### ${\mathbb{F}}_{1}$-schemes and toric varieties.

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Let a reductive group $G$ act on an algebraic variety $X$. We give a Hilbert-Mumford type criterion for the construction of open $G$-invariant subsets $V\subset X$ admitting a good quotient by $G$.

We relate $R$-equivalence on tori with Voevodsky’s theory of homotopy invariant Nisnevich sheaves with transfers and effective motivic complexes.

We present a class of toric varieties V which, over any algebraically closed field of characteristic zero, are defined by codim V +1 binomial equations.

We generalize the work of Jian Song by computing the α-invariant of any (nef and big) toric line bundle in terms of the associated polytope. We use the analytic version of the computation of the log canonical threshold of monomial ideals to give the log canonical threshold of any non-negatively curved singular hermitian metric on the line bundle, and deduce the α-invariant from this.

Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature Kähler metrics in ${c}_{1}\left(\phantom{\rule{-0.166667em}{0ex}}L\right)$ and its automorphism group $\mathrm{Aut}(V\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}},\phantom{\rule{-0.166667em}{0ex}}L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case where $\mathrm{Aut}(V,L)$ is not discrete.

In the spirit of a theorem of Wood, we give necessary and sufficient conditions for a family of germs of analytic hypersurfaces in a smooth projective toric variety $X$ to be interpolated by an algebraic hypersurface with a fixed class in the Picard group of $X$.

Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties, we prove convergence results for sequences of densities $|{\varphi}_{n}{|}^{2}=|{s}_{N}{|}^{2}/\left|\right|{s}_{N}{\left|\right|}_{{L}^{2}}^{2}$ for eigensections ${s}_{N}\in \Gamma (X,{L}^{N})$ approaching a semiclassical ray. Here $X$ is a normal compact toric variety and $L$ is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus. Our work was motivated by and extends that of Shiffman, Tate and Zelditch....

A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational solutions of $A$-hypergeometric systems of Lawrence type. The space of binomial residues of a given degree, modulo those which are polynomial in some variable, has dimension equal to the Euler characteristic of the matroid associated with $A$.