Searcharxiv⌕ Search

arXiv subjects

Dae-Won Lee

Publications and source records attributed to Dae-Won Lee.

17 recordsLinked to original sources
arXiv2026

Anticanonical divisor with good asymptotic base loci

We characterize varieties of Fano type in terms of potentially log canonical pairs and birational Zariski decompositions. We also prove the existence of anticanonical minimal models of a potentially lc pair $(X,Δ)$ under the assumptions that no potentially log canonical centers are contained in the augmented base locus of $-(K_X+Δ)$ and that $-(K_X+Δ)$ is a big $\mathbb{Q}$-Cartier divisor admitting a birational Zariski decomposition with NQC positive part. This can be considered as an anticanonical analogue of the results by Birkar--Hu.

math.AG↗
arXiv2026

Admissible extremal rays and the anticanonical minimal model program

We develop an anticanonical minimal model program for projective $\mathbb{Q}$-factorial klt pairs $(X,Δ)$ such that $-(K_{X}+Δ)$ is pseudoeffective. We first prove the cone and contraction theorems for potentially klt triples and establish the existence of flips for such triples. Using these results, we introduce the notion of admissible extremal rays and show that $(K_{X}+Δ)$-positive admissible extremal rays are precisely the rays that are to be contracted by $-(K_{X}+Δ)$-MMP. The elementary MMP steps associated to such admissible rays preserve klt singularities. For potentially klt pairs $(X,Δ)$ with $\mathbb{Q}$-divisors $Δ$, we prove that every $(K_{X}+Δ)$-positive extremal ray is admissible and obtain a cone theorem for the $(K_{X}+Δ)$-positive part of the Mori cone. Assuming termination of the anticanonical minimal model program, we establish a dichotomy: potentially klt pairs admit anticanonical minimal models, whereas non-potentially klt pairs admit non-admissible models.

math.AG↗
arXiv2026

Semiample perturbations over arbitrary fields

Assuming projective log resolutions, we construct semiample perturbations over arbitrary fields that preserve all log canonical places. As an application, for divisor classes with finitely generated section rings, we characterize the existence of effective divisors yielding lc or klt pairs in terms of asymptotic orders of vanishing. We also derive minimal model and semiampleness results for generalized threefolds over $F$-finite fields of characteristic $p>5$.

math.AG↗
arXiv2026

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 $-(K_X+Δ)$-MMP on the potentially klt pairs.

math.AG↗
arXiv2026

Mori dream fibers and the geometric generic fiber

We construct a smooth projective family of rational surfaces over $\mathbb{G}_{m,\mathbb{Z}}$. The Mori dream property of a fiber is determined by the torsion of the normal bundle of an anticanonical cycle. Over $\mathbb{C}$, the locus of Mori dream fibers is Zariski dense. For every prime $p$, every geometric fiber over a closed point of the reduction modulo $p$ is a Mori dream surface, whereas the geometric generic fiber is not a Mori dream space. In either setting, no restriction to a nonempty open subset is a Mori dream morphism. We also prove that, over any algebraically closed field, a projective fibration becomes a Mori dream morphism after shrinking the base whenever the set of points with Mori dream fibers is not contained in a countable union of proper closed subsets.

math.AG↗
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 $α$- and $δ$-invariants

In this paper, we show that for any projective klt pair $(X,Δ)$ over an algebraically closed field of characteristic \(0\) and any big $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $L$ on $X$, the invariants $α(X,Δ,L)$ and $δ(X,Δ,L)$ 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 $X_{\overlineη}$ is pklt. In particular, for any big divisor $D$ on $X_{\overlineη}$, we can run the anticanonical MMP and $D$-MMP with scaling of an ample divisor on $X_{\overlineη}$.

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↗
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↗