Complete Intersection Singularities Research Articles

5d SCFTs from isolated complete intersection singularities

In this paper, we explore the zoo of 5d superconformal field theories (SCFTs) constructed from M-theory on Isolated Complete Intersection Singularities (ICIS). We systematically investigate the crepant resolution of such singularities, and obtain a classification of rank ⩽ 10 models with a smooth crepant resolution and smooth exceptional divisors, as well as a number of infinite sequences with the same smoothness properties. For these models, we study their Coulomb branch properties and compute the flavor symmetry algebra from the resolved CY3 and/or the magnetic quiver. We check the validity of the conjectures relating the properties of the 5d SCFT and the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT from IIB superstring on the same singularity. When the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT has a Lagrangian quiver gauge theory description, one can obtain the magnetic quiver of the 5d theory by gauging flavor symmetry, which encodes the 5d Higgs branch information. Regarding the smoothness of the crepant resolution and integrality of 4d Coulomb branch spectrum, we find examples with a smooth resolved CY3 and smooth exceptional divisors, but fractional 4d Coulomb branch spectrum. Moreover, we compute the discrete (higher)-symmetries of the 5d/4d SCFTs from the link topology for a few examples.

The Bruce–Roberts Numbers of a Function on an ICIS

Abstract We relate a number of invariants of a function germ with isolated singularity over an isolated complete intersection singularity (ICIS), generalizing our previous results, [17, 18], as a consequence we present a new relatively elementary proof of Greuel’s theorem that for a weighted homogeneous ICIS the Milnor number is equal to Tjurina number. With this relation, we are able to prove that the relative logarithmic characteristic variety is Cohen–Macaulay at any point and the logarithmic characteristic variety is Cohen–Macaulay at any point not in $X\times \{0\}$.

On derivation Lie algebras of isolated complete intersection singularities

On derivation Lie algebras of isolated complete intersection singularities

On the quotient of Milnor and Tjurina numbers for two‐dimensional isolated hypersurface singularities

AbstractIn this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory.

An inequality conjecture and a weak Torelli-type theorem for isolated complete intersection singularities

In this paper, we propose an inequality conjecture for the dimensions of derivation Lie algebras associated to isolated complete intersection singularities. We verify this conjecture for simple and unimodal isolated complete intersection singularities. We also construct several new one-parameter families of solvable Lie algebras from T10, R9, U11, V10, Y11, and M11 singularities and show that the weak Torelli-type theorem holds.

Graded matrix factorizations of size two and reduction

We associate a complete intersection singularity to a graded matrix factorization of size two of a polynomial in three variables. We show that we get an inverse to the reduction of singularities considered by C. T. C. Wall. We study this for the full strongly exceptional collections in the triangulated category of graded matrix factorizations constructed by H. Kajiura, K. Saito, and the second author.

Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

Abstract The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

Families of ICIS with constant total Milnor number

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well-known result of Gabriélov, Lazzeri and Lê for hypersurfaces. We use A’Campo’s theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.

Uniform stable radius and Milnor number for non-degenerate isolated complete intersection singularities

We prove that for two germs of analytic mappings $$f,g:({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}}^p,0)$$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family $$\{f_t\}$$ of analytic maps with $$f_0=f, f_1=g$$ which has a so-called uniform stable radius for the Milnor fibration. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is a generalization of the result by C. Bivia-Ausina. Consequently, we obtain that the Milnor number of a non-degenerate isolated complete intersection singularity is an invariant of Newton boundaries.

Strange Duality Between the Quadrangle Complete Intersection Singularities

There is a strange duality between the quadrangle isolated complete intersection singularities discovered by the first author and Wall. We derive this duality from a variation of the Berglund–Hübsch transposition of invertible polynomials introduced in our previous work about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two.

Weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersections

For a given topological type of a normal surface singularity, there are various types of complex structures which realize it. We are interested in the following problem: Find the maximum of the geometric genus and a condition for that the maximal ideal cycle coincides with the undamental cycle on the minimal good resolution. In this paper, we study weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersection singularities from the perspective of the problem.

Durfee-type inequality for complete intersection surface singularities

We prove that the signature of the Milnor fiber of smoothings of a $2$-dimensional isolated complete intersection singularity does not exceed the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution, which implies Durfee's weak conjecture and a partial answer to Kerner--N\'emethi's conjecture.

Bounds of the multiplicity of abelian quotient complete intersection singularities

In this paper we investigate the multiplicity and the log canonical threshold of abelian quotient complete intersection singularities in terms of the notion of special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.

The Bruce–Roberts Number of A Function on A Hypersurface with Isolated Singularity

AbstractLet $(X,0)$ be an isolated hypersurface singularity defined by $\phi \colon ({\mathbb{C}}^n,0)\to ({\mathbb{C}},0)$ and $f\colon ({\mathbb{C}}^n,0)\to{\mathbb{C}}$ such that the Bruce–Roberts number $\mu _{BR}(f,X)$ is finite. We first prove that $\mu _{BR}(f,X)=\mu (f)+\mu (\phi ,f)+\mu (X,0)-\tau (X,0)$, where $\mu $ and $\tau $ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen–Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.

The nonexistence of negative weight derivations on positive dimensional isolated singularities: Generalized Wahl conjecture

Let $R = \mathbb{C} [ x_1, x_2, \dotsc , x_n ] / (f)$ where $f$ is a weighted homogeneous polynomial defining an isolated singularity at the origin. Then $R$ and $\operatorname{Der} (R,R)$ are graded. It is well-known that $\operatorname{Der} (R,R)$ does not have a negatively graded component. Wahl conjectured that this is still true for $R = \mathbb{C} [ x_1, x_2, \dotsc, x_n] / (f_1, f_2, \dotsc, f_m)$ which defines an isolated, normal and complete intersection singularity and $f_1, f_2, \dotsc, f_m$ weighted homogeneous polynomials with the same weight type $(w_1, w_2, \dotsc, w_n)$. Here we give a positive answer to the Wahl Conjecture and its generalization (without the condition of complete intersection singularity) for $R$ when the degree of $f_i, 1 \leq i \leq m$ are bounded below by a constant $C$ depending only on the weights $w_1, w_2, \dotsc, w_n$. Moreover this bound C is improved when any two of $w_1, w_2, \dotsc, w_n$ are coprime. Since there are counter-examples for the Wahl Conjecture and its generalization when $f_i$ are low degree, our theorem is more or less optimal in the sense that only the lower bound constant can be improved.

Fourier–Mukai and autoduality for compactified Jacobians, II

To every reduced (projective) curve X with planar singularities one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly non-isomorphic) Calabi-Yau projective varieties with locally complete intersection singularities. We define a Poincare' sheaf on the product of any two (possibly equal) fine compactified Jacobians of X and show that the integral transform with kernel the Poincare' sheaf is an equivalence of their derived categories. In particular, any two fine compactified Jacobians are derived equivalent. When applied to the same fine compactified Jacobian, one gets a Fourier-Mukai autoequivalence, which generalizes the classical result of Mukai for Jacobians of smooth curves (or more generally abelian varieties) and of Arinkin for compactified Jacobians of integral curves, thus providing further evidence for the classical limit of the geometric Langlands conjecture (as formulated by R. Donagi and T. Pantev). As a corollary of our main result, we prove an autoduality result for fine compactified Jacobians: there is a natural equivariant open embedding of the connected component of the scheme parametrizing rank-1 torsion-free sheaves on X into the connected component of the algebraic space parametrizing rank-1 torsion-free sheaves on a given fine compactified Jacobian of X.

Birationally rigid complete intersections of high codimension

We prove that a Fano complete intersection of codimension and index in the complex projective space for and with at most multi-quadratic singularities is birationally superrigid. The codimension of the complement of the set of birationally superrigid complete intersections in the natural moduli space is shown to be at least . The proof is based on the technique of hypertangent divisors combined with the recently discovered -inequality for complete intersection singularities.

Finite determinacy of matrices and ideals

The main aim of this paper is to characterize ideals I in the power series ring R=K[[x1,…,xs]] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection singularity, provided dim(R/I)>0 and K is an infinite field (of arbitrary characteristic). Here two ideals I and J are contact equivalent if the local K–algebras R/I and R/J are isomorphic. If I is minimally generated by a1,…,am, we call I finitely contact determined if it is contact equivalent to any ideal J that can be generated by b1,…,bm with ai−bi∈〈x1,…,xs〉k for some integer k. We give also computable and semicontinuous determinacy bounds.The above result is proved by considering left–right equivalence on the ring Mm,n of m×n matrices A with entries in R and we show that the Fitting ideals of a finitely determined matrix in Mm,n have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in R. Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture.

Effective Birational Rigidity of Fano Double Hypersurfaces

We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the set of non-rigid varieties in the natural parameter space of the family. The lower bound is quadratic in the dimension of the variety. The proof is based on the techniques of hypertangent divisors combined with the recently discovered 4n^2-inequality for complete intersection singularities.

Birationally Rigid Complete Intersections with a Singular Point of High Multiplicity

AbstractWe prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of maximal singularities, the generalized 4n2-inequality for complete intersection singularities and the technique of hypertangent divisors.