Algebraic Varieties Research Articles

Cohomological varieties associated to vertex operator algebras

Given a vertex operator algebra V, one can attach a graded Poisson algebra called the C2-algebra R(V). The associate Poisson scheme provides an important invariant for V and has been studied by Arakawa as the associated variety. In this article, we define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties and then construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational C2-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. In this paper, we study R(V) as a commutative algebra only and do not use the property of its Poisson structure, which is expected to provide more refined invariants. The goal of this work is to study the cohomological supports of modules for vertex algebras as the cohomological support varieties for finite groups and restricted Lie algebras.

On conjugacy of additive actions in the affine Cremona group

An additive action on an irreducible algebraic variety X is an effective action with an open orbit of the vector group . Any two additive actions on X are conjugate by a birational automorphism of X. We prove that, if X is the projective space, the conjugating element can be chosen in the affine Cremona group and it is given by so-called basic polynomials of the corresponding local algebra.

𝑉-filtrations and minimal exponents for local complete intersections

Abstract We define and study a notion of minimal exponent for a local complete intersection subscheme 𝑍 of a smooth complex algebraic variety 𝑋, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara–Malgrange 𝑉-filtration associated to 𝑍. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology H Z r ⁢ ( O X ) \mathcal{H}^{r}_{Z}(\mathcal{O}_{X}) , where 𝑟 is the codimension of 𝑍 in 𝑋. We also study its relation to the Bernstein–Sato polynomial of 𝑍. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension 1 case. A key ingredient for our main result is a description of the Kashiwara–Malgrange 𝑉-filtration associated to any ideal ( f 1 , … , f r ) (f_{1},\ldots,f_{r}) in terms of the microlocal 𝑉-filtration associated to the hypersurface defined by ∑ i = 1 r f i ⁢ y i \sum_{i=1}^{r}f_{i}y_{i} .

Line multiview ideals

We study the following problem in computer vision from the perspective of algebraic geometry: Using m pinhole cameras we take m pictures of a line in P 3 . This produces m lines in P 2 and the question is which m-tuples of lines can arise that way. We are interested in polynomial equations and therefore study the complex Zariski closure of all such tuples of lines. The resulting algebraic variety is a subvariety of ( P 2 ) m and is called line multiview variety. In this article, we study its ideal. We show that for generic cameras the ideal is generated by 3 × 3 -minors of a specific matrix. We also compute Gröbner bases and discuss to what extent our results carry over to the non-generic case.

Schmidt's subspace theorem for arbitrary families of moving hypersurface targets

In this article, we mainly obtain a Schmidt's Subspace Theorem for arbitrary families of moving hypersurface targets in algebraic varieties, which is an extension of the Schmidt's Subspace Theorem due to Son-Tan-Thin [17] and Quang [10], etc.

Markov and Division Inequalities on Algebraic Sets

A compact set E⊂CN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E\\subset \\mathbb {C}^N$$\\end{document} satisfies the Markov inequality if the supremum norm on E of the gradient of a polynomial p can be estimated from above by the norm of p multiplied by a constant polynomially depending on the degree of p. This inequality is strictly related to the Bernstein approximation theorem, Schur-type estimates and the extension property of smooth functions. Additionally, the Markov inequality can be applied to the construction of polynomial grids (norming sets or admissible meshes) useful in numerical analysis. We expect such an inequality with similar consequences not only on polynomially determining compacts but also on some nowhere dense sets. The primary goal of the paper is to extend the above definition of Markov inequality to the case of compact subsets of algebraic varieties in CN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^N$$\\end{document}. Moreover, we characterize compact sets satisfying such a Markov inequality on algebraic hypersurfaces as well as on certain varieties defined by several algebraic equations. We also prove a division inequality (a Schur-type inequality) on these sets. This opens up the possibility of establishing polynomial grids on algebraic sets. We also provide examples that complete and ilustrate the results.

On singularity and normality of regular nilpotent Hessenberg varieties

Regular nilpotent Hessenberg varieties form an important family of subvarieties of the flag variety, which are often singular and sometimes not normal varieties. Like Schubert varieties, they contain distinguished points called permutation flags. In this paper, we give a combinatorial characterization for a permutation flag of a regular nilpotent Hessenberg variety to be a singular point. We also apply this result to characterize regular nilpotent Hessenberg varieties which are normal algebraic varieties.

Locally Homogeneous Holomorphic Geometric Structures on Projective Varieties

For any smooth projective variety with holomorphic locally homogeneous structure modelled on a homogeneous algebraic variety, we determine all the subvarieties of it which develop to the model.

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Given a smooth, projective curve \(Y\), a finite group \(G\) and a positive integer $n$ we study smooth, proper families \(X\to Y\times S\to S\) of Galois covers of \(Y\) with Galois group isomorphic to $G$ branched in \(n\) points, parameterized by algebraic varieties \(S\). When \(G\) is with trivial center we prove that the Hurwitz space \(H^G_n(Y)\) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary \(G\) we prove that \(H^G_n(Y)\) is a coarse moduli variety. For families of pointed Galois covers of \((Y,y_0)\) we prove that the Hurwitz space \(H^G_n(Y,y_0)\) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group \(G\). We use classical tools of algebraic topology and of complex algebraic geometry.

Mixed Hodge Structures on Alexander Modules

Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let U U be a smooth connected complex algebraic variety and let f : U → C ∗ f\colon U\to \mathbb {C}^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C ∗ \mathbb {C}^* by f f gives rise to an infinite cyclic cover U f U^f of U U . The action of the deck group Z \mathbb {Z} on U f U^f induces a Q [ t ± 1 ] \mathbb {Q}[t^{\pm 1}] -module structure on H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) . We show that the torsion parts A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) of the Alexander modules H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) carry canonical Q \mathbb {Q} -mixed Hodge structures. We also prove that the covering map U f → U U^f \to U induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when f : U → C ∗ f\colon U\to \mathbb {C}^* is proper, we prove the semisimplicity and purity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , and we compare our mixed Hodge structure on A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) with the limit mixed Hodge structure on the generic fiber of f f .

On state monadic MV-algebras

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. In this paper, we extend monadic MV-algebras with a state operator that describes algebraic properties of states. The resulting variety of algebras will be called state monadic MV-algebras. First, we introduce state monadic MV-algebras and establish a natural equivalence between the category of state monadic MV-algebras and the category of state monadic ℓ-groups with strong units. Moreover, we prove that the class of all state monadic ideals in state monadic MV-algebras is a complete Heyting algebra. In particular, by studying extended state monadic ideals, we prove that the set of all stable state monadic ideals in state monadic MV-algebras is a complete Heyting algebra. Also, the class of all involutory state monadic ideals in state monadic MV-algebras is a complete Boolean algebra. Finally, we introduce and characterize some members in the variety of state monadic MV-algebras, which are subdirectly irreducible, simple, semisimple, local and semilocal, respectively.

Automorphisms of categories of free algebras with unit for classical varieties of non-associative algebras

The goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry.

Discrepancy of rational points in simple algebraic groups

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

Birational Transformations on Irreducible Compact Hermitian Symmetric Spaces

Abstract We construct a sequence of explicit blow-ups and blow-downs on an irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover, this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for $X$ are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of $X$ and its higher secant varieties. The result was known in the special case where $X$ is of rank 2 and could be found in Zak’s monograph “Tangents and secants of algebraic varieties.”

Factor principal congruences and Boolean products in filtral varieties

Motivated by Haviar and Ploščica’s 2021 characterisation of Boolean products of simple De Morgan algebras, we investigate Boolean products of simple algebras in filtral varieties. We provide two main theorems. The first yields Werner’s Boolean-product representation of algebras in a discriminator variety as an immediate application. The second, which applies to algebras in which the top congruence is compact, yields a generalisation of the Haviar–Ploščica result to semisimple varieties of Ockham algebras. The property of having factor principal congruences is fundamental to both theorems. While major parts of our general theorems can be derived from results in the literature, we offer new, self-contained and essentially elementary proofs.

Sequence-regular commutative DG-rings

We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings (A,m¯) such that the maximal ideal m¯⊆H0(A) can be generated by an A-regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular.

Landau Singularities Revisited: Computational Algebraic Geometry for Feynman Integrals.

We reformulate the analysis of singularities of Feynman integrals in a way that can be practically applied to perturbative computations in the standard model in dimensional regularization. After highlighting issues in the textbook treatment of Landau singularities, we develop an algorithm for classifying and computing them using techniques from computational algebraic geometry. We introduce an algebraic variety called the principal Landau determinant, which captures the singularities even in the presence of massless particles or UV/IR divergences. We illustrate this for 114 example diagrams, including a cutting-edge 2-loop 5-point nonplanar QCD process with multiple mass scales.

On Pre-Novikov Algebras and Derived Zinbiel Variety

For a non-associative algebra $A$ with a derivation $d$, its derived algebra $A^{(d)}$ is the same space equipped with new operations $a\succ b = d(a)b$, $a\prec b = ad(b)$, $a,b\in A$. Given a variety ${\rm Var}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in ${\rm Var}$ and for all derivations $d$ of $A$. The same terminology is applied to binary operads governing varieties of non-associative algebras. For example, the operad of Novikov algebras is the derived one for the operad of (associative) commutative algebras. We state a sufficient condition for every algebra from a derived variety to be embeddable into an appropriate differential algebra of the corresponding variety. We also find that for ${\rm Var} = {\rm Zinb}$, the variety of Zinbiel algebras, there exist algebras from the derived variety (which coincides with the class of pre-Novikov algebras) that cannot be embedded into a Zinbiel algebra with a derivation.

On the zero set of the holomorphic sectional curvature

AbstractA notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi‐definite and vanishes along high‐dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real‐valued bihomogeneous polynomial of bidegree (2,2) on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi‐definite, be it negative or positive.

Voronoi cells in metric algebraic geometry of plane curves

Voronoi cells of varieties encode many features of their metric geometry. We prove that each Voronoi or Delaunay cell of a plane curve appears as the limit of a sequence of cells obtained from point samples of the curve. We use this result to study metric features of plane curves, including the medial axis, curvature, evolute, bottlenecks, and reach. In each case, we provide algebraic equations defining the object and, where possible, give formulas for the degrees of these algebraic varieties. We show how to identify the desired metric feature from Voronoi or Delaunay cells, and therefore how to approximate it by a finite point sample from the variety.