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Sukmoon Huh

Publications and source records attributed to Sukmoon Huh.

At least 19 recordsLinked to original sources
arXiv2026

Toric Representation Type of the Veronese Surface

In this article we determine the toric representation type of the Veronese surface $(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(d))$. Based on Klyachko filtrations, we introduce an explicit criterion for a toric vector bundle of arbitrary rank to be arithmetically Cohen--Macaulay. For $d \geq 3$, suitable configurations of partial flags produce stable toric $d$-aCM bundles corresponding to imaginary non-isotropic Schur roots of star-shaped quivers. Their self-extensions give an exact representation embedding of $\operatorname{mod}\mathbb{C}\langle x,y\rangle$, proving that the Veronese surface is toric-wild precisely for $d \geq 3$, while it is toric-finite for $d=1,2$. For $d=3,4$, suitable twists of the basic stable bundles are Ulrich, and the same construction proves that the corresponding Veronese surfaces are toric Ulrich-wild.

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arXiv2025

Virtual Poincare polynomial of moduli space of semistable sheaves of rank two on reducible curves

The main purpose of this paper is to give an explicit description of the moduli space of semistable sheaves of rank two on a stable curve C obtained by gluing two smooth curves at a point. We prove that the moduli space is irreducible and birational to a projective bundle over the moduli space of stable vector bundles on each component curve, independently of the choice of polarization. As an application, we compute the virtual Poincare polynomial of the moduli space.

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arXiv2025

Counting aCM Toric Bundles of Rank Two on the Veronese Surface

We define the isomorphism classes of torus-equivariant rank 2 arithmetically Cohen-Macaulay (aCM) vector bundles on the Veronese surface, up to a twist by the hyperplane class, and count them. Our approach makes use of Klyachko's description of toric vector bundles via filtrations and the associated cohomology computation. We also describe several representative bundles.

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arXiv2025

Net logarithmic tangent sheaves of complete intersections

The main purpose of this paper is to define the {\it net logarithmic tangent sheaf}, as a generalization of the logarithmic tangent sheaf introduced by P.~Deligne, over the field of complex numbers, and prove some basic properties and give some applications. The generalization is valid for the pairs of the smooth complete intersection variety and its complete intersecting subvariety. As applications, we investigate the locus of net logarithmic tangent sheaves on a smooth cubic surface in the corresponding moduli space of semistable sheaves.

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arXiv2023

Logarithmic vector bundles on the blown-up surface

We study the logarithmic vector bundles associated to arrangements of smooth irreducible curves with small degree on the blow-up of the projective plane at one point. We then investigate whether they are Torelli arrangements, that is, they can be recovered from the attached logarithmic vector bundles.

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arXiv2023

Generalized logarithmic sheaf on smooth projective surfaces

We define the notion of generalized logarithmic sheaves on a smooth projective surface, associated to a pair consisting of a reduced curve and some fixed points on it. We then set up the study of the Torelli property in this setting, focusing mostly in the case of the blow-up of the projective plane on a reduced set of points and, in particular, in the case of the cubic surface. We also study the stability property of generalized logarithmic sheaves as well as carrying out the description of their moduli spaces.

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arXiv2023

Rational curves in a quadric threefold via an $\text{SL}(2,\mathbb{C})$-representation

In this paper, we regard the smooth quadric threefold $Q_{3}$ as Lagrangian Grassmannian and search for fixed rational curves of low degree in $Q_{3}$ with respect to a torus action, which is the maximal subgroup of the special linear group $\text{SL}(2,\mathbb{C})$. Most of them are confirmations of very well-known facts. If the degree of a rational curve is $3$, it is confirmed using the Lagrangian's geometric properties that the moduli space of twisted cubic curves in $Q_3$ has a specific projective bundle structure. From this, we can immediately obtain the cohomology ring of the moduli space.

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arXiv2019

Hypersurface arrangements of aCM type

We investigate the arrangement of hypersurfaces on a nonsingular varieties whose associated logarithmic vector bundle is arithmetically Cohen-Macaulay (for short, aCM), and prove that the projective space is the only smooth complete intersection with Picard rank one that admits an aCM logarithmic vector bundle. We also obtain a number of results on aCM logarithmic vector bundles over several specific varieties. As an opposite situation we investigate the Torelli-type problem that the logarithmic cohomology determines the arrangement.

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arXiv2018

Representation type of surfaces in $\mathbb{P}^3$

The goal of this article is to prove that every surface with a regular point in the three-dimensional projective space of degree at least four, is of wild representation type under the condition that either $X$ is integral or $\mathrm{Pic}(X) \cong \langle \Oo_X(1) \rangle$; we construct families of arbitrarily large dimension of indecomposable pairwise non-isomorphic aCM vector bundles. On the other hand, we prove that every non-integral aCM scheme of arbitrary dimension at least two, is also very wild in a sense that there exist arbitrarily large dimensional families of pairwise non-isomorphic aCM non-locally free sheaves of rank one.

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arXiv2018

aCM vector bundles on projective surfaces of nonnegative Kodaira dimension

In this paper we contribute to the construction of families of arithmetically Cohen-Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces $(X,\Oo_X(1))$ for $\Oo_X(1)$ an ample line bundle. In many cases, we show that for every positive integer $r$ there exists a family of indecomposable aCM vector bundles of rank $r$, depending roughly on $r$ parameters, and in particular they are of \emph{wild representation type}. We also introduce a general setting to study the complexity of a polarized variety $(X,\Oo_X(1))$ with respect to its category of aCM vector bundles. In many cases we construct indecomposable vector bundles on $X$ which are aCM for all ample line bundles on $X$.

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arXiv2018

ACM sheaves on the double plane

The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a by-product, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on a multiple hyperplanes and prove the existence of such bundles that do not split, if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most $3/2$ on the double plane and describe the family of isomorphism classes of them.

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arXiv2017

$2$-nilpotent co-Higgs structures

A co-Higgs sheaf is a pair of a torsion-free coherent sheaf $\mathcal{E}$ and a global section of $\mathcal{E}nd(\mathcal{E})\otimes T_X$ with $T_X$ the tangent bundle. We construct $2$-nilpotent co-Higgs sheaves of rank two for some rational surfaces and of rank three for $\mathbb{P}^3$, using the Hartshorne-Serre correspondence. Then we investigate the non-existence, specially over projective spaces.

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arXiv2017

Existence of nontrivial logarithmic co-Higgs structure on curves

We study various aspects on nontrivial logarithmic co-Higgs structure associated to unstable bundles on algebraic curves. We check several criteria for (non-)existence of nontrivial logarithmic co-Higgs structures and describe their parameter spaces. We also investigate the Segre invariants of these structures and see their non-simplicity. In the end we also study the higher dimensional case, specially when the tangent bundle is not semistable.

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arXiv2016

Logarithmic co-Higgs bundles

In this article we introduce a notion of logarithmic co-Higgs sheaves associated to a simple normal crossing divisor on a projective manifold, and show their existence with nilpotent co-Higgs fields for fixed ranks and second Chern classes. Then we deal with various moduli problems with logarithmic co-Higgs sheaves involved, such as coherent systems and holomorphic triples, specially over algebraic curves of low genus.

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arXiv2016

A note on co-Higgs bundles

We show that for any ample line bundle on a smooth complex projective variety with nonnegative Kodaira dimension, the semistability of co-Higgs bundles of implies the semistability of bundles. Then we investigate the criterion for surface $X$ to have $H^0(T_X)=H^0(S^2T_X)=0$, which implies that any co-Higgs structure of rank two is nilpotent.

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arXiv2015

Curves in Segre threefolds

We study locally Cohen-Macaulay curves of low degree in the Segre threefold with Picard number three and investigate the irreducible and connected components respectively of the Hilbert scheme of them. We also discuss the irreducibility of some moduli spaces of purely one-dimensional stable sheaves and apply the similar argument to the Segre threefold with Picard number two.

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