arXiv subjects

Young-Hoon Kiem

Publications and source records attributed to Young-Hoon Kiem.

At least 19 recordsLinked to original sources

arXiv2026

Local geometry for Schmidt number witnesses

Suppose that $F_E$ is the face of the convex set of all $m\otimes n$ bi-partite states which consists of states with ranges contained in a subspace $E$. By the result in [Phys. Rev. A {\bf 112} (2025), 032426], there exist Schmidt number $\ell$ witnesses outside of the face $F_{E^\perp}$ if and only if $\ell\leμ$, where $μ$ is the minimum of Schmidt ranks of nonzero vectors in $E$. In this paper, we show that every neighborhood of the projection state located at the center of $F_{E^\perp}$ contains Schmidt number $\ell$ witnesses if and only if $\ell\geμ$. In order to understand the local shape of the convex sets of $k$-blockpositive matrices around this projection state, we introduce principal supporting hyperplanes, and show that they correspond to vectors in $E$ with Schmidt rank $\le k$. Using techniques from algebraic geometry, we show that the inclusions between these convex sets are detected by principal supporting hyperplanes through the projection state for generic subspaces $E$, while for non-generic subspaces they may not be detected by any supporting hyperplane.

quant-ph↗

arXiv2026

Asymptotic distributions of the Betti and Hodge numbers of $\overline{\mathcal M}_{g,n}$

We study the asymptotic distributions of the Betti and Hodge numbers of the moduli spaces $\overline{\mathcal M}_{g,n}$ of stable pointed curves and the all-light Hassett spaces $\overline{\mathcal M}_{g,0|n}$, for fixed genus $g\ge1$ as the number of markings tends to infinity. We prove central limit theorems for the full Betti distributions, whose variances grow linearly in $n$ with leading coefficients independent of the genus. For the normalized even Betti numbers, we establish refined local limit theorems with estimates uniform in degree. These local estimates yield asymptotic formulas for individual even Betti numbers and determine the sharp central ultra-log-concavity thresholds $\frac{3(e-2)}{2(3-e)}$ and $3$, respectively. For $g\ge2$, the off-diagonal Hodge components form proportions of exact orders $n^{-1/2}$ and $n^{-1}$ of the total cohomology of the two spaces, respectively. The conditional distributions of the Hodge difference $p-q$ given $p\ne q$ converge, for both spaces, to a single discrete law independent of the genus. We also prove that tautological classes account for all but a proportion of exact order $n^{-1}$ of the even cohomology of $\overline{\mathcal M}_{g,n}$. The tautological degree distribution converges to the standard Gaussian under the centering and scaling of the ambient even Betti distribution. An explicit asymptotic formula for the ratio of the ordinary Euler characteristics of the light and heavy spaces answers a question of Kannan, Serpente and Yun. The proofs combine the heavy/light substitution with a stratification of the light spaces by stable cores, singularity analysis of Hodge generating functions, and refined Fourier estimates exploiting Poincaré duality.

math.AG↗

arXiv2026

Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal M}_{1,n}$

By studying wall crossings of Hassett moduli spaces of weighted pointed curves, we prove a recursive formula that reduces the computation of the cohomology of $\overline{\mathcal M}_{g,n}$ to that of the moduli space $\overline{\mathcal M}_{g,(0^+)^n}$ of stable pointed curves of minimal weights. For $g=1$, we further prove an explicit formula for the Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal M}_{1,(0^+)^n}$. Combining these, we prove formulas that enable us to compute the cohomology of $\overline{\mathcal M}_{1,n}$ efficiently.

math.AG↗

arXiv2026

Integral zeros of Krawtchouk polynomials and quantum entanglements

In Heo and Kiem (2019), the authors raised a problem relating integral zeros of binary Krawtchouk polynomials, parametrized by quadruples $(m,n,k,\ell)$ with $m\le n$ and $k\le \ell$, to the existence of pairs of subspaces satisfying the range criterion in quantum information theory, and settled the symmetric case $k=\ell$. In this paper, we solve the problem for the following three asymmetric families: $(m,n,k,\ell)=(3,t^2-1,t(t-1)/2,t(t+1)/2)$, $t\ge2$; $(m,n,k,\ell)=(4,3t^2+2t-1,(3t^2-t)/2,(3t^2+5t+2)/2)$, $t\ge2$; $(m,n,k,\ell)=(4,3s^2+4s,(3s^2+s)/2,(3s^2+7s+4)/2)$, $s\ge1$. The remaining cases are sporadic zeros, with only finitely many for each $m\ge5$.

quant-ph↗

arXiv2026

Hodge--Deligne polynomials and $S_n$-representations on the cohomology of $\overline{\mathcal M}_{2,n}$

We compute the Hodge-Deligne polynomials of the moduli spaces $\overline{\mathcal M}_{2,n}$ of stable $n$-pointed curves of genus two, together with the action of the symmetric group $S_n$ on their cohomology, for all $n$. Using Hassett's moduli spaces of weighted pointed curves, we first show that the generating function of the $S_n$-equivariant Hodge-Deligne polynomials of $\overline{\mathcal M}_{g,n}$ is obtained from that of the moduli spaces of minimally weighted pointed curves by plethysm with the explicit symmetric function $Z=h_1-\sum_{k\ge3}(x+x^2+\dots+x^{k-2})h_k$; numerically, this is an explicit change of variables. The latter spaces map to $\overline{\mathcal M}_g$, and over each boundary stratum they are built from punctured universal curves and Losev-Manin spaces. Combining the stratification by stable graphs with Burnside's formula and twisted characteristics, we obtain in genus two a closed formula whose only non-elementary input is the cohomology of symplectic local systems on $\mathcal M_{1,1}$ and $\mathscr A_2$, known by work of Eichler-Shimura and Petersen. This determines the $S_n$-representations on all $H^{p,q}(\overline{\mathcal M}_{2,n})$ for $n\le 19$ and on all even degree $H^{p,q}(\overline{\mathcal M}_{2,n})$ for every $n$. We carry out the computation in all degrees for $n\le 18$, where it agrees with an independent computation by point counts, and in even degrees for $n\le 30$. As an application, we find that the even degree cohomology of $\overline{\mathcal M}_{2,n}$ is of Hodge-Tate type if and only if $n\le 20$: the space $H^{23,1}(\overline{\mathcal M}_{2,21})$ is the multiplicity free $S_{21}$-representation with Frobenius characteristic $s_1\cdot h_2[e_{10}]$, of dimension $1939938$, so that $H^{24}(\overline{\mathcal M}_{2,21})$ is not spanned by algebraic cycle classes.

math.AG↗

arXiv2026

Cohomology of moduli spaces of pointed curves

In this paper, after reviewing recent progress on the cohomology of $\overline{\cal M}_{0,n}$, we further our investigation on the cohomology of moduli spaces of pointed curves in continuation of [2,4,5,6,7,8]. In particular, we prove that the Betti number distribution of the Fulton-MacPherson compactification $C[n]$ of the space of $n$ ordered distinct points on any smooth projective curve $C$ is asymptotically Gaussian as $n$ goes to infinity.

math.AG↗

arXiv2026

Stratified motivic invariants and bivariate deformations of Poincaré polynomials

For a stratified variety $X$ and a motivic invariant $\mathbb{H}$, we consider the stratified invariant which interpolates the invariant of $X$ and that of its interior $X^{\circ}$. Based on observations in moduli theory, we introduce the notion of an echelon tower of stratified varieties and then prove explicit inductive formulae for the stratified invariants of the moduli spaces $\overline{M}_{0,n}$ of stable curves of genus $0$ and the Fulton-MacPherson varieties $Y[n]$ for any smooth projective variety $Y$. Using these, we show that the mysterious bivariate deformations of the even degree Poincaré polynomials of $\overline{M}_{0,n+1}$ and $\mathbb{P}^1[n]$ in \cite{BercziKiem2026} are nothing but stratified virtual Poincaré polynomials.

math.AG↗

arXiv2026

Real-rootedness of the Poincaré polynomials of $\overline{\mathcal M}_{0,n}$: an AI-assisted proof

We prove real-rootedness for the Poincaré polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli. The proof starts from the Keel--Manin--Getzler recurrence, but its main new idea is a bivariate deformation $F_m(y,t)$ of the Poincaré polynomial. This deformation reveals a hidden interlacing structure not visible in the one-variable recurrence. For fixed $t<0$, the zero set of $F_m$ in the $y$-direction is controlled by a Sturm--Rolle argument on the interval $0<y<1-t$. The original polynomial is recovered on the slice $y=1$, and the ordered crossings of the moving roots through this slice give both real-rootedness and strict interlacing. Consequently, the Betti numbers of $\overline{\mathcal M}_{0,n}$ form an ultra-log-concave sequence. We further prove real-rootedness and ultra-log-concavity for the Poincaré polynomial of the Fulton--MacPherson space $\mathbb{P}^1[n]$ of $n$ ordered points in degenerations of the complex projective line. The proof for $\overline{\mathcal M}_{0,n}$ was obtained through an iterative AI-assisted workflow with Co-Mathematician, an agentic frontier-model system developed by Google DeepMind. Our role was to formulate the problem, evaluate the proposed proof attempts, identify gaps and request corrections, compare the developing argument with the literature, and refine the presentation of the final proof. Our additional human contribution was to observe that a similar residual deformation strategy applies to the Fulton--MacPherson spaces $\mathbb P^1[n]$, yielding the corresponding real-rootedness theorem.

math.AG↗

arXiv2026

Asymptotic distribution of the Betti numbers of $\overline{\mathcal{M}}_{0,n}$

Asymptotic normality is frequently observed in large combinatorial structures, rigorously established for many quantities such as cycles or inversions in random permutations, the number of prime factors of random integers, and various parameters of random graphs. In this paper, we investigate whether this normal limit behavior extends to the topological invariants of geometric spaces. We show that the Betti numbers of the moduli space of rational curves with $n$ marked points $\overline{\mathcal{M}}_{0,n}$ and the Fulton-MacPherson configuration space $\mathbb{P}^1[n]$ are asymptotically normally distributed. Based on numerical evidence and established log-concavity, we conjecture that the Betti numbers of the quotients of these spaces by the symmetric group $\mathbb{S}_n$ are also asymptotically normally distributed. In contrast, we provide examples of geometric spaces that do not follow this Gaussian law.

math.AG↗

arXiv2025

Generalized intersection pairings on moduli spaces of vector bundles over a curve

We introduce the notion of a generalized intersection pairing for an Artin stack with a proper good moduli space and nonempty stable part. For the moduli stack of semistable bundles over a smooth projective curve, there are four known constructions by partial desingularization, parabolic bundles, stable pairs and wall crossing. In this paper, we compare all these generalized intersection pairings by establishing wall crossing formulas between them. Explicit computations for low rank cases are included.

math.AG↗

arXiv2025

Characteristic polynomial of $\overline{\mathcal{M}}_{0,n}$ and log-concavity

Motivated by Stanley's generalization of the chromatic polynomial of a graph to the chromatic symmetric function, we introduce the characteristic polynomial of a representation of the symmetric group, or more generally, of a symmetric function. When the representation arises from geometry, the coefficients of its characteristic polynomial tend to form a log-concave sequence. To illustrate, we investigate explicit examples, including the $n$-fold products of the projective spaces, the GIT moduli spaces of points on $\mathbb{P}^1$ and Hessenberg varieties. Our main focus lies on the cohomology of the moduli space of pointed rational curves, for which we prove asymptotic formulas of its characteristic polynomial and establish asymptotic log-concavity.

math.AG↗

arXiv2025

Cosection localization via shifted symplectic geometry

The purpose of this paper is to shed a new light on classical constructions in enumerative geometry from the view point of derived algebraic geometry. We first prove that the cosection localized virtual cycle of a quasi-smooth derived Deligne-Mumford stack with a $(-1)$-shifted closed $1$-form is equal to the virtual Lagrangian cycle of the degeneracy locus which is $(-2)$-shifted symplectic. We next establish a shifted analogue of the Lagrange multipliers method which gives us the quantum Lefschetz theorems as immediate consequences of the equality of virtual cycles. Lastly we study derived algebraic geometry enhancements of gauged linear sigma models which lead us to the relative virtual cycles in a general and natural form.

math.AG↗

arXiv2024

Recursive algorithm and log-concavity of representations on the cohomology of $\overline{\mathcal M}_{0,n}$

We provide a programmable recursive algorithm for the $\mathbb{S}_n$-representations on the cohomology of the moduli spaces $\overline{\mathcal M}_{0,n}$ of $n$-pointed stable curves of genus 0. As an application, we find explicit inductive and asymptotic formulas for the invariant part $H^*(\overline{\mathcal M}_{0,n}/\mathbb{S}_n)$ and prove that its Poincaré polynomial is asymptotically log-concave. Based on numerical computations with our algorithm, we further conjecture that the sequence $\{H^{2k}(\overline{\mathcal M}_{0,n})\}$ of $\mathbb{S}_n$-modules is equivariantly log-concave.

math.AG↗

arXiv2024

Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture

We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group $S_n$ on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs conjecture. Moreover we present a natural generalization of the Shareshian-Wachs conjecture that involves generalized Hessenberg varieties and provide an elementary proof. As a byproduct, we propose a generalized Stanley-Stembridge conjecture for weighted graphs. Our investigation into the birational geometry of generalized Hessenberg varieties enables us to modify them into much simpler varieties like projective spaces or permutohedral varieties by explicit sequences of blowups or projective bundle maps. Using this, we provide two algorithms to compute the $S_n$-representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties.

math.AG↗

arXiv2024

Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials

Recently, Masuda-Sato and Precup-Sommers independently proved an LLT version of the Shareshian-Wachs conjecture which says that the Frobenius characteristics of the cohomology of the twin manifolds of regular semisimple Hessenberg varieties are unicellular LLT polynomials. The purpose of this paper is to study the geometry of twin manifolds and we prove that they are related by explicit blowups and fiber bundle maps. Upon taking their cohomology, we obtain a direct proof of the modular law which establishes the LLT Shareshian-Wachs conjecture.

math.AG↗

arXiv2023

Representations on the cohomology of $\overline{\mathcal{M}}_{0,n}$

The moduli space $\overline{\mathcal{M}}_{0,n}$ of $n$ pointed stable curves of genus $0$ admits an action of the symmetric group $S_n$ by permuting the marked points. We provide a closed formula for the character of the $S_n$-action on the cohomology of $\overline{\mathcal{M}}_{0,n}$. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of $\overline{\mathcal{M}}_{0,n}$, equivariant with respect to the symmetric group action. Moreover we prove that $H^{2k}(\overline{\mathcal{M}}_{0,n})$ for $k\le 3$ and $H^{2k}(\overline{\mathcal{M}}_{0,n})\oplus H^{2k-2}(\overline{\mathcal{M}}_{0,n})$ for any $k$ are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the $S_n$-representation on the cohomology of the Fulton-MacPherson compactification $\mathbb{P}^1[n]$ of the configuration space of $n$ points on $\mathbb{P}^1$ and more generally on the cohomology of the moduli space $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{m-1},1)$ of stable maps.

math.AG↗

arXiv2022

Generalized Donaldson-Thomas Invariants via Kirwan Blowups

We develop a virtual cycle approach towards generalized Donaldson-Thomas theory of Calabi-Yau threefolds. Let $\mathcal{M}$ be the moduli stack of Gieseker semistable sheaves of fixed topological type on a Calabi-Yau threefold $W$. We construct an associated Deligne-Mumford stack $\widetilde{\mathcal{M}}$ with an induced semi-perfect obstruction theory of virtual dimension zero and define the generalized Donaldson-Thomas invariant of $W$ via Kirwan blowups to be the degree of the virtual cycle $[\widetilde{\mathcal{M}}]^{\mathrm{vir}}$. We show that it is invariant under deformations of the complex structure of $W$.

math.AG↗

arXiv2021

Virtual intersection theories

We construct virtual fundamental classes in all intersection theories including Chow theory, K-theory and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction theories and prove the virtual pullback formula, the virtual torus localization formula and cosection localization principle.

math.AG↗