Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent {\it secondary polytope}, which is a polytope whose vertices correspond to regular triangulations of the associated polytope \cite{KSZ}. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is $H$-semistable if and only if it is $H$-polystable with respect to the standard complex torus action $H$. This \emph{essentially} means that Chow semistability is equivalent to Chow polystability for any (not-necessaliry-smooth) projective toric varieties.
We study the Fulton-Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$-theory defined by Anderson and Payne.
In this short note, we study the asymptotic Chow polystability of toric Del Pezzo surfaces appear in the moduli space of Kähler-Einstein Fano varieties constructed in [OSS16].
For any vector bundle $V$ on a toric Deligne-Mumford stack $\ix$ the formalism of \cite{EJK:16} defines two intertial products $\star_{V^{+}}$ and $\star_{V^{-}}$ on the Chow group of the inertia stack. We give an explicit presentation for the integral $\star_{V^+}$ and $\star_{V^-}$ Chow rings, extending earlier work of Boris-Chen-Smith \cite{BCS:05} and Jiang-Tsen \cite{JiTs:10} in the orbifold Chow ring case, which corresponds to $V = 0$. We also describe an {\em asymptotic} product on the rational Chow group of the inertia stack obtained by letting the rank of the bundle $V$ go to infinity…
In this paper, we present explicit toric construction of moduli space of quasi maps from $\mathbb{P}^1$ with two marked points to $\mathbb{P}^1 \times \mathbb{P}^1$, which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases.
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 \cite{botero}, a top intersection product of toric b-divisors on a smooth complete toric variety is defined. It is shown that a nef toric b-divisor corresponds to a convex set and that its top intersection number equals the volume of this convex set. The goal of this article is to extend this result and define an intersection product of sufficiently positive toric b-classes of arbitrary codimension. For this, we extend the polytope algebra of McMullen (\cite{McM1}, \cite{McM2}, \cite{McM3}) to the so called \emph{convex-set algebra} and we show that it embeds in the toric b-Chow group. In t…
Topological invariants such as characteristic classes are an important tool to aid in understanding and categorizing the structure and properties of algebraic varieties. In this note we consider the problem of computing a particular characteristic class, the Chern-Schwartz-MacPherson class, of a complete simplicial toric variety X defined by a fan from the combinatorial data contained in the fan. Specifically, we give an effective combinatorial algorithm to compute the Chern-Schwartz-MacPherson class of X, in the Chow ring (or rational Chow ring) of X. This method is formulated by combining, a…
The Chow quotient of a toric variety by a subtorus, as defined by Kapranov-Sturmfels-Zelevinsky, coarsely represents the main component of the moduli space of stable toric varieties with a map to a fixed projective toric variety, as constructed by Alexeev and Brion. We show that, after endowing both spaces with the structure of a logarithmic stack, the resulting spaces are isomorphic. Along the way, we construct the Chow quotient stack and demonstrate several properties that it satisfies.
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.
We generalize the Künneth formula for Chow groups to an arbitrary OBM-homology theory satisfying descent (e.g. algebraic cobordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a description for the homology groups of smooth toric varieties, that allows the calculation of algebraic cobordism groups of singular toric varieties. Some computations are carried out.
This article is a continuation of [Kub18], which proves that if a $3$-dimensional affine normal quasihomogeneous $SL(2)$-variety $E$ is toric, then it has an equivariant resolution of singularities given by an invariant Hilbert scheme $\mathcal H$. In this article, we consider the case where $E$ is non-toric and show that the Hilbert-Chow morphism $γ: \mathcal H \to E$ is a resolution of singularities and that $\mathcal H$ is isomorphic to the minimal resolution of a weighted blow-up of $E$.
This paper presents an algebraic construction of Euler-Maclaurin formulas for polytopes. The formulas obtained generalize and unite the previous lattice point formulas of Morelli and Pommersheim-Thomas, and the Euler-Maclaurin formulas of Berline-Vergne While the approach of this paper originates in the theory of toric varieties, and recovers previous results about characteristic classes of toric varieties, the present paper is self-contained and does not rely on results from toric geometry. We aim in particular to exhibit in a combinatorial way ingredients such as such Todd classes and cycle-…
In this paper, we prove formulas that represent two-pointed Gromov-Witten invariant <O_{h^a}O_{h^b}>_{0,d} of projective hypersurfaces with d=1,2 in terms of Chow ring of Mbar_{0,2}(P^{N-1},d), the moduli spaces of stable maps from genus 0 stable curves to projective space P^{N-1}. Our formulas are based on representation of the intersection number w(O_{h^a}O_{h^b})_{0,d}, which was introduced by Jinzenji, in terms of Chow ring of Mp_{0,2}(N,d), the moduli space of quasi maps from P^1 to P^{N-1} with two marked points. In order to prove our formulas, we use the results on Chow ring of Mbar_{0,…
