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3-dimensional sundials

Enrico Carlini, Maria Catalisano, Anthony Geramita (2011)

Open Mathematics

R. Hartshorne and A. Hirschowitz proved that a generic collection of lines on ℙn, n≥3, has bipolynomial Hilbert function. We extend this result to a specialization of the collection of generic lines, by considering a union of lines and 3-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).

Courbes lisses sur les surfaces rationnelles génériques : un lemme d'Horace différentiel

Thierry Mignon (2000)

Annales de l'institut Fourier

Nous démontrons un lemme permettant d’étudier l’irréductibilité et la lissité (hors des singularités prescrites) de la courbe plane générique de degré d passant par r points génériques avec des multiplicités m 1 , ... , m r fixées par avance. Ce lemme repose sur la “méthode d’Horace”, introduite par A. Hirschowitz. Il est appliqué ici à l’étude des courbes de genre inférieur ou égal à 4 .

Factorization of point configurations, cyclic covers, and conformal blocks

Michele Bolognesi, Noah Giansiracusa (2015)

Journal of the European Mathematical Society

We describe a relation between the invariants of n ordered points in projective d -space and of points contained in a union of two linear subspaces. This yields an attaching map for GIT quotients parameterizing point configurations in these spaces, and we show that it respects the Segre product of the natural GIT polarizations. Associated to a configuration supported on a rational normal curve is a cyclic cover, and we show that if the branch points are weighted by the GIT linearization and the rational...

Freeness of hyperplane arrangements and related topics

Masahiko Yoshinaga (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

These are the expanded notes of the lecture by the author in “Arrangements in Pyrénées”, June 2012. We are discussing relations of freeness and splitting problems of vector bundles, several techniques proving freeness of hyperplane arrangements, K. Saito’s theory of primitive derivations for Coxeter arrangements, their application to combinatorial problems and related conjectures.

Geometric structures on the complement of a projective arrangement

Wim Couwenberg, Gert Heckman, Eduard Looijenga (2005)

Publications Mathématiques de l'IHÉS

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We find a finite subset of this interval for...

Hyperplane arrangements and Milnor fibrations

Alexander I. Suciu (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 local systems on the complement of the arrangement to gain information on the homology of the other three spaces, and on the monodromy operators of the various fibrations.

K ( π , 1 ) conjecture for Artin groups

Luis Paris (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The purpose of this paper is to put together a large amount of results on the K ( π , 1 ) conjecture for Artin groups, and to make them accessible to non-experts. Firstly, this is a survey, containing basic definitions, the main results, examples and an historical overview of the subject. But, it is also a reference text on the topic that contains proofs of a large part of the results on this question. Some proofs as well as few results are new. Furthermore, the text, being addressed to non-experts, is as...

On the diffeomorphic type of the complement to a line arrangement in a projective plane

Fedor Bogomolov, Viktor Kulikov (2012)

Open Mathematics

We show that the diffeomorphic type of the complement to a line arrangement in a complex projective plane P 2 depends only on the graph of line intersections if no line in the arrangement contains more than two points in which at least two lines intersect. This result also holds for some special arrangements which do not satisfy this property. However it is not true in general, see [Rybnikov G., On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal. Appl., 2011,...

On the genus of reducible surfaces and degenerations of surfaces

Alberto Calabri, Ciro Ciliberto, Flaminio Flamini, Rick Miranda (2007)

Annales de l’institut Fourier

We deal with a reducible projective surface X with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the ω -genus p ω ( X ) of X , i.e. the dimension of the vector space of global sections of the dualizing sheaf ω X . Then we prove that, when X is smoothable, i.e. when X is the central fibre of a flat family π : 𝒳 Δ parametrized by a disc, with smooth general fibre, then the ω -genus of the fibres of π is constant.

Polytopes, quasi-minuscule representations and rational surfaces

Jae-Hyouk Lee, Mang Xu, Jiajin Zhang (2017)

Czechoslovak Mathematical Journal

We describe the relation between quasi-minuscule representations, polytopes and Weyl group orbits in Picard lattices of rational surfaces. As an application, to each quasi-minuscule representation we attach a class of rational surfaces, and realize such a representation as an associated vector bundle of a principal bundle over these surfaces. Moreover, any quasi-minuscule representation can be defined by rational curves, or their disjoint unions in a rational surface, satisfying certain natural...

Predegree Polynomials of Plane Configurations in Projective Space

Tzigantchev, Dimitre (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 14N10, 14C17.We work over an algebraically closed field of characteristic zero. The group PGL(4) acts naturally on PN which parameterizes surfaces of a given degree in P3. The orbit of a surface under this action is the image of a rational map PGL(4) ⊂ P15→PN. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general Pj,...

Salvetti complex, spectral sequences and cohomology of Artin groups

Filippo Callegaro (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The aim of this short survey is to give a quick introduction to the Salvetti complex as a tool for the study of the cohomology of Artin groups. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively the cohomology of Artin groups, simplifying many computations. In the last section some examples of applications are presented.

Springer fiber components in the two columns case for types A and D are normal

Nicolas Perrin, Evgeny Smirnov (2012)

Bulletin de la Société Mathématique de France

We study the singularities of the irreducible components of the Springer fiber over a nilpotent element N with N 2 = 0 in a Lie algebra of type A or D (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen–Macaulay, and have rational singularities.

Syzygies and logarithmic vector fields along plane curves

Alexandru Dimca, Edoardo Sernesi (2014)

Journal de l’École polytechnique — Mathématiques

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a plane curve C and the stability of the sheaf of logarithmic vector fields along C , the freeness of the divisor C and the Torelli properties of C (in the sense of Dolgachev-Kapranov). We show in particular that curves with a small number of nodes and cusps are Torelli in this sense.

The automorphism group of M ¯ 0 , n

Andrea Bruno, Massimiliano Mella (2013)

Journal of the European Mathematical Society

The paper studies fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov’s description we are able to prove that the only such morphisms are forgetful maps. This allows us to show that the automorphism group of M ¯ 0 , n is the permutation group on n elements as soon as n 5 .

Toric and tropical compactifications of hyperplane complements

Graham Denham (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel ' fand-MacPherson construction, Kapranov’s visible contours compactification, and De Concini and Procesi’s wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry.

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