arXiv subjects

Wanseok Lee

Publications and source records attributed to Wanseok Lee.

6 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 $\beta_{p,1}(X)$ 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 $\beta_{p,1}(X)$ detects containment in a variety of minimal degree at each hierarchy.

math.AG↗

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 $n \geq 3$.

math.AG↗

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 $r \geq 6$.

math.AG↗

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.

math.AG↗

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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arXiv2013

Projective 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 $\reg(C)$ of a general hyperplane section curve $C = X \cap \mathbb{P}^{r-1}$ takes the maximally possible value $d-r+3$. We show that each of these surfaces is either a cone over a curve $C \subset \mathbb{P}^{r-1}$ of maximal regularity or else a birational outer linear projection of a smooth rational surface scroll $\widetilde{X} \subset \mathbb{P}^{d+1}$. We prove that the Castelnuovo-Mumford regularity of these surfaces satisfies the equality $\reg(X) = d-r+3$ and we compute or estimate various of their cohomological invariants as well as their Betti numbers. We study the the extremal variety $\mathbb{F}(X)$ of these surfaces $X$, that is the closed union of the extremal secant lines of all smooth hyperplane section curves of $X$. We show that $\mathbb{F}(X)$ is either a plane or that otherwise $r =5$ and $\mathbb{F}(X)$ is a rational smooth threefold scroll $S(1,1,1) \subset \mathbb{P}^5$.

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