The space of holomorphic maps from $S^2$ to a complex algebraic variety $X$, i.e. the space of parametrized rational curves on $X$, arises in several areas of geometry. It is a well known problem to determine an integer $n(D)$ such that the inclusion of this space in the corresponding space of continuous maps induces isomorphisms of homotopy groups up to dimension $n(D)$, where $D$ denotes the homotopy class of the maps. The solution to this problem is known for an important but special class of varieties, the generalized flag manifolds: such an integer may be computed, and $n(D)\to\infty$ as $D\to\infty$. We consider the problem for another class of varieties, namely, toric varieties. For smooth toric varieties and certain singular ones, $n(D)$ may be computed, and $n(D)\to\infty$ as $D\to\infty$. For other singular toric varieties, however, it turns out that $n(D)$ cannot always be made arbitrarily large by a suitable choice of $D$.
This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal rational parametrizations of certain projective varieties. We give numerous examples and then discuss what happens in the singular case. We also describe rational maps to smooth toric varieties.
We shall show that the variety of minimal rational tangents on a complete toric manifold X is linear and minimal components in RatCurves^n(X) corresponds bijectively to some special primitive collections in the fan defining X.
We explicitly describe cohomology of the sheaf of differential forms with poles along a semiample divisor on a complete simplicial toric variety. As an application, we obtain a new vanishing theorem which is an analogue of the Bott-Steenbrink-Danilov vanishing theorem.
Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every "minimal embedded toric resolution". We prove that this correspondence is bijective except for the $E_8$ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.
In this paper, we prove that: For any given finitely many distinct points $P_1,...,P_r$ and a closed subvariety $S$ of codimension $\geq 2$ in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there exists a rational curve $f:\mathbb{P}^1\to X$ passing through $P_1,...,P_r$, disjoint from $S\setminus \{P_1,...,P_r\}$ (see Main Theorem). As a corollary, we prove that the smooth loci of complete toric varieties are strongly rationally connected.
The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a moduli space of real degenerations with the secondary polytope. A configuration A of real vectors gives an irrational projective toric variety in a simplex. We identify a space of translations and degenerations of the irrational projective toric variety with the secondary polytope of A. For this, we develop a theory of irrational toric varieties associated to arb…
We write the equivariant Todd class of a general complete toric variety as an explicit combination of the orbit closures, the coefficients being analytic functions on the Lie algebra of the torus which satisfy Danilov's requirement.
We explain a method for calculating the cohomology of line bundles on a toric variety in terms of the cohomology of certain constructible sheaves on the polytope. We show its effective use by means of some examples.
The Nash problem on arc families is affirmatively answered for a toric variety by Ishii and Kollar's paper which also shows the negative answer for general case. The Nash problem is one of questions about the relation between arc families and valuations. In this paper, the relation is described clearly for a toric variety. The arc space of a toric variety admits an action of the group scheme determined by the torus. Each orbit on the arc space corresponds to a lattice point in the cone and therefore corresponds to a toric valuation. The dominant relation among the orbits is described in terms …
We study equivariant bi-vector fields on a toric variety. We prove that, on a smooth toric variety of dimension $n$, the locus where the rank of an equivariant bi-vector field is $\leq 2k$ is not empty and has at least a component of dimension $\geq 2k+1$, for all integers $k> 0$ such that $2k < n$. The same is true also for $k=0$, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.
The semi-simplicity of the Hodge group is proved for a simple Abelian variety with a stable reduction of odd toric (reductive) rank. If, besides, the dimension of the Abelian variety is an odd integer, then the Hodge conjecture on algebraic cycles holds for it.
We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.
Let X be a complete toric variety and Aut(X) be the automorphism group. We give an explit description of Aut(X)-orbits on X. In particular, we show that Aut(X) acts on X transitively if and only if X is a product of projective spaces.
We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, \mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, \mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ to see that, in general, the subalgebra $L$ does not have Poincare duality.
In this paper we construct a degeneration of Bott-Samelson-Demazure-Hansen varieties to toric varieties in an algebraic family and study the geometry of the resulting toric varieties. We give a natural set of torus invariant curves that generate the Chow group of $1$-cycles of the limiting toric variety and express the ample cone of this toric variety as a sub-cone of the ample cone of the corresponding Bott-Samelson-Demazure-Hansen variety. We also give a description of Extremal and Mori rays and determine when this toric variety is Fano.
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…
In this paper we prove a formula for the number of rational points of bounded height relative to all the generators of the cone of effective divisor for a toric variety over a number field.
The Cox ring provides a coordinate system on a toric variety analogous to the homogeneous coordinate ring of projective space. Rational maps between projective spaces are described using polynomials in the coordinate ring, and we generalise this to toric varieties, providing a unified description of arbitrary rational maps between toric arieties in terms of their Cox coordinates. Introducing formal roots of polynomials is necessary even in the simplest examples.
In this paper we study the geometry of toric degeneration of a Bott-Samelson-Demazure-Hansen (BSDH) variety. We give some applications to BSDH varieties. Precisely, we classify Fano, weak Fano and log Fano BSDH varieties and their toric limits in Kac-Moody setting. We prove some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties. We also recover the results in arXiv:1604.01998, by toric methods.
For a fan $Δ$, we introduce Grothendieck weights as a ring of functions from $Δ$ to $\mathbb{Z}$ that form a K-theoretic analogue of Minkowski weights and describe the operational $K$-theory of a complete toric variety. We give an explicit balancing condition and product formula for these weights, and describe relationships with other fan-based invariants. Applications are given to vector bundles on a toric surface, and to the calculation of Euler characteristics on schon subvarieties.
Let X be an affine toric variety. The total coordinates on X provide a canonical presentation of X as a quotient of a vector space 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.
Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta's conjecture. More generally, we show that if $X$ is a $\mathbb{Q}$-factorial terminal split toric variety of arbitrary dimension, then $P$ is better approximated by points on a rational curve than by any Zariski dense sequence.
We define a quasi--projective reduction of a complex algebraic variety $X$ to be a regular map from $X$ to a quasi--projective variety that is universal with respect to regular maps from $X$ to quasi--projective varieties. A toric quasi--projective reduction is the analogous notion in the category of toric varieties. For a given toric variety $X$ we first construct a toric quasi--projective reduction. Then we show that $X$ has a quasi--projective reduction if and only if its toric quasi--projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a…
Using the method of degenerating a Grassmannian into a toric variety, we calculate recursive formulas for the dimensions of the eigenspaces of the action of an n-dimensional torus on a Grassmannian of planes in an n-dimensional space. In order to verify our result we compare it with the polynomial describing the Euler characteristic of invertible sheaves on a projective space with four blown-up points.
We calculate the automorphism group of a complete toric variety $X$ with torus $T_M$. We prove that the radical unipotent of $Aut_k^0X$ is a semidirect product of additive groups, the reductive part is a quotient of a product of lineal groups and we give the action of linear groups on the additive groups. We also prove that $Aut_kX/Aut_k^0X$ is a quotient of the group of automorphisms of $M$ leaving invariant the fan.
In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even when it is singular) in terms of combinatorial data coming from the fan of the toric variety.
We consider the action of a subtorus of the big torus on a toric variety. The aim of the paper is to define a natural notion of a quotient for this setting and to give an explicit algorithm for the construction of this quotient from the combinatorial data corresponding to the pair consisting of the subtorus and the toric variety. Moreover, we study the relations of such quotients with good quotients. We construct a good model, i.e. a dominant toric morphism from the given toric variety to some ``maximal'' toric variety having a good quotient by the induced action of the given subtorus.
The real solutions to a system of sparse polynomial equations may be realized as a fiber of a projection map from a toric variety. When the toric variety is orientable, the degree of this map is a lower bound for the number of real solutions to the system of equations. We strengthen previous work by characterizing when the toric variety is orientable. This is based on work of Nakayama and Nishimura, who characterized the orientability of smooth real toric varieties.
In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product we assign a rational number m(s) to each rational polyhedral cone s in the lattice, such that for any toric variety X with fan S, the Todd class of X is the sum over all cones s in S of m(s)[V(s)]. This constitutes an improved answer to an old question of Danilov. In a similar way, beginning with the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli. Our construction is based on an intersection product on c…
From a rational convex polytope of dimension $r\ge 2$ J.P. Hansen constructed an error correcting code of length $n=(q-1)^r$ over the finite field $\fq$. A rational convex polytope is the same datum as a normal toric variety and a Cartier divisor. The code is obtained evaluating rational functions of the toric variety defined by the polytope at the algebraic torus, and it is an evaluation code in the sense of Goppa. We compute the dimension of the code using cohomology. The minimum distance is estimated using intersection theory and mixed volumes, extending the methods of J.P. Hansen for plane…
We study smoothness of toric quiver varieties. When a quiver $Q$ is defined with the identity dimension vector, the corresponding quiver variety is also a toric variety. So it has both fan representation and quiver representation. We work only on quivers with canonical weight and we classify smooth such toric quiver varieties. We show that a variety corresponding to a quiver with the identity dimension vector and the canonical weight is smooth if and only if it is a product of projective spaces or their blowups.
Classical toric varieties are among the simplest objects in algebraic geometry. They arise in an elementary fashion as varieties parametrized by monomials whose exponents are a finite subset $\mathcal{A}$ of $\mathbb{Z}^n$. They may also be constructed from a rational fan $Σ$ in $\mathbb{R}^n$. The combinatorics of the set $\mathcal{A}$ or fan $Σ$ control the geometry of the associated toric variety. These toric varieties have an action of an algebraic torus with a dense orbit. Applications of algebraic geometry in geometric modeling and algebraic statistics have long studied the nonnegative r…
For an ample line bundle $\mathcal{L}$ on a complete toric surface $X$, we consider the subset $V_{\mathcal{L}} \subset \vert \mathcal{L} \vert$ of irreducible, nodal, rational curves contained in the smooth locus of $X$. We study the monodromy map from the fundamental group of $V_{\mathcal{L}}$ to the permutation group on the set of nodes of a reference curve $C \in V_{\mathcal{L}}$. We identify a certain obstruction map $\varPsi_{X}$ defined on the set of nodes of $C$ and show that the image of the monodromy is exactly the group of deck transformations of $\varPsi_{X}$, provided that $\mathc…
We study the Berkovich analytification of the space of genus $0$ logarithmic stable maps to a toric variety $X$ and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps -- (1) as an explicit toroidal modification of $\overline M_{0,n}\times X$ and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus $0$ logarithmic stable maps coincides with the space of tropical stable maps, giving a large new…
We consider the notions of Groebner fan and Newton non-degeneracy for an ideal on a toric variety, extending the two existing notions for ideals on affine spaces. We prove, without assumptions on the characteristic of the base fields, that the "Groebner fan" of such an ideal is actually a polyhedral fan and that a sub-variety defined by a Newton non-degenerate ideal on a toric variety $X_σ$ admits a toric embedded resolution of singularities $Z\longrightarrow X_σ.$
The higher Nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non-singular points. In the case of normal toric varieties we give a combinatorial description of the higher Nash blowup in terms of a Gröbner fan. This description will allow us to prove the analogue of Nobile's theorem on the usual Nash blowup in this context. More precisely, we prove that for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular.
The quasisymmetric flag variety: a toric complex on noncrossing partitions
York University（约克大学）；Universite Claude Bernard Lyon 1, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208（里昂第一大学、法国国家科学研究中心、里昂中央理工学院、里昂国立应用科学学院、让·莫内大学、ICJ联合研究实验室）；University of Toronto（多伦多大学）；University of Toronto Mississauga（多伦多大学密西沙加分校）
Let X be a smooth projective toric surface, and H^d(X) the Hilbert scheme parametrising the length d zero-dimensional subschemes of X. We compute the rational Chow ring A^*(H^d(X))\_Q. More precisely, if T is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring A\_T^*(H^d(X))\_Q and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.
An infinite type toric variety is a normal toric variety given by a fan with infinitely many cones. We construct examples in this paper coming from representation theory of loop groups. The fans that appear are cones on Voronoi tilings on a vector space equipped with an inner product. We also explain the affine analogue of the connection between a generic torus orbit closure in a flag variety and the closure a maximal torus in the wonderful compactification.
We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a categorical quotient in the category of divisorial varieties. Our result generalizes previous statements for the quasiprojective case. An important tool for the proof is a universal reduction of an arbitrary toric variety to a divisorial one. This is done in terms of support maps, a notion generalizing support functions on a polytopal fan. A further essential …
We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to…
