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For any two nef line bundles F and G on a toric variety X represented by lattice polyhedra P respectively Q, we present the universal equivariant extension of G by F under use of the connected components of the set theoretic difference of Q and P.
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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.
There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $Delta^+-Delta^-$ of polyhedra in the character space $M_{mathbb R}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference $Delta^- setminus Delta^+$. Here, we provide a short and direct proof of this formula.
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This article and the first two that preceded it are the first in a series aimed to explore this new approach. In Part I, we set up the combinatorial framework and showed how graphs weighted with integer lengths associated to the edges provide tilings of Euclidean spaces by certain polytopes associated to the graph itself and to its subgraphs. In Part II, we described the arrangements of toric varieties associated to the tilings of Part I in several ways: using normal fans, as unions of orbits, by equations and as degenerations of tori. In the present Part III, we show how these combinatorial and toric frameworks allow us to describe all stable limits of a family of line bundles along a degenerating family of curves. Our main result asserts that the collection of all these limits is parametrized by a connected 0-dimensional closed substack of the Artin stack of all torsion-free rank-one sheaves on the limit curve. Moreover, we thoroughly describe this closed substack and all the closed substacks that arise in this way as certain torus quotients of the arrangements of toric varieties of Part II determined by the Voronoi tilings of Euclidean spaces studied in Part I.
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This article and its first and third part companion parts are the first in a series aimed to explore this new approach. In the first part, we set up the combinatorial framework and showed how graphs weighted with integer lengths associated to the edges provide tilings of Euclidean spaces by polytopes associated to the graph itself and to its subgraphs. In this part, we describe the arrangements of toric varieties associated to these tilings. Roughly speaking, the normal fan to each polytope in the tiling corresponds to a toric variety, and these toric varieties are glued together in an arrangement according to how the polytopes meet. We provide a thorough description of these toric arrangements from different perspectives: by using normal fans, as unions of torus orbits, by describing the (infinitely many) polynomial equations defining them in products of doubly infinite chains of projective lines, and as degenerations of algebraic tori. These results will be of use in the third part to achieve our goal of describing all stable limits of a family of line bundles along a degenerating family of curves.
A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in ${rm U(1)}$. By using the theory of indigenous bundles, we construct on a compact Riemann surface $X$ of genus $g_X geq 1$ a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in $2 pi mathbb{Z}_{>1}$, which is generically injective in the algebro-geometric sense as $g_X geq 2$. As an application, we prove the following two results about irreducible metrics: $bullet$ as $g_X geq 2$ and $d$ is even and greater than $12g_X - 7$, the effective divisors of degree $d$ which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension $geq 2(d+3-3g_X)$ in ${rm Sym}^d(X)$; $bullet$ as $g_X geq 1$, for almost every effective divisor $D$ of degree odd and greater than $2g_X-2$ on $X$, there exist finitely many cone spherical metrics representing $D$.
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