We present and expand some existing results on the Zariski closure of cyclic groups and semigroups of matrices. We show that, with the exclusion of isolated points, their irreducible components are toric varieties. Additionally, we demonstrate how every toric variety can be realized as the Zariski closure of a cyclic matrix group. Our paper includes a number of explicit examples and a note on existing computational results.
The main result of this paper is that every (separated) toric variety which has a semigroup structure compatible with multiplication on the underlying torus is necessarily affine. In the course of proving this statement, we also give a proof of the fact that every separated toric variety may be constructed from a certain fan in a Euclidean space. To our best knowledge, this proof differs essentially from the ones which can be found in the literature.
An ind-variety is an inductive limit of closed embeddings of algebraic varieties and an ind-group is a group object in the category of ind-varieties. These notions were first introduced by Shafarevich in the study of the automorphism group of affine spaces and have been studied by many authors afterwards. An ind-torus is an ind-group obtained as an inductive limit of closed embeddings of algebraic tori that are also algebraic group homomorphisms. In this paper, we introduce the natural definition of toric ind-varieties as ind-varieties having an ind-torus as an open set and such that the actio…
In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions. In particular, we describe to which components each such deformation maps, show how to induce each deformation from a versal family, give explicit equations for each deformation, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions.
Anderson proved that the finite generation of the value semigroup $Γ_{Y_\bullet}(D)$ in the construction of the Newton-Okounkov body $Δ_{Y_\bullet}(D)$ induces a toric degeneration of the corresponding variety $X$ to some toric variety $X_0$. In this case the normalization of $X_0$ is the normal toric variety corresponding to the rational polytope $Δ_{Y_\bullet}(D)$. Since $X_0$ is not normal in general this correspondence is rather implicit. In this article we investigate in conditions to assure that $X_0$ is normal, by comparing the Hilbert polynomial with the Ehrhart polynomial. In the case…
Let $R$ be a complete equicharacteristic noetherian local domain and $ν$ a valuation of its field of fractions whose valuation ring dominates $R$ with trivial residue field extension. The semigroup of values of $ν$ on $R\setminus \{0\}$ is not finitely generated in general. We produce equations in an appropriate generalized power series ring for the algebra encoding the degeneration of $R$ to the toric graded algebra ${\rm gr}_νR$ associated to the filtration defined by $ν$. We apply this to represent $ν$ as the limit of a sequence of Abhyankar semivaluations (valuations on quotients) of $R$ w…
P-resolutions of two-dimensional, cyclic quotient singularities have been introduced to study deformation theory. Those P-resolutions (as well as the singularities themselves) are toric varieties. In the present paper we give a straight, elementary description of them just by their defining fans.
We study the $k[G]$-module structure of the space of holomorphic differentials of a curve defined over an algebraically closed field of positive characteristic, for a cyclic group $G$ of order $p^\ell n$. We also study the relation to the Weierstrass semigroup for the case of Galois Weierstrass points.
The Okounkov body is a construction which, to an effective divisor D on an n-dimensional algebraic variety X, associates a convex body in the n-dimensional Euclidean space R^n. It may be seen as a generalization of the moment polytope of an ample divisor on a toric variety, and it encodes rich numerical information about the divisor D. When constructing the Okounkov body, an intermediate product is a lattice semigroup, which we will call the Okounkov semigroup. Recently it was discovered that finite generation of the Okounkov semigroup has interesting geometric implication for X regarding tori…
Hille and Perling associate to every cyclic full strongly exceptional sequence of line bundles on a toric weak Fano surface a toric system, which defines a new toric surface. In this note we interprete this construction as an instance of mirror symmetry and extend it to a duality on the set toric weak Fano surfaces equiped with a cyclic full strongly exceptional sequence.
A toric del Pezzo surface $X_P$ with cyclic quotient singularities determines and is determined by a Fano polygon $P$. We construct an affine manifold with singularities that partially smooths the boundary of $P$; this a tropical version of a Q-Gorenstein partial smoothing of $X_P$. We implement a mild generalization of the Gross-Siebert reconstruction algorithm - allowing singularities that are not locally rigid - and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half…
We find presentations for subalgebras of invariants of the coordinate algebras of binary symmetric models of phylogenetic trees studied by Buczynska and Wisniewski in \cite{BW}. These algebras arise as toric degenerations of rings of global sections of weight varieties of the Grassmanian of two planes associated to the Plücker embedding, and as toric degenerations of rings of invariants of Cox-Nagata rings.
We generalize the Bernstein-Sato polynomials of Budur, Mustata and Saito to ideals in normal semigroup rings. In the case of monomial ideals, we also relate the roots of the Bernstein-Sato polynomial to the jumping coefficients of the corresponding multiplier ideals. In order to prove the latter result, we obtain a new combinatorial description for the multiplier ideals of a monomial ideal in a normal semigroup ring.
A non-degenerate toric variety $X$ is called $S$-homogeneous if the subgroup of the automorphism group $\text{Aut}(X)$ generated by root subgroups acts on $X$ transitively. We prove that maximal $S$-homogeneous toric varieties are in bijection with pairs $(P,\mathcal{A})$, where $P$ is an abelian group and $\mathcal{A}$ is a finite collection of elements in $P$ such that $\mathcal{A}$ generates the group $P$ and for every $a\in\mathcal{A}$ the element $a$ is contained in the semigroup generated by $\mathcal{A}\setminus\{a\}$. We show that any non-degenerate homogeneous toric variety is a big o…
We prove that every toric quiver flag variety $Y$ is isomorphic to a fine moduli space of cyclic modules over the algebra $\text{End}(T)$ for some tilting bundle $T$ on $Y$. This generalises the well known fact that $\mathbb{P}^n$ can be recovered from the endomorphism algebra of $\bigoplus_{0\leq i\leq n} \mathcal{O}_{\mathbb{P}^n}(i)$.
We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k-forms of projective spaces when K/k is cyclic, and we also study k-forms of surfaces.
The arithmetic motivic Poincaré series of a variety $V$ defined over a field of characteristic zero, is an invariant of singularities which was introduced by Denef and Loeser by analogy with the Serre-Oesterlé series in arithmetic geometry. They proved that this motivic series has a rational form which specializes to the Serre-Oesterlé series when $V$ is defined over the integers. This invariant, which is known explicitly for a few classes of singularities, remains quite mysterious. In this paper we study this motivic series when $V$ is an affine toric variety. We obtain a formula for the rati…
We construct endomotives associated to toric varieties, in terms of the decomposition of a toric variety into torus orbits and the action of a semigroup of toric morphisms. We show that the endomotives can be endowed with time evolutions and we discuss the resulting quantum statistical mechanical systems. We show that, in particular, one can construct a time evolution related to the logarithmic height function. We discuss relations to F1-geometry.
We explicitly describe infintesimal deformations of cyclic quotient singularities that satisfy one of the deformation conditions introduced by Wahl, Kollár-Shepherd-Barron and Viehweg. The conclusion is that in many cases these three notions are different from each other. In particular, we see that while the KSB and the Viehweg versions of the moduli space of surfaces of general type have the same underlying reduced subscheme, their infinitesimal structures are different.
A subgroup H of a reductive group G is horospherical if it contains a maximal unipotent subgroup. We describe the Grothendieck semigroup of invariant subspaces of regular functions on G/H as a semigroup of convex polytopes. From this we obtain a formula for the number of solutions of a generic system of equations on G/H in terms of mixed volume of polytopes. This generalizes Bernstein-Kushnirenko theorem from toric geometry.
We prove that the wreath product $C=A wr B$ of a semigroup $A$ with zero and an infinite cyclic semigroup $B$ is $q_ω$-compact (logically Noetherian). Our result partially solves the Plotkin`s problem about wreath products
We prove that special ample line bundles on toric varieties arising from root systems are projectively normal. Here the maximal cones of the fans correspond to the Weyl chambers, and special means that the bundle is torus-equivariant such that the character of the line bundle that corresponds to a maximal Weyl chamber is dominant with respect to that chamber. Moreover, we prove that the associated semigroup rings are quadratic.
Normal affine algebraic varieties in characteristic 0 are uniquely determined (up to isomorphism) by the Lie algebra of derivations of their coordinate ring. This is not true without the hypothesis of normality. But, we show that (in general, non-normal) toric varieties defined by simplicial affine semigroups are uniquely determined by their Lie algebra if they are supposed to be Cohen-Macaulay of dimension at least 2 or Gorenstein of dimension 1. Moreover, every automorphism of the Lie algebra is induced from a unique automorphism of the variety and every derivation of the Lie algebra is inne…
We study Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. For zero dimensional valuations we prove that whenever the ring is complete and the semigroup of values taken by the valuation is finitely generated (which implies that the valuation is Abhyankar) the valuation can be uniformized in an embedded way by a birational map which is monomial with respect to a suitable system of generators of the maximal ideal. We prove that conversely if a valuation is Abhyankar after a birational modification and localization at the point picked b…
The relation of the Weierstrass semigroup with several invariants of a curve is studied. For Galois covers of curves with group $G$ we introduce a new filtration of the group decomposition subgroup of $G$. The relation to the ramification filtration is investigated in the case of cyclic covers. We relate our results to invariants defined by Boseck and we study the one point ramification case. We also give applications to Hasse-Witt invariant and symmetric semigroups.
On a Computational Approach to the Nash Blowup Problem
Centro de Investigación en Matemáticas（数学研究中心）；Centro de Ciencias Matematicas, UNAM（墨西哥国立自治大学数学科学中心）；Instituto de Matemáticas, Universidad de Talca（塔尔卡大学数学研究所）
We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop for monoid schemes many notions from classical algebraic geometry, such as separated and proper maps.
Let $K/k$ be a finite Galois extension, $G=\text{Gal}(K/k)$, $Σ$ be a fan in a lattice $N$ and $X_Σ$ be an associated toric variety over $k$. It is well known that the set of $K/k$-forms of $X_Σ$ is in bijection with $H^1(G,\text{Aut}_Σ^T)$, where $\text{Aut}_Σ^T$ is an algebraic group of toric automorphisms of $X_Σ$. In this paper, we suggest an algorithm to compute $H^1(G,\text{Aut}_Σ^T)$ and find that followings can be classified via this algorithm : $K/k$-forms of all toric surfaces, $K/k$-forms of all 3-dimensional affine toric varieties with no torus factor, $K/k$-forms of all 3-dimensio…
The mirror of a toric orbifold is an affine curve called the mirror curve. In this paper, firstly, we recall the basic tools in tropical geometry and give a definition of the mirror curve. Then we calculate the tropical spine of the mirror curve for a smooth toric Calabi-Yau 3-fold near the large radius limit. Finally, we recall a special kind of real algebraic curves called the cyclic-M curve and show that under some special choices of parameters near the large radius limit, the mirror curve is a cyclic-M curve. Applying Mikhalkin's result of cyclic-M curves, we show that the mirror curve is …
We define the notion of a holomorphic bundle on the noncommutative toric orbifold $T_θ/G$ associated with an action of a finite cyclic group $G$ on an irrational rotation algebra. We prove that the category of such holomorphic bundles is abelian and its derived category is equivalent to the derived category of modules over a finite-dimensional algebras $Λ$. As an application we finish the computation of $K_0$-groups of the crossed product algebras describing the above orbifolds initiated by Kumjian, Walters and Buck. Also, we describe a torsion pair in the category of $Λ$-modules, such that th…
A smooth scheme X over a field k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, first we prove that smooth toric varieties are strongly liftable. As a corollary, we obtain the Kawamata-Viehweg vanishing theorem for smooth projective toric varieties. Second, we prove the Kawamata-Viehweg vanishing theorem for normal projective surfaces which are birational to a strongly liftable smooth projective surface. Finally, we deduce the cyclic cover trick over W_2(k), which can be used to construct a l…
Using aritmethic conditions on affine semigroups we prove that for a simplicial toric variety of codimension 2 the property of being a set-theoretic complete intersection on binomials in characteristic $p$ holds either for all primes $p$, or for no prime $p$, or for exactly one prime $p$.
We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results on the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Co…
We define a moduli space of rational curves with finite-order automorphism and weighted orbits, and we prove that the combinatorics of its boundary strata are encoded by a particular polytopal complex that also captures the algebraic structure of a complex reflection group acting on the moduli space. This generalizes the situation for Losev-Manin's moduli space of curves (whose boundary strata are encoded by the permutohedron and related to the symmetric group) as well as the situation for Batyrev-Blume's moduli space of curves with involution, and it extends that work beyond the toric context…
The holonomic rank of the A-hypergeometric system M_A(β) is the degree of the toric ideal I_A for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of M_A(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of M_A(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the pa…
We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started in our paers "Polytopal linear groups" (J. Algebra 219 (1999), 715-737) and "Polyhedral algebras, toric arrangements, and their groups" (preprint), where we have i…
I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use of the recent theory of noncommutative motives, we extend Panin and Merkurjev-Panin computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneou…
Our main goal is to show that the Gelfand--Tsetlin toric degeneration of the type A flag variety can be obtained within a degenerate representation-theoretic framework similar to the theory of PBW degenerations. In fact, we provide such frameworks for all Gröbner degenerations intermediate between the flag variety and the GT toric variety. These degenerations are shown to induce filtrations on the irreducible representations and the associated graded spaces are acted upon by a certain associative algebra. To achieve our goal, we construct embeddings of our Gröbner degenerations into the projec…
Given two torus invariant Weil divisors $D$ and $D'$ on a two-dimensional cyclic quotient singularity $X$, the groups $\mathop{Ext}\nolimits^i_{X}(\mathcal{O}(D),\mathcal{O}(D'))$, $i>0$, are naturally $\mathbb{Z}^2$-graded. We interpret these groups via certain combinatorial objects using methods from toric geometry. In particular, it is enough to give a combinatorial description of the $\mathop{Ext}\nolimits^1$-groups in the polyhedra of global sections of the Weil divisors involved. Higher $\mathop{Ext}\nolimits^i$-groups are then reduced to the case of $\mathop{Ext}\nolimits^1$ via a quive…
