovo Lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension $c$ is at most ${{c+1} \choose {2}}$ and the equality is attained if and only if the variety is of minimal degree. Also G. Fano's generalization of Castelnuovo Lemma implies that the next case occurs if and only if the variety is a del Pezzo variety. Recently, these results are extended to the next case. This paper is intended to complete the classification of varieties satisfying at least ${{c+1} \choose {2}}-3$ linearly independent quadratic equations. Also we investigate the zero set of those quadratic equations and apply our results to projective varieties of degree $\geq 2c+1$.

math.AG↗

arXiv2019

On the structures of hive algebras and tensor product algebras for general linear groups of low rank

The tensor product algebra TA(n) for the complex general linear group GL(n), introduced by Howe et al., describes the decomposition of tensor products of irreducible polynomial representations of GL(n). Using the hive model for the Littlewood-Richardson coefficients, we provide a finite presentation of the algebra TA(n) for n=2, 3, 4 in terms of generators and relations, thereby giving a description of highest weight vectors of irreducible representations in the tensor products. We also compute the generating function of certain sums of Littlewood-Richardson coefficients.

math.RT↗

arXiv2018

On curves lying on a rational normal surface scroll

In this paper, we study the minimal free resolution of non-ACM divisors $X$ of a smooth rational normal surface scroll $S=S(a_1 ,a_2 ) \subset \mathbb{P}^r$. Our main result shows that for $a_2 \geq 2a_1 -1$, there exists a nice decomposition of the Betti table of $X$ as a sum of much simpler Betti tables. As a by-product of our results, we obtain a complete description of the graded Betti numbers of $X$ for the cases where $S=S(1,r-2)$ for some $r \geq 3$ and $S=S(2,r-3)$ for some $r \geq 6$.

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arXiv2015

Projective varieties of maximal sectional regularity

We study projective varieties $X \subset \mathbb{P}^r$ of dimension $n \geq 2$, of codimension $c \geq 3$ and of degree $d \geq c + 3$ that are of maximal sectional regularity, i.e. varieties for which the Castelnuovo-Mumford regularity $\reg (\mathcal{C})$ of a general linear curve section is equal to $d -c+1$, the maximal possible value (see \cite{GruLPe}). As one of the main results we classify all varieties of maximal sectional regularity. If $X$ is a variety of maximal sectional regularity, then either (a) it is a divisor on a rational normal $(n+1)$-fold scroll $Y \subset \mathbb{P}^{n+3}$ or else (b) there is an $n$-dimensional linear subspace $\mathbb{F} \subset \mathbb{P}^r$ such that $X \cap \mathbb{F} \subset \mathbb{F}$ is a hypersurface of degree $d-c+1$. Moreover, suppose that $n = 2$ or the characteristic of the ground field is zero. Then in case (b) we obtain a precise description of $X$ as a birational linear projection of a rational normal $n$-fold scroll.

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arXiv2015

On surfaces of maximal sectional regularity

We study projective surfaces $X \subset \mathbb{P}^r$ (with $r \geq 5$) of maximal sectional regularity and degree $d > r$, hence surfaces for which the Castelnuovo-Mumford regularity $\reg(\mathcal{C})$ of a general hyperplane section curve $\mathcal{C} = X \cap \mathbb{P}^{r-1}$ takes the maximally possible value $d-r+3$. We use the classification of varieties of maximal sectional regularity of \cite{BLPS1} to see that these surfaces are either particular divisors on a smooth rational $3$-fold scroll $S(1,1,1)\subset \mathbb{P}^5$, or else admit a plane $\mathbb{F} = \mathbb{P}^2 \subset \mathbb{P}^r$ such that $X \cap \mathbb{F} \subset \mathbb{F}$ is a pure curve of degree $d-r+3$. We show that our surfaces are either cones over curves of maximal regularity, or almost non-singular projections of smooth rational surface scrolls. We use this to show that the Castelnuovo-Mumford regularity of such a surface $X$ satisfies the equality $\reg(X) = d-r+3$ and we compute or estimate various of the cohomological invariants as well as the Betti numbers of such surfaces. We also study the geometry of extremal secant lines of our surfaces $X$, more precisely the closure $Σ(X)$ of the set of all proper extremal secant lines to $X$ in the Grassmannian $\mathbb{G}(1, \mathbb{P}^r).$

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