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Dae-Won Lee

Publications and source records attributed to Dae-Won Lee.

14 recordsLinked to original sources
arXiv2026

Coupled Pklt Tuples and Varieties of Pklt Type

We introduce asymptotic multiplier ideal sheaves and log canonical thresholds associated with tuples of pseudoeffective divisors on a projective klt pair. We prove that the threshold of a coupled potentially klt tuple is computed by a quasi-monomial valuation. For varieties of potentially klt type, we prove that every big divisor admits a birational Zariski decomposition with semiample positive part. We also prove finite generation of multisection rings of big divisors and give a criterion for a variety of potentially klt type to be a Mori dream space.

math.AG↗
arXiv2026

Hilbert schemes of low-degree rational curves on a prime Fano threefold of degree $22$

In this paper, we give a complete description of the Hilbert schemes of rational curves up to degree $6$ on a prime Fano threefold $X$ of degree $22$. One of the key ingredients in the geometry of these Hilbert schemes is the geometry of bisecant conics associated with rational curves on $X$. As applications, we describe additional geometric features of these moduli spaces and derive Donaldson--Thomas (DT) type invariants.

math.AG↗
arXiv2026

Flexibility of affine cones over blow-ups of weighted projective planes

Let $S_m^n$ be the surface obtained by blowing up $\mathbb{P}(1,1,m)$ at $n$ general smooth points, where $m\geq2$. For every very ample Cartier divisor $H$ on $S_m^n$, we prove that $\mathrm{Affcone}_H(S_m^n)$ is flexible when $n\leq m+1$. For $n=m+2$ and $n=m+3$, we prove that $\mathrm{Affcone}_H(S_m^n)$ is generically flexible.

math.AG↗
arXiv2026

On the quasi-monomiality of the $\alpha$- and $\delta$-invariants

In this paper, we show that for any projective klt pair $(X,\Delta)$ over an algebraically closed field of characteristic \(0\) and any big $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $L$ on $X$, the invariants $\alpha(X,\Delta,L)$ and $\delta(X,\Delta,L)$ are computed by quasi-monomial valuations, without any uncountability assumption on the base field.

math.AG↗
arXiv2026

Structure of the Anticanonical Minimal Model Program for Potentially klt Pairs

We give an alternative proof of the existence of the anticanonical minimal model program for potentially klt pairs, assuming the anticanonical divisor admits a birational Zariski decomposition. Moreover, we establish a structure theorem showing that any partial anticanonical MMP starting from a potentially klt pair can be lifted to a compatible sequence of nonpositive maps between the $\mathbb{Q}$-factorial terminalizations of its successive steps.

math.AG↗
arXiv2025

Minimal model program on the generic fiber of log Calabi-Yau type fibration

We study the minimal model program on the geometric generic fiber of a fibration $f:X\to S$ such that for a Zariski dense subset $S'\subseteq S$, $X_s$ is an $\varepsilon$-lc log Calabi--Yau type for every $s\in S'$. We prove that for a fibration $f:X\to S$ of varieties, if the fibers are of $\varepsilon$-lc log Calabi--Yau type, then the geometric generic fiber $X_{\overline{\eta}}$ is pklt. In particular, for any big divisor $D$ on $X_{\overline{\eta}}$, we can run the anticanonical MMP and $D$-MMP with scaling of an ample divisor on $X_{\overline{\eta}}$.

math.AG↗
arXiv2025

A valuative approach to the anticanonical minimal model program

In this paper, we show that the log canonical threshold of a potentially klt triple can be computed by a quasi-monomial valuation. The notion of potential triples provides a larger and more flexible framework to work with than that of generalized pairs. Our main result can be considered as an extension to the result of Xu on klt pairs. As an application of the main result, we show that we can run the MMP on any potentially klt triples and $-(K_X+\Delta)$-MMP on the potentially klt pairs.

math.AG↗
arXiv2025

On minimal model program and Zariski decomposition of potential triples

In this paper, we investigate properties of potential triples $(X,Δ,D)$ which consists of a pair $(X,Δ)$ and a pseudoeffective $\mathbb{R}$-Cartier divisor $D$. In particular, we show that if $D$ admits a birational Zariski decomposition, then one can associate a generalized pair structure to the potential triple $(X,Δ,D)$. Moreover, we can run the generalized MMP on $(K_X+Δ+D)$ as special cases. As an application, we also show that for a pklt pair $(X,Δ)$, if $-(K_X+Δ)$ admits a birational Zariski decomposition with $\mathrm{NQC}$ positive part, then there exists a $-(K_X+Δ)$-minimal model.

math.AG↗
arXiv2024

Anticanonical divisor with good asymptotic base loci

In this paper, we give a characterization of Fano type varieties in terms of the asymptotic base loci of $-(K_X+\Delta)$. We also show that for a potentially lc pair $(X,\Delta)$, if no plc centers are contained in the augmented base locus $\mathbf{B}_{+}(-(K_X+\Delta))$, then $(X,\Delta)$ has a good $-(K_X+\Delta)$-minimal model. This gives an analogous result of Birkar--Hu on the existence of good minimal models.

math.AG↗
arXiv2017

Set-valued mapping and Rough Probability

In 1982, the theory of rough sets proposed by Pawlak and in 2013, Luay concerned a rough probability by using the notion of Topology. In this paper, we study the rough probability in the stochastic approximation spaces by using set-valued mapping and obtain results on rough expectation, and rough variance.

math.GM↗