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Register a new userVoronoi tilings, toric arrangements and degenerations of line bundles I
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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 paper and its second and third companion parts are the first in a series aimed to explore this new approach. In the present article, we set up the combinatorial framework and show how graphs with integer lengths associated to the edges provide tilings of Euclidean spaces by certain polytopes associated to the graph itself and to certain of its subgraphs. We further provide a description of the combinatorial structure of these polytopes and the way they are glued together in the tiling. In the second part of the series, we describe the arrangements of toric varieties associated to these tilings. 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.
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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.
We prove Anzis and Tohaneanu conjecture, that is the Dirac-Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free arrangements of lines contain at least one double point, that can be regarded as the Sylvester-Gallai theorem for some free arrangements. This is a corollary of a general result that if you add a line to a free projective line arrangement, then that line has to contain at least one double point. Also we prove some conjectures and one open problems related to supersolvable line arrangements and the number of double points.
Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation ${J_s}$ of the complex structure on $X$ and bases $mathcal{B}_s$ of $H^0(X,L, J_s)$ so that $J_0$ is the standard complex structure and, in the limit as $s to infty$, the basis elements approach dirac-delta distributions centered at Bohr-Sommerfeld fibers of a moment map associated to $X$ and its toric degeneration. The theory of Newton-Okounkov bodies and its associated toric degenerations shows that the technical hypotheses mentioned above hold in some generality. Our results significantly generalize previous results in geometric quantization which prove independence of polarization between Kahler quantizations and real polarizations. As an example, in the case of general flag varieties $X=G/B$ and for certain choices of $lambda$, our result geometrically constructs a continuous degeneration of the (dual) canonical basis of $V_{lambda}^*$ to a collection of dirac delta functions supported at the Bohr-Sommerfeld fibres corresponding exactly to the lattice points of a Littelmann-Berenstein-Zelevinsky string polytope $Delta_{underline{w}_0}(lambda) cap mathbb{Z}^{dim(G/B)}$.
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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