arXiv subjects
Publications and source records attributed to In-Kyun Kim.
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.
We completely characterize the $n$-fold connected sums of $S^3\times S^4$ that admit Sasaki-Einstein structures by proving that such a structure exists if and only if $n$ is even.
We study polarized cylinders in certain rational surfaces arising from blow-ups of weighted projective planes. In particular, we consider the surfaces obtained by blowing up $m+4$ points in general position on the weighted projective plane $\mathbb{P}(1,1,m)$. These surfaces appear naturally as weighted hypersurfaces or quasi-smooth complete intersections.
Cylinders in Fano varieties receives a lot of attentions recently from the viewpoints of birational geometry and unipotent geometry. In this article, we provide a survey of several known et new results concerning the anti-canonically polar cylindricity of quasi-smooth, well-formed weighted Fano complete intersections in weighted projective spaces.
We develop a one-stage Bayesian framework for quantifying issue-specific legislative alignment in multi-party systems. The approach integrates a Latent Space Item Response Model (LSIRM), which embeds legislators and bills in a shared Euclidean space, with Bayesian beta regression using text-derived topic proportions as bill-level covariates. This yields posterior distributions of legislator- and issue-specific coefficients, enabling coherent comparison of polarization and cohesion across policy domains. Uncertainty is propagated through a one-stage MCMC sampler that jointly updates the latent-space and regression components. Application to the 17th Korean National Assembly reveals substantial heterogeneity in partisan conflict: fiscal domains such as Taxation and Grants and Local Government Budget show sharp polarization with tight within-party clustering, whereas Armed Services, Patriots, and Veterans exhibits weak party structuring and greater intra-party variability. The Democratic Labor Party (DLP) forms a coherent and distinct cluster on several issues even where the two major parties are not strongly polarized, confirming that important dimensions of legislative conflict are not captured by a single left-right ordering. The framework provides a principled tool for analyzing issue-structured voting behavior in legislatures where one-dimensional ideal point models yield unreliable estimates.
In this paper, we study the cylindricity of $\Bbbk$-forms of singular del Pezzo surfaces obtained by blowing up weighted projective planes $\mathbb{P}(1,1,m)$ over an arbitrary field $\Bbbk$ of characteristic zero. As an application, we obtain vertical cylinders on higher-dimensional fibrations whose generic fibers are such $\Bbbk$-forms.
For a locally defined real analytic function $f$, we study the relation between the oscillation index of oscillatory integrals and the real log canonical threshold. The former is always negative, and its absolute value is greater than or equal to the latter. They coincide very often, but there are certain exceptional cases, and it is not very clear when the equality holds. In this note we give some sufficient conditions for the coincidence to hold or to fail. In the Newton-nondegenerate convenient homogeneous case, we show that the strict inequality holds if the number of variables $n$ is even and smaller than the degree $d$ of $f$ (or $f^{-1}(0)=\{0\}$), and the equality holds if $n$ is odd and $f^{-1}(0)=\{0\}$ (in particular, $d$ is even). The first assertion does not seem to be compatible with some standard formula in the literature, and there must be some error somewhere, although it does not seem easy to detect it inside this paper.
It is well known that there are totally 130 deformation families of quasi-smooth terminal weighted hypersurface Fano threefolds and all members belonging to 95 families of Fano indices one are birationally rigid. Among remaining $35$ families, $15$ families have the property that general members are irrational, in particular, any of them has a finite automorphism group and is not cylindrical. In the present paper, we will observe the cylindricity and the full automorphism groups of every member in the remaining $20$ families. Moreover, we deal with the cylindricity of their forms.
In this paper, we study the K-stability of del Pezzo surfaces with a single quotient singularity whose minimal resolution admits exactly two exceptional curves \(E_1\) and \(E_2\) with \(E_{1}^2=-n\), \(E_{2}^2=-m\) for \(n,m\geq 2\).
We completely determine the existence of anticanonical polar cylinders in quasi-smooth log del Pezzo surfaces of index one.
For a hypersurface isolated singularity defined by a convergent power series $f$, the Steenbrink spectrum can be defined as the Poincar\'e polynomial of the graded quotients of the $V$-filtration on the Jacobian ring of $f$. The Tjurina subspectrum is defined by replacing the Jacobian ring with its quotient by the image of the multiplication by $f$. We prove that their difference (consisting of missing spectral numbers) has a canonical graded symmetry. This follows from the self-duality of the Jacobian ring, which is compatible with the action of $f$ as well as the $V$-filtration. It implies for instance that the number of missing spectral numbers which are smaller than $(n{+}1)/2$ (with $n$ the number of variables) is bounded by $[(\mu{-}\tau)/2]$. We can moreover improve the estimate of Brian\c{c}on-Skoda exponent in the semisimple monodromy case.
We describe the K-moduli spaces of weighted hypersurfaces of degree $2(n+3)$ in $\mathbb{P}(1,2,n+2,n+3)$. We show that the K-polystable limits of these weighted hypersurfaces are also weighted hypersurfaces of the same degree in the same weighted projective space. This is achieved by an explicit study of the wall crossing for K-moduli spaces $M_w$ of certain log Fano pairs with coefficient $w$ whose double cover gives the weighted hypersurface. Moreover, we show that the wall crossing of $M_w$ coincides with variation of GIT except at the last K-moduli wall which gives a divisorial contraction. Our K-moduli spaces provide new birational models for some natural loci in the moduli space of marked hyperelliptic curves.
We study the original version of the generalized Hertling conjecture on the variance of the Tjurina spectral numbers, which was proposed by Shi, Wang, and Zuo, and provide a sufficient condition for the original conjecture to fail, employing a theorem of Hertling in an essential way. We calculate certain examples using some codes in Singular.
The first secant variety $\Sigma$ of a rational normal curve of degree $d \geq 3$ is known to be a $\mathbf{Q}$-Fano threefold. In this paper, we prove that $\Sigma$ is K-polystable, and hence, $\Sigma$ admits a weak K\"{a}hler-Einstein metric. We also show that there exists a $(-K_{\Sigma})$-polar cylinder in $\Sigma$.
We prove the existence of singular del Pezzo surfaces that are neither K-semistable nor contain any anticanonical polar cylinder.
We prove that closed simply connected $5$-manifolds $2(S^2\times S^3)\# nM_2$ allow Sasaki-Einstein structures, where $M_2$ is the closed simply connected $5$-manifold with $\mathrm{H}_2(M_2,\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$, $nM_2$ is the $n$-fold connected sum of $M_2$, and $2(S^2\times S^3)$ is the two-fold connected sum of $S^2\times S^3$.
We completely classify K-stability of log del Pezzo hypersurfaces with index 2.
We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal $V$-filtration modulo the Jacobian ideal. Via the Tjurina subspectrum, we can compare the Hodge ideal spectrum with the Steenbrink spectrum which can be defined by the microlocal $V$-filtration. As a consequence of a formula of Mustata and Popa, these two spectra coincide in the weighted homogeneous case. We prove sufficient conditions for their coincidence and non-coincidence in some non-weighted-homogeneous cases where the defining function is semi-weighted-homogeneous or with non-degenerate Newton boundary in most cases. We also show that the convenience condition can be avoided in a formula of Zhang for the non-degenerate case, and present an example where the Hodge ideals are not weakly decreasing even modulo the Jacobian ideal.