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We develop an analogue of Eisenbud-Floystad-Schreyers Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier-Mukai transform, works quite generally and provides a new perspective on the relationship between Tate resolutions and Beilinsons resolution of the diagonal. We also develop a Beilinson-type resolution of the diagonal for toric varieties and use it to generalize Eisenbud-Floystad-Schreyers computationally effective construction of Beilinson monads.
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In this paper we give a geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to $2n-2$, is exactly $2n-2$ if and only if the cone is a bipyramidal cone, where $n>1$ is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones.
We study standard monomial bases for Richardson varieties inside the flag variety. In general, writing down a standard monomial basis for a Richardson variety can be challenging, as it involves computing so-called defining chains or key tableaux. However, for a certain family of Richardson varieties, indexed by compatible permutations, we provide a very direct and straightforward combinatorial rule for writing down a standard monomial basis. We apply this result to the study of toric degenerations of Richardson varieties. In particular, we provide a new family of toric degenerations of Richardson varieties inside flag varieties.
For a toric pair $(X, D)$, where $X$ is a projective toric variety of dimension $d-1geq 1$ and $D$ is a very ample $T$-Cartier divisor, we show that the Hilbert-Kunz density function $HKd(X, D)(lambda)$ is the $d-1$ dimensional volume of ${overline {mathcal P}}_D cap {z= lambda}$, where ${overline {mathcal P}}_Dsubset {mathbb R}^d$ is a compact $d$-dimensional set (which is a finite union of convex polytopes). We also show that, for $kgeq 1$, the function $HKd(X, kD)$ can be replaced by another compactly supported continuous function $varphi_{kD}$ which is `linear in $k$. This gives the formula for the associated coordinate ring $(R, {bf m})$: $$lim_{kto infty}frac{e_{HK}(R, {bf m}^k) - e_0(R, {bf m}^k)/d!}{k^{d-1}} = frac{e_0(R, {bf m})}{(d-1)!}int_0^inftyvarphi_D(lambda)dlambda, $$ where $varphi_D$ (see Proposition~1.2) is solely determined by the shape of the polytope $P_D$, associated to the toric pair $(X, D)$. Moreover $varphi_D$ is a multiplicative function for Segre products. This yields explicit computation of $varphi_D$ (and hence the limit), for smooth Fano toric surfaces with respect to anticanonical divisor. In general, due to this formulation in terms of the polytope $P_D$, one can explicitly compute the limit for two dimensional toric pairs and their Segre products. We further show that (Theorem~6.3) the renormailzed limit takes the minimum value if and only if the polytope $P_D$ tiles the space $M_{mathbb R} = {mathbb R}^{d-1}$ (with the lattice $M = {mathbb Z}^{d-1}$). As a consequence, one gets an algebraic formulation of the tiling property of any rational convex polytope.
We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional sequences, investigating the diagonal property, or the toric Frobenius morphism. In the present paper we focus on line bundles on toric varieties. First, we present a possibility of understanding their cohomology in terms of their (generalized) momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This will be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans. The locus of immaculate line bundles contains several linear strata of varying dimensions. We introduce a notion of relative immaculacy with respect to certain contraction morphisms. This notion will be stronger than plain immaculacy and provides an explanation of some of these linear strata.
We study the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety $X$ with branching set the invariant divisor under the action of $(mathbb{C}^*)^n$. This is the proalgebraic toric-completion $X_{mathbb{Q}}$ of $X$. The ramification over the invariant divisor and the singular invariant divisors of $X$ impose topological constraints on the automorphisms of $X_{mathbb{Q}}$. Considering this proalgebraic space as the toric functor on the adelic complex plane multiplicative semigroup, we calculate its automorphic group. Moreover we show that its vector bundle category is the direct limit of the respective categories of the finite toric varieties coverings defining the proalgebraic toric-completion.
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