The purpose of this work is to collect in one place available information on line arrangements known in the literature as braid, monomial, Ceva or Fermat arrangement. They have been studied for a long time and appeared recently in connection with highly interesting problems, namely: the containment problem between symbolic and ordinary powers of ideals and the existence of unexpected hypersurfaces. We also study also derived configurations of points (or more general: linear flats) which arise by intersecting hyperplanes in Fermat arrangements or by taking duals of these hyperplanes. Furthermore we discuss briefly higher dimensional generalizations and present results arising by applying this approach to problems mentioned above. Some of our results are original and appear for the first time in print.
In the present note we study certain arrangements of codimension $2$ flats in projective spaces, we call them "Fermat arrangements". We describe algebraic properties of their defining ideals. In particular, we show that they provide counterexamples to an expected containment relation between ordinary and symbolic powers of homogeneous ideals.
The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of $6$-division points on the Fermat cubic $F$ and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction for searching for conceptual proofs of facts stated here and their generalisations. Extensions in several directions seem possible (taking curves of higher degree and contact to $F$, studying higher degree curves passing through higher order division points on $F$, studying…
Unexpected hypersurfaces are a brand name for some special linear systems. They were introduced around 2017 and are a field of intensive study since then. They attracted a lot of attention because of their close tights to various other areas of mathematics including vector bundles, arrangements of hyperplanes, geometry of projective varieties. Our research is motivated by the what is now known as the BMSS duality, which is a new way of deriving projective varieties out of already constructed. The last author coined the concept of companion surfaces in the setting of unexpected curves admitted …
The purpose of this note is to present and study a new series of the so-called unexpected curves. They enjoy a surprising property to the effect that their degree grows to infinity, whereas the multiplicity at a general fat point remains constant, equal $3$, which is the least possible number appearing as the multiplicity of an unexpected curve at its singular point. We show that additionally the BMSS dual curves inherits the same pattern of behaviour.
The purpose of this note is to show a new series of examples of homogeneous ideals $I$ in ${\mathbb K}[x,y,z,w]$ for which the containment $I^{(3)}\subset I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb P}^3$, which resemble Fermat configurations of points in ${\mathbb P}^2$, see \cite{NagSec16}. All examples exhibiting the failure of the containment $I^{(3)}\subseteq I^2$ constructed so far have been supported on points or cones over configurations of points. Apart of providing new counterexamples, these ideals seem quite interesting on their own.
In this note we show that there are algebraic families of hyperbolic, Fermat-Waring type hypersurfaces in P^n of degree 4(n-1)^2, for all dimensions n>1. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in P^n of degree 4n^2-2n+1 possessing complete hyperbolic, hyperbolically embedded complements.
In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to the one from Fermat's last theorem. We begin by finding local coordinates for the standard affine cover of the plane, and then prove that the curve is smooth. This is done by pulling the curve up to the surface's desingularization. Then a map from the curve to ${\mathbb P^1}$ is constructed, and it's ramification divisor is determined. We conclude by applying …
In this paper we study properties of the locus of second type lines of a general cubic threefold and fourfold. By analysing the geometry of the Fano scheme of lines of the Fermat cubic fourfold and in particular giving an explicit description of the locus of second type lines, we deduce that the Voisin map is birational over the second type locus. For a general cubic threefold, by studying properties of the second type locus again, we compute that various natural geometric monodromy groups are the full symmetric group.
For a polynomial $f = x_1^n + \dots + x_N^n$ let $G_f$ be the non--abelian maximal group of symmetries of $f$. This is a group generated by all $g \in \mathrm{GL}(N,\mathbb{C})$, rescaling and permuting the variables, so that $f(\mathbf{x}) = f(g \cdot \mathbf{x})$. For any $G \subseteq G_f$ we compute explicitly Hochschild cohomology of the category of $G$--equivarint matrix factorizations of $f$. We introduce the pairing on it showing that it is a Frobenius algebra.
A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $n,p \geq 2$ are integers such that $(p-1)(n-1)>2$, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms with quotient orbifold $S/H$ of genus zero with exactly $n+1$ cone points, each one of order $p$; in this case $H$ is called a generalized Fermat group of type $(p,n)$. In this case, it is known that $S$ is non-hyperelliptic and that $H$ is its unique generalized Fermat group of type $(p,n)$. Also, explicit equations for them, as a fiber product of classical Fermat curves of deg…
Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F\subset S$ is the set of fixed points of the non-trivial elements of the group $H$, then $F$ is exactly the set of hyperosculating points of the standard embedding $S\hookrightarrow {\mathbb{P}}^{n}$. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.
Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F\subset S$ is the set of fixed points of the non-trivial elements of the group $H$, then $F$ is exactly the set of hyperoscualting points of the standard embedding $S\hookrightarrow {\mathbb{P}}^{n}$. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.
Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of $\mathbb{F}_q$-rational points on the affine hypersurface $\mathcal X$ given by $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$, where $b\in\mathbb{F}_q^*$. A classic well-konwn result from Weil yields a bound for such number of points. This paper presents necessary and sufficient conditions for the maximality and minimality of $\mathcal X$ with respect to Weil's bound.
Let $K$ be an algebraically closed field of characteristic $p \geq 0$. A generalized Fermat curve of type $(k,n)$, where $k,n \geq 2$ are integers (for $p \neq 0$ we also assume that $k$ is relatively prime to $p$), is a non-singular irreducible projective algebraic curve $F_{k,n}$ defined over $K$ admitting a group of automorphisms $H \cong {\mathbb Z}_{k}^{n}$ so that $F_{k,n}/H$ is the projective line with exactly $(n+1)$ cone points, each one of order $k$. Such a group $H$ is called a generalized Fermat group of type $(k,n)$. If $(n-1)(k-1)>2$, then $F_{k,n}$ has genus $g_{n,k}>1$ and it i…
Let $\mathbb{K}$ be an algebraically closed field of characteristic $p>0$. A pressing problem in the theory of algebraic curves is the determination of the $p$-rank of a (nonsingular, projective, irreducible) curve $\mathcal{X}$ over $\mathbb{K}$, This birational invariant affects arithmetic and geometric properties of $\mathcal{X}$, and its fundamental role in the study of the automorphism group $\operatorname{Aut}(\mathcal{X})$ has been noted by many authors in the past few decades. In this paper, we provide an extensive study of the $p$-rank of curves of Fermat type $y^m = x^n + 1$ over $\m…
Let K be an arithmetic function field, that is, a field of finite type over the rational number field. In this note, as an application of the height theory due to Chen-Moriwaki, we would like to show that the solutions of Fermat's curve X^N + y^N = 1 of degree N over K consist of only either 0 or roots of unity for almost positive integers N. More precisely, the density of such N is 1.
In this paper we give a construction of Lagrangian torus fibration for Fermat type quintic \cy hypersurfaces via the method of gradient flow. We also compute the monodromy of the expected special Lagrangian torus fibration and discuss structures of singular fibers.
The Fermat type Calabi-Yau $n$-fold, denoted by $\mathscr{F}_n$, is the hypersurface of $\mathbb{P}^{n+1}$ defined by $\sum_{i=0}^{n+1}x_i^{n+2}=0$, which is the smooth fiber over the Fermat point $ψ=0$ of the Fermat pencil $$ \sum_{i=0}^{n+1} x^{n+2}_i -(n+2)\, ψ\, \prod_{i=0}^{n+1} x_i =0. $$ The nowhere vanishing holomorphic $n$-form on $\mathscr{F}_n$ defines an $n+1$ dimensional sub-Hodge structure of $(H^n(\mathscr{F}_n,\mathbb{Q}),F_p)$. In this paper, we will formulate a conjecture which says that this $n+1$ dimensional sub-Hodge structure splits completely into the direct sum of pure …
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.
In this work, we investigate the positivity of logarithmic and orbifold cotangent bundles along hyperplane arrangements in projective spaces. We show that a very interesting example given by Noguchi (as early as in 1986) can be pushed further to a very great extent. Key ingredients of our approach are the use of Fermat covers and the production of explicit global symmetric differentials. This allows us to obtain some new results in the vein of several classical results of the literature on hyperplane arrangements. These seem very natural using the modern point of view of augmented base loci, a…
The first part of this paper is a survey on algebro-geometric aspects of sheaves of logarithmic vector fields of hyperplane arrangements. In the second part we prove that the relative de Rham cohomology (of degree two) of ADE-type adjoint quotient map is naturally isomorphic to the module of certain multiderivations. The isomorphism is obtained by the Gauss-Manin derivative of the Kostant-Kirillov form.
We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.
Kapranov and schechtman gave quiver description of perverse sheaves on real hyperplane arrangements. We used this description to relate the perverse sheaves on Coxeter hyperplane arrangements of type $\mathcal A_n$ for different values of $n$. As a consequence we prove that the simple perverse sheaves whose stalk on open cells are zero are induced from the perverse sheaves on lower dimension arrangements.
We consider deformations of the Weyl arrangement of type $A_2$, which include the extended Shi and Catalan arrangements. These last ones are well-known to be free. We study their sheaves of logarithmic vector fields in all other cases, and show that they are Steiner bundles. Also, we determine explicitly their unstable lines. As a corollary, some counter-examples to the shift isomorphism problem are given.
We investigate the arrangement of hypersurfaces on a nonsingular varieties whose associated logarithmic vector bundle is arithmetically Cohen-Macaulay (for short, aCM), and prove that the projective space is the only smooth complete intersection with Picard rank one that admits an aCM logarithmic vector bundle. We also obtain a number of results on aCM logarithmic vector bundles over several specific varieties. As an opposite situation we investigate the Torelli-type problem that the logarithmic cohomology determines the arrangement.
We establish a new relationship (the MLK correspondence) between twisted FJRW theory and local Gromov-Witten theory in all genera. As a consequence, we show that the Landau-Ginzburg/Calabi-Yau correspondence is implied by the crepant transformation conjecture for Fermat type in genus zero. We use this to then prove the Landau-Ginzburg/Calabi-Yau correspondence for Fermat type, generalizing the results of A. Chiodo and Y. Ruan.
We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally symmetric variety of type IV determines such a completion canonically. This completion admits a natural contraction that leaves the complement of the arrangement untouched. The resulting completion of the arrangement complement is very much like a Baily-Borel compactification: it is the proj of an algebra of meromorphic automorphic forms. When that complement has a moduli space in…
Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in particular, to homology planes arising from arrangements of lines. This permits to determine the infinitude of the fundamental group of homology planes of log-general type that may be the first examples of this type.
Let $\mathcal{A} $ be a complexified-real arrangement of lines in $\mathbb{C}^2.$ Let $H$ be any line in $ \mathcal{A} $. Then, form a new complexified-real arrangement $ \mathcal{B}_H = \mathcal{A} \cup \mathcal{C} $ where $ \mathcal{C} \cup \{H\} $ is a pencil of lines with multiplicity $ m\geq 3 $, the intersection point in the pencil is not a multiple point in $ \mathcal{A}, $ and every line in $ \mathcal{C} $ intersects every line in $ \mathcal{A}\setminus \{H\} $ in points of multiplicity two. In this article, we show that for $ H_1, H_2 \in \mathcal{A} $ we may have that $ \mathcal{B}_{…
In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated discriminantal arrangement on the other hand. The type of hyperplane arrangements considered and the isomorphism classes have been defined precisely. As a consequence we enumerate such isomorphism classes by computing the characteristic polynomial of the discriminantal arrangement. With a certain restriction, the enumerated value is shown to be independent of the discri…
The complete classification of (3,3)-nets and of (3,4)-nets with only double and triple points is given. Up to lattice isomorphism, there are exactly 3 effective possibilities in each case, and some of these provide new examples of pencil-type line arrangements. For arrangements consisting of at most 14 lines and having points of multiplicity at most 5, we show that the non-triviality of the monodromy on the first cohomology H^1(F) of the associated Milnor fiber F implies the arrangement is of reduced pencil-type. In particular, the monodromy is determined by the combinatorics in such cases.
We study the homotopy types of complements of arrangements of n transverse planes in R^4, obtaining a complete classification for n <= 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R^4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.
Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this subspace arrangement are related to those of the complement of the original hyperplane arrangement. In particular, if the hyperplane arrangement is fiber-type, then, apart from grading, the Lie algebra obtained from the descending central series for the fundamental group of the complement of the hyperplane arrangement is isomorphic to the Lie algebra of primitive…
The main goal of this note is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erdős-type inequality and Hirzebruch-type inequality for a certain class of conic-line arrangements having ordinary singularities. We will also study, in detail, certain conic-line arrangements in the context of the geography of log-surfaces and free divisors in the sense of Saito.
Combining recent results by A. Macinic, S. Papadima and R. Popescu with a spectral sequence and computer aided computations, we determine the monodromy action on $H^1(F,\mathbb{C})$, where $F$ denotes the Milnor fiber of the hyperplane arrangement associated to the exceptional irreducible complex reflection group $G_{31}$. This completes the description given by the first author of such monodromy operators for all the other irreducible complex reflection groups.
We prove similar theorems concerning the structure of bundles involving complements of fiber-type hyperplane arrangements and orbit configuration spaces. These results facilitate analysis of the fundamental groups of these spaces, which may be viewed as generalizations of the Artin pure braid group. In particular, we resolve two disparate conjectures. We show that the Whitehead group of the fundamental group of the complement of a fiber-type arrangement is trivial, as conjectured by Aravinda, Farrell, and Rouchon. For the orbit configuration space corresponding to the natural action of a finit…
