On algebraic semigroups and monoids, II

We study 2-cocycle twists, or equivalently Zhang twists, of semigroup algebras over a field ${\mathbb K}$ . If the underlying semigroup is affine, that is abelian, cancellative and finitely generated, then $\mathsf {Spec}~{\mathbb K}[S]$ is an affine toric variety over ${\mathbb K}$ , and we refer to the twists of ${\mathbb K}[S]$ as quantum affine toric varieties. We show that every quantum affine toric variety has a “dense quantum torus”, in the sense that it has a localization isomorphic to a quantum torus. We study quantum affine toric varieties and show that many geometric regularity properties of the original toric variety survive the deformation process.

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\bar X \to X$$ of X as a quotient of a vector space $$\bar X$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

Let [Formula: see text] be a [Formula: see text]-equivariant algebraic principal [Formula: see text]-bundle over a normal complex affine variety [Formula: see text] equipped with an action of [Formula: see text], where [Formula: see text] and [Formula: see text] are complex linear algebraic groups. Suppose [Formula: see text] is contractible as a topological [Formula: see text]-space with a dense orbit, and [Formula: see text] is a [Formula: see text]-fixed point. We show that if [Formula: see text] is reductive, then [Formula: see text] admits a [Formula: see text]-equivariant isomorphism with the product principal [Formula: see text]-bundle [Formula: see text], where [Formula: see text] is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal [Formula: see text]-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal [Formula: see text]-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].

Let X be an affine toric variety and let D(X) be the set of weights of all root subgroups. It is known that D(X) together with its embedding into the character group determines X as a toric variety. In this article we prove that X is already determined by the abstract set D(X) together with some additional combinatorial data.

We describe maximal commutative unipotent subgroups of the automorphism group \mathrm{Aut}(X) of an irreducible affine variety X . Furthermore, we show that a group isomorphism \mathrm{Aut}(X) \allowbreak\to \mathrm{Aut}(Y) maps unipotent elements to unipotent elements, where Y is irreducible and affine. Using this result, we show that the automorphism group detects sphericity and the weight monoid.As an application, we show that an affine toric variety different from an algebraic torus is determined by its automorphism group among normal irreducible affine varieties, and we show that a smooth affine spherical variety different from an algebraic torus is determined by its automorphism group (up to an automorphism of the base field) among smooth irreducible affine varieties. This generalizes results obtained by Cantat, Kraft, Liendo, Urech, Xie and the authors.

Any map of schemes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X\to Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defines an equivalence relation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R equals upper X times Subscript upper Y Baseline upper X right-arrow upper X times upper X"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:mi>X</mml:mi> <mml:msub> <mml:mo> × </mml:mo> <mml:mi>Y</mml:mi> </mml:msub> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> <mml:mo> × </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">R=X\times _Y X\to X\times X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the relation of “being in the same fiber”. We have shown elsewhere that not every equivalence relation has this form, even if it is assumed to be finite. By contrast, we prove here that every toric equivalence relation on an affine toric variety does come from a morphism and that quotients by finite toric equivalence relations always exist in the affine case. In special cases, this result is a consequence of the vanishing of the first cohomology group in the Amitsur complex associated to a toric map of toric algebras. We prove more generally the exactness of the Amitsur complex for maps of commutative monoid rings.

Let $X$ be a scheme of finite type over a finite field $k$, and let ${\cal L} X$ denote its arc space; in particular, ${\cal L} X(k)=X(k[[t]])$. Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularities of ${\cal L} X$ in the neighborhood of non-degenerate arcs, we show that a canonical ``basic function'' can be defined on the non-degenerate locus of ${\cal L} X(k)$, which corresponds to the trace of Frobenius on the stalks of the intersection complex of any finite-dimensional model. We then proceed to compute this function when $X$ is an affine toric variety or an ``$L$-monoid''. Our computation confirms the expectation that the basic function is a generating function for a local unramified $L$-function; in particular, in the case of an $L$-monoid we prove a conjecture formulated by the second author.

The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vector space. All components of the scheme are toric varieties, and among them, there is a fairly well understood coherent component. It is unknown whether toric Hilbert schemes are always connected. In this chapter we illustrate the use of Macaulay 2 for exploring the structure of toric Hilbert schemes. In the process we will encounter algorithms from commutative algebra, algebraic geometry, polyhedral theory and geometric combinatorics.KeywordsToric VarietyHilbert SeriesLocal EquationMonomial IdealSemigroup AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Different commutative semigroups may have a common saturation. We consider distinguishing semigroups with a common saturation based on their ``sparsity''. We propose to qualitatively describe sparsity of a semigroup by considering which faces of the corresponding rational polyhedral cone have saturation points. For a commutative semigroup we give a necessary and sufficient condition for determining which faces have saturation points. We also show that we can construct a commutative semigroup with arbitrary consistent patterns of faces with saturation points.

A partial semimetric on a set X is a function $(x, y) \mapsto p(x, y) \in \RR_{\geq 0}$ satisfying $p(x, y) = p(y, x)$, $p(x, y) \geq p(x, x)$ and $p(x, z) \leq p(x, y) + p(y, z) − p(y, y)$ for all $x, y, z \in X$. We study here the polyhedral convex cone $PSMET_n$ of all partial semimetrics on $n$ points, using computations done for $n \leq 6$. We present data on those cones and their relatives: the number of facets, of extreme rays, of their orbits, incidences, characterize ${0, 1}$- valued extreme rays, etc.

In this chapter we describe a resolution process for the singularity of the normalization of the surface in A3 defined by the equation Z P = XY q over an algebraically closed field k of characteristic zero; here 0 < p < q are integers and gcd(p, q) = 1. These singularities arise in a natural way: In section 1 we show in (1.6) that if L is a finite extension of Q = k((U, V)), S is the integral closure of R = k 〚U, V〛 in L, and the only prime ideals of R which are ramified in S are at most RU or RV, then L = k((X,Y))[Z]/(Z P − XY q ), and that S is the integral closure of S0 = k 〚X,Y 〛[Z]/(Z p − XY q ). Also, they arise as the only singular point of the normalization of a toric variety associated with a not nonsingular strongly convex rational polyhedral cone of dimension 2 [cf. section 4].

On a finite-dimensional real vector space, we give a microlocal characterization of (derived) piecewise linear sheaves (PL sheaves) and prove that the triangulated category of such sheaves is generated by sheaves associated with convex polyhedra. We then give a similar theorem for PL $\gamma $-sheaves, that is, PL sheaves associated with the $\gamma $-topology, for a closed convex polyhedral proper cone $\gamma $. Our motivation is that convex polyhedra may be considered as building blocks for higher dimensional barcodes.

Syzygies of affine toric varieties

The assembly plan from observation (APO) system observes a human operator perform an assembly task, analyzes the observations, models the task and generates the programs for the robot to perform the same task. A major component of the APO system is the task recognition module, which models the observed task. The task model in the APO context is defined as a sequence of assembly states of the part being assembled and the actions which cause the transition between the states. The state of the assembled part is based on its freedom, which can be computed from the geometry of the contacts between the part and its environment. This freedom can be represented as a polyhedral convex cone (PCC) in screw space. We show that any contact configuration can be classed into a finite number of contact states. These contact states correspond to typologically distinct intersections of the PCC with a linear subspace T in screw space. The models of any observed task can be represented as a path in a transition graph obtained from these contact states. We illustrate the theory by implementing the APO system for polygonal objects assembled in a plane using rotation and translation.

In this paper, we consider a special class of conic optimization problems, consisting of set-semidefinite(or K-semidefinite) programming problems, where the set K is a polyhedral convex cone.&#x0D; For these problems, we introduce the concept of immobile indices and study the properties of the set of normalized immobile indices and the feasible set. This study provides the main result of the paper, which is to formulate and prove the new first-order optimality conditions in the form of a criterion. The optimality conditions are explicit and do not use any constraint qualifications. For the case of a linear cost function, we reformulate the K-semidefinite problem in a regularized form and construct its dual. We show that the pair of the primal and dual regularized problems satisfies the strong duality relation which means that the duality gap is vanishing.