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Seung-Jo Jung

Publications and source records attributed to Seung-Jo Jung.

14 recordsLinked to original sources
arXiv2026

Defect of projective hypersurfaces with isolated singularities

Let $X$ be a hypersurface with isolated singularities defined by $f$ in ${\bf P^{n+1}}$ with $n>1$. The difference ${\rm def}(X):=h^{n+1}(X)-h^{n-1}(X)$ is called the defect of $X$ (for self-duality of the cohomology of $X$). It is known that its vanishing is closely related to ${\bf Q}$-factoriality of $X$ without assuming rational singularities when $n=3$. This number coincides with the dimension of the cokernel of the inclusion $H^{n-1}(X)\to{\rm IH}^{n-1}(X)$, the rank of the morphism from the vanishing cohomologies of $X$ to $H^{n+1}(X)$ for a one-parameter smoothing of $X$ with total space smooth, and also with the dimension of the unipotent monodromy part of the Milnor fiber cohomology of $f$ with degree $n$. In the case $X$ has only weighted homogeneous isolated singularities, the defect ${\rm def}(X)$ is then given by the $E_2$-term of the spectral sequence of the double complex with differentials ${\rm d}f\wedge$ and $\rm d$ by the $E_2$-degeneration of the pole order spectral sequence. It can be calculated explicitly using a computer even for analogues of the Hirzebruch quintic threefold with more than one hundred ordinary double points found by B.\ van Geemen and J.\ Werner in a compatible way with their computation. We give also an example with ${\rm def}(X)>0$ and $|{\rm Sing}\,X|=1$ in the non-projective cone case where $n=3$.

math.AG↗
arXiv2026

Factoriality of normal projective varieties

For a normal projective variety $X$, the $\mathbb Q$-factoriality defect $σ(X)$ is defined to be the rank of the quotient of the group of Weil divisors by the subgroup of Cartier ones. We prove an improvement of a topological formula of S.G. Park and M. Popa asserting that $σ(X)\le h^{2n-2}(X)-h^2(X)$ by assuming only 1-semi-rational singularities instead of rational singularities, and the equality holds in the 2-semi-rational case. Here the singularities are called $j$-semi-rational if $R^kπ_*{\mathcal O}_{\widetilde{X}}=0$ for any $k\in[1,j]$ with $π:\widetilde{X}\to X$ a desingularization, $h^k(X):=\dim H^k(X,{\mathbb Q})$, and $n:=\dim X\ge2$. We also show (a slight generalization of) the assertion that $\mathbb Q$-factoriality implies factoriality if $X$ is a local complete intersection whose singular locus has at least codimension three. We then get a new proof for the projective case of Grothendieck's theorem asserting that $X$ is factorial if it is a local complete intersection whose singular locus has at least codimension four. We also show that a local complete intersection $X$ of dimension 3 having only isolated singularities is factorial if the (topological) defect ${\rm def}(X):=h^4(X)-h^2(X)$ vanishes, without any assumption on rational singularities.

math.AG↗
arXiv2026

Roots of Bernstein-Sato polynomials for projective hypersurfaces with ordinary double points

Let $X\subset{\mathbb P}^{n-1}$ be a hypersurface of degree $d\ge3$ with ordinary double points, where $n\ge3$. The roots of Bernstein-Sato polynomial of its defining polynomial $f$ are given up to sign by 1, $(n-1)/2$, and $j/d$ for $j\in{\mathbb Z}\cap[n,nd-n-p_f]$ with $p_f$ a positive integer. Here $p_f$ is bounded above by the minimal positive integer $q_s$ satisfying $\binom{q_s+n-1}{n-1}>s:=|{\rm Sing}\,X|$, and we can verify that $p_f$ coincides with $q_s$ in the case the singular points of $X$ are in ``general position". We show that this upper bound is sharp in the case $\binom{\lfloor d/2\rfloor+n-2}{n-1}\ge s$ or $\binom{d+n-3}{n-1}\ge sn$ by providing a homogeneous polynomial of degree $d$ such that the associated projective hypersurface has ordinary double points at given $s$ points in sufficiently general position and is nonsingular outside them (using a theorem of Alexander and Hirschowitz for the second case). It is conjectured that the above sharp bound under the first hypothesis can be extended naturally to the case where $X$ has only $A_2$-singularities instead of ordinary double points.

math.AG↗
arXiv2026

Some remarks on the generalized Hertling conjecture for Tjurina spectrum

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.

math.AG↗
arXiv2026

Tjurina spectrum and graded symmetry of missing spectral numbers

For a hypersurface isolated singularity defined by a convergent power series $f$, the Steenbrink spectrum can be defined as the Poincaré 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 $[(μ{-}τ)/2]$. We can moreover improve the estimate of Briançon-Skoda exponent in the semisimple monodromy case.

math.AG↗
arXiv2025

Spectrum of cones of projective hypersurfaces with singularities isolated

Let $Z$ be a projective hypersurface such that its underlying reduced variety has only isolated singularities. In case its irreducible components have constant multiplicities, for instance if $\dim Z>1$, we show that the spectrum of its cone can be described by using the spectral numbers at singular points of the reduced hypersurface and the global degree. In the non-reduced plane curve case, assuming that the underlying reduced curve has only semi-weighted-homogeneous singularities, we express the spectrum of the cone in terms of the local weights and the weighted degrees and multiplicities of local irreducible components together with the degrees and multiplicities of global ones. These generalize a formula for reduced line arrangements. In the non-reduced ordinary (that is, semi-homogeneous) singularity case, the second formula is essentially equivalent to the one obtained by the third-named author.

math.AG↗
arXiv2023

Spectrum of non-degenerate functions with simplicial Newton polytopes

We show a precise proof of Steenbrink's formula for the spectrum of convenient Newton non-degenerate functions, and prove the symmetry of combinatorial polynomials in the simplicial case. Combined with the modified Steenbrink conjecture for spectral pairs (that is, weighted spectrum) which is recently proved in that case, this simplifies quite a lot of their calculations in such a case. We also introduce the $Γ$-spectrum of simplicial convenient non-degenerate functions as a first approximation of the spectrum, generalizing Arnold's picture in the 2 variable case. Analyzing their difference, we can find simple formulas for weighted spectrum in the 3 or 4 variable case. This is proved by using the symmetry of combinatorial polynomials, and fails in the non-simplicial case. Combining these with the Yomdin-Steenbrink formula for the spectrum, we can prove a formula for the spectrum of certain non-isolated surface singularities with simplicial non-degenerate Newton boundaries. As a byproduct of these arguments, we find an example where the Yomdin-Steenbrink formula for spectral pairs does not hold because of fusions of compact faces under projections.

math.AG↗
arXiv2022

Briançon-Skoda exponents and the maximal root of reduced Bernstein-Sato polynomials

For a holomorphic function $f$ on a complex manifold $X$, the Briançon-Skoda exponent $e^{\rm BS}(f)$ is the smallest integer $k$ with $f^k\in(\partial f)$ (replacing $X$ with a neighborhood of $f^{-1}(0)$), where $(\partial f)$ denotes the Jacobian ideal of $f$. It is shown that $e^{\rm BS}(f)\le d_X$ $(:=\dim X)$ by Brian\c con-Skoda. We prove that $e^{\rm BS}(f)\le[d_X-2\widetildeα_f]+1$ with $-\widetildeα_f$ the maximal root of the reduced Bernstein-Sato polynomial $b_f(s)/(s+1)$, assuming the latter exists (shrinking $X$ if necessary). This implies for instance that $e^{\rm BS}(f)\le d_X-2$ in the case $f^{-1}(0)$ has only rational singularities, that is, if $\widetildeα_f>1$.

math.CV↗
arXiv2022

Higher Du Bois singularities of hypersurfaces

For a complex algebraic variety $X$, we introduce higher $p$-Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms $Ω_X^q$ and the shifted graded pieces of the Du Bois complex $\underlineΩ_X^q$ for $q\le p$. If $X$ is a reduced hypersurface, we show that higher $p$-Du~Bois singularity coincides with higher $p$-log canonical singularity, generalizing a well-known theorem for $p=0$. The assertion that $p$-log canonicity implies $p$-Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two theorems asserting that the sheaves of reflexive differential forms $Ω_X^{[q]}$ ($q\le p$) coincide with $Ω_X^q$ and $\underlineΩ_X^q$ respectively, and these are shown by calculating the depth of the latter two sheaves. We construct explicit isomorphisms between $Ω_X^q$ and $\underlineΩ_X^q$ applying the acyclicity of a Koszul complex in a certain range. We also improve some non-vanishing assertion shown by them using mixed Hodge modules and the Tjurina subspectrum in the isolated singularity case. This is useful for instance to estimate the lower bound of the maximal root of the reduced Bernstein-Sato polynomial in the case where a quotient singularity is a hypersurface and its singular locus has codimension at most 4.

math.AG↗
arXiv2022

Hodge ideals and spectrum of isolated hypersurface singularities

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.

math.AG↗
arXiv2016

On the Craw-Ishii conjecture

Craw and Ishii proved that for a finite abelian group G in SL_3(C) every (projective) relative minimal model of C^3/G is isomorphic to the fine moduli space of θ-stable G-constellations for some GIT parameter θ. In this article, we conjecture that the same is true for a finite group G in GL_3(C) if a relative minimal model Y of X=C^3/G is smooth. We prove this for some abelian groups.

math.AG↗
arXiv2015

Terminal Quotient Singularities in Dimension three via variation of GIT

A 3-fold terminal quotient singularity X=C^3/G admits the economic resolution Y-> X, which is "close to being crepant". This paper proves that the economic resolution Y is isomorphic to a distinguished component of a moduli space of certain G-equivariant objects using the the King stability condition $θ$ introduced by Kędzierski.

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