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Register a new userOn complex supersolvable line arrangements
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We show that the number of lines in an $m$--homogeneous supersolvable line arrangement is upper bounded by $3m-3$ and we classify the $m$--homogeneous supersolvable line arrangements with two modular points up-to lattice-isotopy. A lower bound for the number of double points $n_2$ in an $m$--homogeneous supersolvable line arrangement of $d$ lines is also considered. When $3 leq m leq 5$, or when $m geq frac{d}{2}$, or when there are at least two modular points, we show that $n_2 geq frac{d}{2}$, as conjectured by B. Anzis and S. O. Tohu aneanu. This conjecture is shown to hold also for supersolvable line arrangements obtained as cones over generic line arrangements, or cones over arbitrary line arrangements having a generic vertex.
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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 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.
By way of Ziegler restrictions we study the relation between nearly free plane arrangements and combinatorics and we give a Yoshinaga-type criterion for plus-one generated plane arrangements.
We introduce the package textbf{arrangements} for the software CoCoA. This package provides a data structure and the necessary methods for working with hyperplane arrangements. In particular, the package implements methods to enumerate many commonly studied classes of arrangements, perform operations on them, and calculate various invariants associated to them.
In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Teraos celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally free arrangement of hyperplanes, which can be constructed from the empty arrangement by using only the addition theorem. Then we can show that Teraos conjecture is true in this class. As an application, we can show that every ideal-Shi arrangement is additionally free, implying that their freeness is combinatorial.
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.
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