arXiv subjects
Publications and source records attributed to Huaiqing Zuo.
For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of f. This uncovers a new, unexpected difficulty with proving the monodromy conjecture.
This paper studies the poles of the real Archimedean zeta function for a weighted homogeneous polynomial $f \in \mathbb{R}[x, y]$ with an isolated singularity at the origin. By applying a weighted blow-up, we derive the meromorphic continuation of $Z_{f,φ}$ to $\text{Re }s > -1$. This explicit expression yields a necessary and sufficient condition for a root $s \in (-1, 0)$ of the Bernstein-Sato polynomial $b_f(s)$ to be a pole of $Z_{f,φ}$. Unlike the complex case established by F. Loeser (1985), this condition may fail in certain obvious cases -- such as when $f$ is odd or even in $x$, $y$, or $(x, y)$ -- so not all such roots necessarily become poles.
Let $ (X,0) $ denote an isolated singularity defined by a weighted homogeneous polynomial $ f $. Let $ \mathcal{O}$ be the local algebra of holomorphic function germs at the origin, with the maximal ideal $m $. We study the $k$-th Tjurina algebra, defined by $ A_k(f): = \mathcal{O} / \left( f , m^k J(f) \right) $, where $J(f)$ denotes the Jacobian ideal of $ f $. The zeroth Tjurina algebra is well known to represent the tangent space of the base space of the semi-universal deformation of $(X, 0)$. Motivated by this observation, we explore the deformation of $(X,0)$ with respect to a fixed $k$-residue point. We show that the tangent space of the corresponding deformation functor is a subspace of the $k$-th Tjurina algebra. Explicit calculation of the $k$-th Tjurina numbers, which correspond to the dimensions of the $k$-th Tjurina algebras, plays a crucial role in understanding these deformations. According to the results of Milnor and Orlik, the zeroth Tjurina number can be expressed explicitly in terms of the weights of the variables in $f$. However, we observe that for values of $k$ exceeding the multiplicity of $X$, the $k$-th Tjurina number becomes more intricate and is not solely determined by the weights of the variables. In this paper, we introduce a novel complex derived from the classical Koszul complex and obtain a computable formula for the $k$-th Tjurina numbers for all $ k \geqslant 0 $. As an application, we calculate the $k$-th Tjurina numbers for all weighted homogeneous singularities in three variables.
We resolve the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for Pfaffian ideals.
This paper presents a proof of the monodromy conjecture for determinantal varieties. Our strategy centers on an in-depth analysis of monodromy zeta functions, leveraging a generalized A'Campo formula, an examination of multiple contact loci, and the exploitation of the intrinsic symmetric structures inherent to these varieties. Furthermore, we prove the holomorphy conjecture for determinantal varieties and the monodromy conjecture for Brill-Noether loci of generic curves. Keywords. monodromy conjecture, determinantal varieties, monodromy zeta function.
We determine, up to exponentiating, the polar locus of the multivariable archimedean zeta function associated to a finite collection of polynomials F. The result is the monodromy support locus of F, a topological invariant. We give a relation between the multiplicities of the irreducible components of the monodromy support locus and the polar orders. These generalize results of Barlet for the case when F is a single polynomial. Our result determines the slopes of the polar locus of the zeta function of F, closing a circle of results of Loeser, Maisonobe, Sabbah. We apply our main result to elucidate the topological information contained by the oblique part of the zero locus of any ideal of Bernstein-Sato type.
The jet closure and jet support closure were first introduced by de Fernex, Ein and Ishii to solve the local isomorphism problem. In this paper, we introduce two local algebras associated to jet closure and jet support closure respectively. We show that these two algebras are invariants of the singularities. We compute and investigate these invariants for some interesting cases, such as the cases of monomial ideals and homogeneous ideals. We also introduce a new filtration and jet index to jet closures. The jet index describes which jet scheme recover the information of base scheme. Moreover, we obtain some properties of the jet index. Keywords:jet closure, jet support closure and filtration.
In this paper, we will investigate the jet schemes of determinantal varieties. It is quite often the case that the geometric information concerning the jet schemes of an algebraic variety can be described, but the more refined algebraic information is quite mysterious. For example, it is known that computing the Hilbert function associated to a natural grading on these jet schemes is a very hard problem. The present paper handles a few such computations. It succeeds in computing the Hilbert functions of the second order jet schemes in the case of maximal minors of a $2\times n$ matrix.
Determinantal varieties are important objects of study in algebraic geometry. In this paper, we will investigate them using the jet scheme approach. We have found a new connection for the Hilbert series between a determinantal variety and its jet schemes. We denote the $k$-th order jet scheme of the determinantal variety defined by $r$-minors in an $m \times n$ matrix as $\mathscr{L}^{m,n}_{r,k}$. For the special case where $m$, $n$, and $r$ are equal, and $m$ and $r$ are 3 while $k$ is 1, we establish a correspondence between the defining ideals of $\mathscr{L}^{m,n}_{r,k}$ and abstract simplicial complexes, proving their shellability and obtaining the Hilbert series of $\mathscr{L}^{m,n}_{r,k}$ accordingly. Moreover, for general $\mathscr{L}^{m,n}_{r,k}$, \cite{12} provides its irreducible decomposition. We further provide a specific polynomial family defining its irreducible components. Keywords. Determinantal varieties, jet schemes, shellability, Hilbert series.
For an ideal of a regular $\cc$-algebra, its Bernstein-Sato polynomial is the monic polynomial of the lowest degree satisfying an Bernstein-Sato functional equation. We generalize the notion of Bernstein-Sato functional equations to the case of ideals in an $F$-finite ring of positive characteristic $p$, and show the relationship between these equations and Bernstein-Sato roots. By applying this theory, we provide an explicit description of Bernstein-Sato roots of a weighted homogeneous polynomial with an isolated singularity at the origin in characteristic $p$. Moreover, we give multiplicative and additive Thom-Sebastiani properties for the set of Bernstein-Sato roots, which prove the characteristic $p$ analogue of Budur and Popa's question.
We compute the stringy E-functions of determinantal varieties and establish that the stringy E-function of a determinantal variety coincides with the E-function of the product of a Grassmannian and an affine space. Furthermore, a similar result holds for the projectivization of determinantal varieties. As an application, we verify the non-negativity conjecture for the stringy Hodge numbers of these varieties.
For $f_1,...,f_r\in \mathbb C[z_1,...,z_n]\setminus \mathbb C$, we introduce the variation of archimedean zeta function. As an application, we show that the $n/d$-conjecture, proposed by Budur, Mustaţă, and Teitler, holds for generic multiplicities. Consequently, strong monodromy conjecture holds for hyperplane arrangements with generic multiplicities as well.
The well-known Nakai Conjecture concerns a very natural question: For an algebra of finite type over a characteristic zero field, if the ring of its differential operators is generated by the first order derivations, is the algebra regular? And it is natural to extend the Nakai Conjecture to local domains, in this paper, we verify it for isolated hypersurface singularities of modality $\le 2$, this extends the existing works.
The notion of the Yau sequence was introduced by Tomaru, as an attempt to extend Yau's elliptic sequence for (weakly) elliptic singularities to normal surface singularities of higher fundamental genera. In this paper, we obtain the canonical cycle using the Yau cycle for certain surface singularities of degree two. Furthermore, we obtain a formula of arithmetic genera and an upper bound of geometric genera for these singularities. We also give some properties about the classification of weighted dual graphs of certain surface singularities of degree two.
We provide the formula of motivic zeta function for semi-quasihomogeneous singularities and in dimension two, we determine the poles of zeta functions. We also give another formula for stringy E-function using embedded $\mathbb{Q}$-resolution, and we utilize it to calculate the stringy E-function for semi-quasihomogeneous polynomials and non-degenerate polynomials.
K. Saito's classification of simple elliptic singularities includes three families of weighted homogeneous singularities: $ \tilde{E}_{6}, \tilde{E}_7$, and $ \tilde{E}_8 $. For each family, the isomorphism classes can be distinguished by K. Saito's $j$-functions. By applying the Mather-Yau theorem, which states that the isomorphism class of an isolated hypersurface singularity is completely determined by its $k$-th moduli algebra, M. Eastwood demonstrated explicitly that one can directly recover K. Saito's $j$-functions from the zeroth moduli algebras. This research aims to generalize M. Eastwood's result through meticulous computation of the groupoids associated with simple elliptic singularities. We not only directly retrieve K. Saito's $j$-functions from the $k$-th moduli algebras but also elucidate the automorphism structure within the $k$-th moduli algebras. We derive the automorphisms using the methodology of the $k$-th Yau algebra and establish a Torelli-type theorem for the $\tilde{E}_7 $-family when $k=1$. In contrast, we find that the Torelli-type theorem is inapplicable for the first Yau algebra in the $ \tilde{E}_6 $-family. By considering the first Yau algebra as a module rather than solely as a Lie algebra, we can impose constraints on the coefficients of the transformation matrices, which facilitates a straightforward identification of all isomorphisms. Our new approach also provides a simple verification of the result by Chen, Seeley, and Yau concerning the zeroth moduli algebras.
We classify three fold isolated quotient Gorenstein singularity $C^3/G$. These singularities are rigid, i.e. there is no non-trivial deformation, and we conjecture that they define 4d $\mathcal{N}=2$ SCFTs which do not have a Coulomb branch.
We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N=2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.