Algebraic Varieties Research Articles

Maple-based introductory visual guide to Gröbner bases

In 1975 the Consejo Superior de Investigaciones Científicas (the main Spanish institution for scientific research) published the monograph [14] by the second author (by the way, father and Ph.D. advisor of the first author). Its title could be translated as "Geometric Interpretation of Ideal Theory" (nowadays Ideal Theory is not normally used, in favour of Commutative Algebra). It somehow illustrated the geometric ideas underlying the basics of the classic books of the period (like [2, 11, 16]) and was a success: although written in Spanish, the edition was sold out.Of course there are much more modern books on ideals and varieties than [2, 11, 16], such as the famous [7] or [8], that illustrate the theory with images. Moreover, there are introductory works to Gröbner bases such as [3, 9, 12, 13, 15], as well as books on the topic like [1], and articles about applications, like the early [4]. Even a summary in English of the original Ph.D. Thesis by Bruno Buchberger is available [5].
 Nevertheless, we believe that there is a place for a visual guide to Gröbner bases, as there was a place for [14].
 For instance, statistical packages are probably the pieces of mathematical software best known by non-mathematicians, and they are frequently used as black boxes by users with a slight knowledge of the theory behind. Meanwhile, Gröbner bases, the most common exact method behind non-linear polynomial systems (algebraic systems) solving, although incorporated to all computer algebra systems, are only known by a relatively small ratio of the members of the scientific community, most of them mathematicians. This article presents in an intuitive and visual way an illustrative selection of ideals and their Gröbner bases, together with the plots of the (real part) of their corresponding algebraic varieties, computed and plotted with Maple [6, 10]. A minimum amount of theoretical details is given. We believe that exact algebraic systems solving could also be used as a black box by non-mathematicians just understanding the basic ideas underlying commutative algebra and computer algebra.

Idempotent Varieties of Incidence Monoids and Bipartite Posets

The algebraic variety defined by the idempotents of an incidence monoid is investigated. Its irreducible components are determined. The intersection with an antichain submonoid is shown to be the union of these irreducible components. The antichain monoids of bipartite posets are shown to be orthodox semigroups. The Green’s relations are explicitly determined, and applications to conjugacy problems are described. In particular, it is shown that two elements in the antichain monoid are primarily conjugate in the monoid if and only if they belong to the same -class and their multiplication by an idempotent of the same -class gives conjugate elements in the group.

Moduli of Stokes torsors and singularities of differential equations

Let \mathcal{M} be a meromorphic connection with poles along a smooth divisor D in a smooth algebraic variety. Let \operatorname{Sol} \mathcal{M} be the solution complex of \mathcal{M} . We prove that the good formal structure locus of \mathcal{M} coincides with the locus where the restrictions to D of \operatorname{Sol} \mathcal{M} and \operatorname{Sol} \operatorname{End}\mathcal{M} are local systems. Despite the very different natures of these loci (the first one is defined via algebra, and the second via analysis), the proof of their coincidence is geometric. It relies on moduli of Stokes torsors.

Finite Generation of Cox Rings

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.

Homogeneous Algebraic Varieties and Transitivity Degree

Homogeneous Algebraic Varieties and Transitivity Degree

Symplectic reduction along a submanifold

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$-space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$, where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

The moment map for the variety of 3-Lie algebras

For the moment map m:PVn→iu(n) for the action of GL(n) on Vn=Λ3(Cn)⁎⊗Cn, we will study the critical points of Fn=‖m‖2:PVn→R in this paper. We first prove that [μ]∈PVn is a critical point if and only if m([μ])=cμI+Dμ for some cμ∈R and Dμ∈Der(μ), and that there exists a constant c>0 such that the eigenvalues of cDμ are integers prime to each other and some eigenvalues can be strictly negative. Then we give a description of the maxima and minima of Fn:Ln→R, where Ln is the projectivization of all n-dimensional 3-Lie algebras. Furthermore, the structure of the critical points of Fn is discussed. Finally, as an application, we show that every three-dimensional 3-Lie algebra is isomorphic to a critical point of F3; and there exists a curve of nonisomorphic four-dimensional 3-Lie algebras which are not isomorphic to any critical point of F4.

The numerical algebraic geometry of bottlenecks

This is a computational study of bottlenecks on algebraic varieties. The bottlenecks of a smooth variety X⊆Cn are the lines in Cn which are normal to X at two distinct points. The main result is a numerical homotopy that can be used to approximate all isolated bottlenecks. This homotopy has the optimal number of paths under certain genericity assumptions. In the process we prove bounds on the number of bottlenecks in terms of the Euclidean distance degree. Applications include the optimization problem of computing the distance between two real varieties. Also, computing bottlenecks may be seen as part of the problem of computing the reach of a smooth real variety and efficient methods to compute the reach are still to be developed. Relations to triangulation of real varieties and meshing algorithms used in computer graphics are discussed in the paper. The resulting algorithms have been implemented with Bertini [4] and Macaulay2 [17].

The Geometry of Polynomial Representations

Abstract We define a GL-variety to be an (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used to study asymptotic properties of invariants like strength and tensor rank and played a key role in two recent proofs of Stillman’s conjecture. We initiate a systematic study of $\textbf {GL}$-varieties and establish a number of foundational results about them. For example, we prove a version of Chevalley’s theorem on constructible sets in this setting.

On images of subshifts under embeddings of symbolic varieties

AbstractWe show that the image of a subshift X under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, respectively a sofic subshift, if and only if so is X. Similarly, let G be a countable monoid and let A, B be Artinian modules over a ring. We prove that for every closed subshift submodule $\Sigma \subset A^G$ and every injective G-equivariant uniformly continuous module homomorphism $\tau \colon \! \Sigma \to B^G$ , a subshift $\Delta \subset \Sigma $ is of finite type, respectively sofic, if and only if so is the image $\tau (\Delta )$ . Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.

Structural Completeness in Many-Valued Logics with Rational Constants

The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal.

On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0

AbstractLet K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.

Notions of representability for cylindric algebras: some algebras are more representable than others

The theory of cylindric algebras was introduced by Tarski in the fifties of the twentieth century, and its intensive study was further pursued by pioneers such as Henkin and Monk and, by the Hungarian mathematicians Andréka, Németi and Sain, and many of their students; to name only a few: Madarász, Marx, Kurucz, Simon, Mikulás, and Sági and many others outside Hungary including the author of this paper. Here we introduce and investigate new notions of representability for cylindric algebras and investigate various connections between such notions. Let 2<nle l<mle omega . Let textsf {CA}_n denote the variety of cylindric algebras of dimension n and let textsf {RCA}_n denote the variety of representable textsf {CA}_ns. We say that an atomic algebra {{mathfrak {A}}}in textsf {CA}_n has the complex neat embedding property up to l and m if {{mathfrak {A}}}in textsf {RCA}_ncap textsf {Nr}_ntextsf {CA}_l and {{{mathfrak {C}}}{{mathfrak {m}}}}textsf {At}{{mathfrak {A}}}in mathbf {S}textsf {Nr}_ntextsf {CA}_m. Fixing the prarameters l at the value n, this is a measure of how much the algebra is representable. The yardstick is how far can its Dedekind–MacNeille completion be dilated, that is to say, counting the number of more extra dimensions its Dedekind–MacNeille completion neatly embeds into. If {{mathfrak {A}}}, {{mathfrak {B}}}in textsf {RCA}_n are atomic, {{{mathfrak {C}}}{{mathfrak {m}}}}textsf {At}{{mathfrak {B}}}in Stextsf {Nr}_ntextsf {CA}_l and {{{mathfrak {C}}}{{mathfrak {m}}}}textsf {At}{{mathfrak {A}}}in Stextsf {Nr}_ntextsf {CA}_m, then we say that {{mathfrak {A}}} is more representable than {{mathfrak {B}}}. When m=omega , we say that {{mathfrak {A}}} is strongly representable; this is the maximum degree of representability; the algebra in question cannot be ‘more representable’ than that. In this case the atom structure of {{mathfrak {A}}}, namely textsf {At}{{mathfrak {A}}}, is strongly representable in the sense of Hirsch and Hodkinson. This notion gives an infinite potential spectrum of ‘degrees’ of representability. In this connection, we exhibit various atomic algebras in textsf {RCA}_ncap textsf {Nr}_ntextsf {CA}_l that do no not have the complex neat embedding property for infinitely many values of l and m. It is known that the class of Kripke frames textsf {Str}(textsf {RCA}_n)={{{mathfrak {F}}}: {{{mathfrak {C}}}{{mathfrak {m}}}}{{mathfrak {F}}}in textsf {RCA}_n} is not elementary. From this it follows that there is some n<m<omega such that textsf {Str}(mathbf {S}textsf {Nr}_ntextsf {CA}_m)={{{mathfrak {F}}}: {{{mathfrak {C}}}{{mathfrak {m}}}}{{mathfrak {F}}}in mathbf {S}textsf {Nr}_ntextsf {CA}_m} is not elementary. Replacing mathbf {S} by mathbf {S}_c (forming complete subalgebras), mathbf {S}_d (forming dense subalgebras) and mathbf {I} (forming isomorphic copies), respectively, we show that for any mathbf {O}in {mathbf {S}_c, mathbf {S}_d, mathbf {I}}, the class of frames textsf {Str}(mathbf {O}textsf {Nr}_ntextsf {CA}_{n+3})={{{mathfrak {F}}}: {{{mathfrak {C}}}{{mathfrak {m}}}}{{mathfrak {F}}}in mathbf {O}textsf {Nr}_ntextsf {CA}_{n+3}} is not elementary. Metalogical applications are given to n-variable fragments of first-order logic endowed with so-called clique guarded semantics. The last semantics capture the new notions of representations introduced and studied in this paper.

Zeta statistics and Hadamard functions

We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any “natural” sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.

Fields over Fields

We investigate certain arithmetic properties of field theories. In particular, we study the vacuum structure of supersymmetric gauge theories as algebraic varieties over number fields of finite characteristic. Parallel to the Plethystic Programme of counting the spectrum of operators from the syzygies of the complex geometry, we construct, based on the zeros of the vacuum moduli space over finite fields, the local and global Hasse-Weil zeta functions, as well as develop the associated Dirichlet expansions. We find curious dualities wherein the geometrical properties and asymptotic behaviour of one gauge theory is governed by the number theoretic nature of another.

Rigid Analytic p-Adic Simpson Correspondence for Line Bundles

The p-adic Simpson correspondence due to Faltings (Adv Math 198(2):847–862, 2005) is a p-adic analogue of non-abelian Hodge theory. The following is the main result of this article: The correspondence for line bundles can be enhanced to a rigid analytic morphism of moduli spaces under certain smallness conditions. In the complex setting, Simpson shows that there is a complex analytic morphism from the moduli space for the vector bundles with integrable connection to the moduli space of representations of a finitely generated group as algebraic varieties. We give a p-adic analogue of Simpson’s result.

On the nilpotent orbits arising from admissible affine vertex algebras

We give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals.

On uniqueness of additive actions on complete toric varieties

By an additive action on an algebraic variety X we mean a regular effective action Gan×X→X with an open orbit of the commutative unipotent group Gan. In this paper, we give a uniqueness criterion for additive action on a complete toric variety.

A variety of algebras closely related to subordination algebras

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras.

Regulator maps for higher Chow groups via current transforms

We show how to use equidimensional algebraic correspondences between complex algebraic varieties to construct pull-backs and transforms of certain classes of geometric currents. Using this construction we produce explicit formulas at the level of complexes for a regulator map from the higher Chow groups of smooth complex quasi-projective algebraic varieties to Deligne–Beilinson cohomology with integral coefficients. A distinct aspect of our approach is the use of Suslin's complex n ↦ Z Δ , eq p ( X , n ) $n \mapsto \mathcal {Z}^p_{\Delta , \text{eq}}(X,n)$ of equidimensional cycles over Δ n $ \Delta ^n$ to compute Bloch's higher Chow groups. We calculate explicit examples involving the Mähler measure of Laurent polynomials.