Geradenkonfigurationen und Algebraische Flächen
Chapters (6)
Wir betrachten (komplexe) algebraische Flächen, d.h. zweidimensionale komplex-algebraische Mannigfaltigkeiten, die wir meistens als kompakt voraussetzen. Die Untersuchung der algebraischen Flächen ist eines der klassischen Themen der algebraischen Geometrie; insbesondere ist die Suche nach einer befriedigenden Klassifikation seit dem Ende des 19. Jahrhunderts eines der wesentlichen Probleme gewesen. Wir wollen zum besseren Verständnis zunächst an die Situation im eindimensionalen Fall erinnern.
Wir betrachten Überlagerungen von algebraischen Flächen, die längs endlich vieler glatter Kurven in der Basisfläche mit konstanter lokaler Verzweigungsordnung zyklisch verzweigt sind (siehe 1.1,B bzw. 1.2,C-D für die genaue Beschreibung in lokalen Koordinaten). Dabei interessieren wir uns besonders für die Beziehungen zwischen den charakteristi-schen Zahlen der beiden Flächen (CHERNsche Zahlen c
12 und c2) bzw. der Verzweigungskurven (EULER- und Selbstschnittzahl) und für die Frage, wann die Überlagerungsfläche ein Ballquotient ist (vgl. Einführung).
In diesem Kapitel diskutieren wir Konfigurationen von Geraden (englisch “arrangements of lines”) in der Ebene. Wir behandeln in Abschnitt 2.1 zunächst die Aspekte, die in beliebigen projektiven Ebenen gelten. Insbesondere betrachten wir dort neben den grundlegenden kombinatorischen Invarianten auch die speziellen Typen von singulären Konfigurationen, die bei der Klassifikation der KUMMER — Überlagerungen in Abschnitt 3.2 eine Ausnahmerolle spielen.
Nach der Diskussion von Geradenkonfigurationen in der projektiven Ebene kehren wir zur Betrachtung von algebraischen Flächen zurück. Dazu greifen wir die Konstruktion der KUMMERschen Überlagerungen der komplex-projektiven Ebene aus Abschnitt 1.5 auf.
Wir wollen in den folgenden Kapiteln die Untersuchung verzweigter Überlagerungen algebraischer Flächen fortsetzen. Die Voraussetzungen an die Basisfläche und die Verzweigungskonfiguration bleiben ungeändert; auch die Forderung an das Verzweigungsverhalten bleibt im wesentlichen gleich. Als einzige Änderung lassen wir die Forderung konstanter lokaler Verzweigungsordnung fallen und betrachten den etwas allgemeineren Fall, daß die lokale Verzweigungsordnung längs verschiedener Komponenten der Verzweigungskonfiguration in der Basisfläche verschieden sein kann. Die Komponenten der Konfiguration werden also durch die vorgeschriebene lokale Verzweigungsordnung „gewichtet“. Da wir auch wie in Abschnitt 1.2 Konfigurationen mit singulären Schnittpunkten betrachten wollen und über den beim Aufblasen eingesetzten Ausnahmekurven ebenfalls Verzweigung vorliegen kann, müssen auch diese Schnittpunkte gewichtet werden.
Wir betrachten die in Abschnitt 4.1 beschriebene Situation für eine Konfiguration L von Geraden L1,..., Lk in der komplex-projektiven Ebene S = ℙ2: Zu jeder Geraden Lj und zu jedem singulären (d.h. mehrfachen) Schnittpunkt pv
ist ein positives ganzzahliges Gewicht nj bzw. mv
gegeben.
... A class of complex hyperbolic lattices in PU (2, 1) called the Deligne-Mostow lattices has been reinterpreted by Hirzebruch (see Hirzebruch, in: Arithmetic and geometry, vol II, volume 36 of Progress in Mathematics, Birkhäuser, Boston, pp 113-140, 1983;Barthel et al., in: Aspects of mathematics, D4. Friedrich Vieweg & Sohn, Braunschweig, 1987 and Tretkoff, in: Complex ball quotients and line arrangements in the projective plane, volume 51 of mathematical notes, Princeton University Press, Princeton, 2016) in terms of line arrangements. ...
... These lattices also have a third interpretation, introduced by Hirzebruch (see, for example, [10]) for one case and generalised in [1] (see also [24]) for all of them. This construction is explained in Sect. ...
... This is done using the equality case in Bogomolov-Miyaoka-Yau inequality, which guarantees that a compact complex surface of general type (which is our case), satisfying c 2 1 − 3c 2 = 0, is complex hyperbolic. In fact, one proves [1] that the surface is the quotient of (a torsion-free subgroup of) a Deligne-Mostow lattice, denoted as ( p, k) or ( p, k, p ) for three or twofold symmetry (see Sect. 2.2 for more details). ...
A class of complex hyperbolic lattices in PU (2, 1) called the Deligne-Mostow lattices has been reinterpreted by Hirzebruch (see [1, 10] and [24]) in terms of line arrangements. They use branched covers over a suitable blow up of the complete quadrilateral arrangement of lines in P 2 to construct the complex hyperbolic surfaces over the orbifolds associated to the lattices. In [18] and [19], fundamental domains for these lattices have been built by Pasquinelli. Here we show how the fundamental domains can be interpreted in terms of line arrangements as above. This parallel is then applied in the following context. Wells in [25] shows that two of the Deligne-Mostow lattices in PU (2, 1) can be seen as hybrids of lattices in PU (1, 1). Here we show that he implicitly uses the line arrangement and we complete his analysis to all possible pairs of lines. In this way, we show that three more Deligne-Mostow lattices can be given as hybrids.
... It is by now well known that there exist non-arithmetic lattices in PU(2, 1), the group of biholomorphisms of the complex hyperbolic plane. The first examples were due to Mostow [22], and his construction was later reformulated and generalized in several ways, see the work of Deligne-Mostow [8], Thurston [29], Barthel-Hirzebruch-Höfer [2] , for in- stance. The Deligne-Mostow examples are very special kinds of lattices (for one thing they are generated by complex reflections), and they turn out to yield relatively few commensurability classes of examples. ...
... Later, we will describe the relevant birational transformations directly on the level of P(2, 3, 7), see section 3.1, but for now we give an equivariant description on the level of P 2 C , seen as a branched covering of P(2, 3, 7) of degree 168. It is well known that the configuration of mirrors of the group G in P 2 C has 49 singular points, that come into 21 quadruple points (all in the same G-orbit), and 28 triple points (also forming a G-orbit), see [2], for instance. Definition 1. ...
... By pulling-back the orbifold structure (V 0 , D 0 ) under the covering P 2 C → P(2, 3, 7) (or Y → Y , Z → Z respectively) of degree 168, we get a precise relation of some of our ball quotients with the ball quotients constructed by Barthel, Hirzebruch and Höfer from the Klein configuration of lines in P 2 C . Recall from [2] that Barthel, Hirzebruch and Höfer produced several ball quotients from line arrangements P 2 C , one of which is given by the configuration K of mirrors of the automorphism group of the Klein quartic (see page 215 of [2]). These groups also appear in work of Couwenberg, Heckman and Looijenga, who give a generalization of the Barthel- Hirzebruch-Höfer construction (see Table 5on p. 160 of [6]). ...
We give an algebro-geometric description of the non-arithmetic ball quotients constructed by the author, Parker and Paupert. For three of the six corresponding lattices, we exhibit congruence subgroups of index 168 that are conjugate to the lattices constructed by Barthel-Hirzebruch-H\"ofer from branched coverings of \CP^2, ramified along the configuration of mirrors of reflections in the automorphism group of the Klein quartic.
... Mostow Rigidity Theorem then implies that O is a complex hyperbolic orbifold as well. A bit more streamlined version of this argument was later developed by Barthel-Hirzebruch-Hofer, [6], and Holzapfel, [51], who defined orbifold-characteristic classes directly computable from lines arrangement A in P 2 C (as well as P 1 C × P 1 C ) and the orbifold-ramification numbers assigned to rational curves in the corresponding post-blow-up divisor. The corresponding orbifold B has three singular points a 2 , a 3 , b 1 , with local isotropy groups Z 5 for each of them. ...
... (b) By constructing the corresponding complex hyperbolic orbifolds M Γ whose underlying space is a blown-up P n , see [6,84,26,32]. ...
This survey of discrete subgroups of isometries of complex hyperbolic spaces is aimed to discuss interactions between function theory on complex hyperbolic manifolds and the theory of discrete groups. We present a number of examples and basic results about complex-hyperbolic Kleinian groups. The appendix to the paper written by Mohan Ramachandran includes a proof of a result known as “Burns’ Theorem” about ends of complex-hyperbolic manifolds.
... For some decades, the Deligne-Mostow examples were the only known examples, even though some alternative constructions were given, see [22] for instance. To this day, it is still unknown whether there exist non-arithmetic lattices in SU(n, 1) for any n > 3. A slightly different construction was given by Hirzebruch (see [1]), based on the equality case in the Miyaoka-Yau inequality, i.e. an orbifold version of the fact that a compact complex surface X of general type with c 2 1 (X) = 3c 2 (X) is covered by the ball. Given such an X, the existence of a lattice Γ in P U(2, 1) such that X = Γ \ B 2 is guaranteed, but it is not obvious how to describe the lattice explicitly (the existence of a Kähler-Einstein metric is obtained by showing existence of a solution to a Monge-Ampère equation). ...
... It is unclear how often these conditions are satisfied, but there is a somewhat large list of examples associated to finite unitary groups generated by complex reflections (these were classified by Shephard and Todd [20]). That list contains a lot of the previously known examples of reflective lattices in P U(n, 1), namely the Deligne-Mostow lattices [7], as well as the ones constructed by Barthel, Hirzebruch and Höfer [1]. Note that some examples in [11] are still not covered by the Couwenberg-Heckman-Looijenga construction, see [10]. ...
We study the arithmeticity of the Couwenberg-Heckman-Looijenga lattices in PU(n,1), and show that they contain a non-arithmetic lattice in PU(3,1) which is not commensurable to the non-arithmetic Deligne-Mostow lattice in PU(3,1). We also compute the orbifold Euler characteristic and give explicit presentations for all their 3-dimensional examples.
... Example 1: (5,4,1,1,1)/6 (nonuniform, arithmetic) This example is noncompact, arithmetic, and commensurable with the Picard modular group over the field Q(ζ 3 ). The curve A in Figure 2 has orbifold weight 6 and the curve B has weight 3. The point of tangency between A and B is a cusp with Euclidean cusp group 3 [4]6. See the appendix to [43] for more on the orbifold structure and details concerning the computations that follow. ...
This paper studies residual finiteness of lattices in the universal cover of and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in or a finite covering of it. First, we prove that certain lattices in the universal cover of are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in .
... Just as in [10] , the results of this paper give an alternative construction of certain nonarithmetic ball quotients, whose existence was known so far only by giving explicit matrix generators and constructing a fundamental domain for their action (see [12] and [11]). The analysis in [10] shows that some of the non-arithmetic lattices in [11], even though they are not commensurable to Deligne-Mostow lattices (see [8], [21]), are commensurable to Couwenberg-Heckman-Looijenga lattices (see [7], which was inspired in part by [3]). For brevity, we refer to these two classes of lattices as DM and CHL lattices, respectively (note that DM lattices are special cases of CHL lattices). ...
We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with Parker and Paupert.
... It was recently observed [8] that some of these non-arithmetic lattices actually appear in a list of lattices constructed by Couwenberg, Heckman and Looijenga [4], that gave a common generalization of work of Barthel-Hirzebruch-H?fer [1] and Deligne-Mostow [5]. We refer to these lattices as CHL lattices. ...
We describe a general procedure to produce fundamental domains for complex hyperbolic triangle groups, a class of groups that contains a representative of the commensurability class of every known non-arithmetic lattice in . We discuss several commensurability invariants for lattices, and show that some triangle groups yield new commensurability classes, bringing the number of known non-arithmetic commensurability classes to 22.
... The key idea of Hirzebruch, which enabled constructing these new ball-quotients, is that one can consider abelian covers of the complex projective plane branched along line configurations. Let us recall briefly how the celebrated construction of Hirzebruch works (for more details please consult for instance [1]). ...
In this note we show that there are no real configurations of lines on the projective plane such that the associated Kummer covers of order are ball-quotients. Moreover, we show that there exists only one configuration of real lines such that the associated Kummer cover of order is a ball-quotient. In the second part we consider the so-called topological -configurations and we show, using Shnurnikov`s inequality, that for there do not exist -configurations and and for there do not exist -configurations.
... the boundary divisor we get the Hirzebruch-Höfer proportionality theorem (K X + S)C + 2C 2 + ρ(C) = 4δ by adjusting the proof in [9] to the non compact case. Therefore this becomes ...
We study the bounded negativity conjecture for non-quaternionic Hilbert
modular surfaces and give an explicit bound for the special case of
Hirzebruch-Zagier curves on Hilbert modular surfaces.
... Whereas we focus in this note on the geometry side, let us briefly explain these motivations. Configurations of lines have been present in algebraic geometry ever since Hirzebruch constructions of examples of surfaces of general type X with Chern classes satisfying c 2 1 (X) = 3c 2 (X) or c 2 1 (X) close to 3c 2 (X), see [17], [2] and [22] for recent results along these lines. Recently configurations of lines have prominently appeared in two new areas of research. ...
In the present work we study moduli spaces of two line point configurations
introduced by B\"or\"oczky. These configurations are extremal from the point of
view of Dirac-Motzkin Conjecture settled recently by Green and Tao. They have
appeared also recently in commutative algebra in connection with the
containment problem for symbolic and ordinary powers of homogeneous ideals and
in algebraic geometry in considerations revolving around the Bounded Negativity
Conjecture. Our main results are Theorem A and Theorem B. We show that the
moduli space of what we call B12 configurations is a three dimensional
rational variety. As a consequence we derive the existence of a three
dimensional family of rational B12 configurations. On the other hand the
moduli space of B15 configurations is shown to be an elliptic curve with only
finitely many rational points, all corresponding to degenerate configurations.
Thus, somewhat surprisingly, we conclude that there are no rational B15
configurations.
... Arrangements of lines were introduced to algebraic geometry by Hirzebruch in his works concerning the geography of surfaces (i.e. construction of surfaces X with prefixed invariants c 2 1 (X) and c 2 (X)), see [6], [1]. Multiplier ideals defined by arrangements of lines were studied by Teitler [10] and Mustat¸˘Mustat¸˘ a [7]. ...
In the present note we study absolute linear Harbourne constants. These are
invariants which were introduced in order to relate the lower bounds on the
selfintersection of negative curves on birationally equivalent surfaces to the
complexity of the birational map between them. We provide various lower and
upper bounds on Harbourne constants and give their values for the number of
lines s of the form for any prime number p and also for all
values of s up to 31. This extends considerably earlier results of the
third author.
... The topic discussed here is often framed as pertaining to arrangements of lines—but arrangements include the edges and faces determined by the set of lines, and these are not relevant in the present context, except for their brief mention in section 4. If a better term can be suggested, we would be happy to replace " aggregate, " which was the best we could come up with. It may be noted that in German the situation is even worse, since there is no accepted translation for " arrangement, " and the term " configuration " is used instead (see, for example, [2]) despite its well-known designation for a special class of families of points and lines. (ii) It is clear that projective transformations applied to any family L of lines apply the same transformations to the set O(L) of omittable lines. ...
Given a collection of n lines in the real projective plane, a line l is said to be omittable if l is free of ordinary points of intersection-in other words, if all the intersection points of l with other lines from the collection come at the intersection of three or more lines. Given a collection of lines L, denoting by O(L) the set of all omittable lines in the collection and by g(L) the cardinality of O(L), we describe three infinite families of lines that can serve as O(L) for suitable L and also display a finite set of sporadic additional examples such that O(L) does not fall into any of the three families. We derive bounds on the size of g(L) in case O(L) falls into one of the three infinite families and weaker bounds for the more general case.
... This is a generalization of the methods described in [BHH,Chapter 1.2] to arbitrary dimension and (algebraically closed) base field. ...
In the present note, we study combinatorial and algebraic properties of cubic-line arrangements in the complex projective plane admitting nodes, ordinary triple and A 5 singular points. We deliver a Hirzebruch-type inequality for such arrangement and study the freeness of such arrangements providing an almost complete classification.
For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizers of (reprentatives of each group orbit of) mirrors of complex reflections. We give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice.
For n≥2, we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps between complex hyperbolic manifolds, which is related to a question of Siu, that certain hyperbolic 3-manifolds cannot be totally geodesic submanifolds of complex hyperbolic manifolds, and that arithmeticity of complex hyperbolic manifolds is detected purely by the topology of the underlying complex variety, which is related to a question of Margulis. Our results also provide some evidence for a conjecture of Klingler that is a broad generalization of the Zilber–Pink conjecture.
We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp.
As already indicated, from the late 1930s onward, algebraic geometry underwent a complete reformulation, when it was realized that the ‘classical’ language was inadequate and that a thorough rewriting in algebraic terms was needed. In Sect. 3.3, we saw how Zariski and Weil considered their own rewriting projects with commutative algebra. However, after the reshaping of algebraic geometry in terms of the language of commutative algebra, in the 1950s and 1960s, algebraic geometry was reshaped again via the language of category theory, schemes and sheaves; the classification of surfaces project was extended to surfaces over fields with characteristic p > 0, and a new classification of surfaces slowly emerged (that of Enriques–Kodaira), extending the existing classification of Castelnuovo and Enriques to non-algebraic compact complex surfaces, now classifying surfaces according to the Kodaira dimension with the help of Chern classes. The research on coverings was generalized: a definition of ramified covers was first given in algebraic terms (see Zariski, Sects. 1.2 and 3.3) and later formulated in terms of schemes (with Grothendieck); the algebraic fundamental group was defined and the relations between this group and the topological fundamental group became clearer and clearer. The ramification (and branch) varieties (or schemes) were defined with the terms of this new language, which did not resemble Zariski’s algebraic definition from his 1958 paper. But one cannot say that these new definitions, results and revolutions reshaped the research on branch curves. Indeed, in 1971, in a new appendix to the second edition of Zariski’s book Algebraic Surfaces, David Mumford notes that the “classification of plane curves C with d nodes and k cusps and the computation of the fundamental group π1(ℂℙ2 − C) has unfortunately not been pursued.” Recalling that the classification of these curves was one of Zariski’s main research themes in the 1920s, one may be tempted to conclude that the new techniques of algebraic geometry prompted stagnation in the research on branch curves, or simply, that the theme of branch curves was no longer considered epistemically, as something from which new research directions could emerge.
We study the geometry of -conic arrangements in the complex projective plane. These are arrangements consisting of smooth conics and they admit certain quasi-homogeneous singularities. We show, to our surprise, that such -conic arrangements are never free. Moreover, we provide combinatorial constraints of the weak combinatorics of such arrangements.
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.
We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those root system hypergeometric functions found by Heckman–Opdam, and in view of the work of Couwenberg–Heckman–Looijenga on the geometric structures on projective arrangement complements, we consider a family of connections on a total space, namely, a -bundle on the complement of a toric mirror arrangement (=finite union of hypertori, determined by a root system). We prove that these connections are torsion free and flat, and hence define a family of affine structures on the total space, which is equivalent to a family of projective structures on the toric arrangement complement. We then determine a parameter region for which the projective structure admits a locally complex hyperbolic metric. In the end, we show that the space in question can be biholomorphically mapped onto a divisor complement of a ball quotient if the Schwarz conditions are invoked.
We produce an example of a rigid, but not infinitesimally rigid smooth compact complex surface with ample canonical bundle using results about arrangements of lines inspired by work of Hirzebruch, Kapovich & Millson, Manetti and Vakil.
In diesem Kapitel stelle ich sehr einfach zu formulierende offene mathematische Probleme zu Punkt-Geraden-Konfigurationen in der Ebene vor. Ich starte mit einer Anwendung: Vor den Wahlen eines Präsidenten in Slowenien gab es acht Kandidaten. Der nationale Fernsehsender beabsichtigte, an acht Tagen je eine Fernsehdebatte mit jeweils drei Kandidaten zu senden. Jeder Kandidat bekam drei Termine. Es sollten nie zwei Kandidaten mehrfach zusammentreffen. Wir deuten die n = 8 Kandidaten als Punkte und die n = 8 Fernsehdebatten als Geraden. Jede Gerade enthält k = 3 Punkte, und durch jeden Punkt gehen k = 3 Geraden. Wir sprechen dann von einer (nk)-Punkt-Geraden-Konfiguration. Diese Punkt-Geraden-Konfigurationen wurden in den letzten 30 Jahren erneut in der mathematischen Forschung aufgegriffen. Im Fall k = 4 wurde für alle natürlichen Zahlen n geklärt, ob es eine entsprechende Punkt-Geraden-Konfiguration gibt, nur der Fall n = 23 blieb auf überraschende Weise bisher offen.
The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian.
Its automorphism group is the symmetric group . We give canonical explicit -invariant Pfaffian equations through a 66 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of . Finally, we give -invariant equations for the embedding of Y inside , and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.
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.
Let X be a smooth projective surface and let be an arrangement of curves on X. The Harbourne constant of was defined as a way to investigate the occurrence of curves of negative self-intersection on blow ups of X. This is related to the bounded negativity conjecture which predicts that the self-intersection number of all reduced curves on a surface is bounded below by a constant. We consider a geometrically ruled surface X over a smooth curve and give lower bounds for the Harbourne constants of transversal arrangements of curves on X. We also define a global Harbourne constant as the infimum of Harbourne constants for arrangements of a specific type and give a lower bound for it.
We prove the equisingular rigidity of the singular Hirzebruch–Kummer coverings X(n, ) of the projective plane branched on line configurations , satisfying some technical condition. In the case, the complete quadrangle, we give explicit equations of the Hirzebruch–Kummer covering the minimal desingularisation of in a product of four Fermat curves of degree n. Since is the covering of the Del Pezzo surface of degree 5 branched on the 10 lines, these equations are derived from explicit equations of the image of in .We describe more generally determinantal equations for all Del Pezzo surfaces of degree as subvarieties of the k-fold product of the projective line.
Exploring the classical Ceva configuration in a Desarguesian projective plane, we construct two families of minimal blocking sets as well as a new family of blocking semiovals in PG(2, 32h). Also, we show that these blocking sets of PG(2, q²), regarded as pointsets of the derived André plane (Formula presented.), are still minimal blocking sets in (Formula presented.). Furthermore, we prove that the new family of blocking semiovals in PG(2, 32h) gives rise to a family of blocking semiovals in the André plane (Formula presented.) as well.
We define Picard-Einstein metrics on complex algebraic surfaces as Kähler-Einstein metrics with negative constant sectional curvature pushed down from the complex unit ball allowing degenerations along cycles. We demonstrate how the tool of orbital heights, especially the Proportionality Theorem presented in [H98], works for detecting such orbital cycles on the projective plane. The simplest cycle we found on this way is supported by a quadric and three tangent lines (Apollonius configuration) with at most 3 cusp points sitting on the double points of the configuration. We determine precisely the uniformizing ball lattices in the case of 3, 2, 1 or 0 cusp(s) respectively. The corresponding orbital planes are (leveled) Shimura surfaces corresponding to Jacobian varieties of certain families of plane genus 3, 6, 5 or 13 genus respectively. We present many examples of plane orbital surfaces with quadrics, and determine for them precisely the uniformizing ball lattices. By the way we check that %precisely these two cases are some of them are Galois quotients of celebrated 27 orbital planes with line arrangements occurring in the PTDM-list (Picard-Terada-Mostow-Deligne) which we will call also BHH-list (Barthel-Hirzebruch-Höfer) because it is most convenient to get it from [BHH]. The others are quotients of Mostow's [M2] half-integral arrangements. Proofs are based on the Proportionality Theorem and classification results for hermitian lattices and algebraic surfaces.
We present some simple examples of smooth projective varieties in positive characteristic, arising from linear algebra, which do not admit a lifting neither to characteristic zero, nor to the ring of Witt vectors of length 2. Our first construction is the blow-up of the graph of the Frobenius morphism of a homogeneous space. The second example is a blow-up of ℙ³ in a ‘purely characteristic-p’ configuration of points and lines.
Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality \cite[Theorem 0.1]{Langer} we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.
This article is devoted to examples of (orbifold) K\"ahler groups from the perspective of the so-called Shafarevich conjecture on holomorphic convexity. It aims at pointing out that every quasi-projective complex manifold with an 'interesting' fundamental group gives rise to interesting instances of this long-standing open question. Complements of line arrangements are one of the better known classes of quasi-projective complex surfaces with an interesting fundamental group. We solve the corresponding instance of the Shafarevich conjecture partially giving a proof that the universal covering surface of a Hirzebruch's covering surface with equal weights is holomorphically convex. The final section reduces the Shafarevich conjecture to a question related to the Serre problem.
Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. The paper contains some new results but is mostly a survey paper, dealing with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (new results obtained with Ingrid Bauer) projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance the BCDH surfaces, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces.
In this paper we come back to a problem proposed by F. Hirzebruch in the 1980's, namely whether there exists a configuration of smooth conics in the complex projective plane such that the associated desingularization of the Kummer extension is a ball quotient. We extend our considerations to the so-called d-configurations of curves on the projective plane and we show that in most cases for a given configuration the associated desingularization of the Kummer extension is not a ball quotient. Moreover, we provide improved versions of Hirzebruch-type inequality for d-configurations. Finally, we show that the so-called characteristic numbers (or numbers) for d-configurations are bounded from above by 8/3. At the end of the paper we give some examples of surfaces constructed via Kummer extensions branched along conic configurations.
This article investigates the subject of rigid compact complex manifolds. First of all we investigate the different notions of rigidity (local rigidity, global rigidity, infinitesimal rigidity, etale rigidity and strong rigidity) and the relations among them. Only for curves these notions coincide and the only rigid curve is the projective line. For surfaces we prove that a rigid surface which is not minimal of general type is either a Del Pezzo surface of degree >= 5 or an Inoue surface. We give examples of rigid manifolds of dimension n >= 3 and Kodaira dimensions 0, and 2 <=k <= n. Our main theorem is that the Hirzebruch Kummer coverings of exponent n >= 4 branched on a complete quadrangle are infinitesimally rigid. Moreover, we pose a number of questions.
A line arrangement of 3n lines in satisfies Hirzebruch property if each line intersect others in n+1 points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in is real, confirming that there exist exactly four such arrangements.
In a previous paper we characterized those H ( Λ , n ) H(\Lambda ,n) (compact complex surfaces constructed by Hirzebruch) that have nef cotangent bundle. In this article we extend the methods to study more general branched coverings with regard to nefness of their cotangent bundles.
Let ΓΚ = double-struck U sign ((2, 1), D fraktur signΚ) be the full Picard modular group of the imaginary quadratic number field Κ. For all natural congruence subgroups ΓΚ (m), m ≥ 3, acting freely on the two-dimensional complex unit ball, we prove an explicit polynomial formula for the dimensions of spaces of cusp forms of weight n ≥ 2. The coefficients of these polynomials in the natural variables m, n are expressed by higher third and first Bernoulli numbers of the Dirichlet character χΚ of Κ and by values of Euler factors of the Riemann Zeta function and such factors of the L-series of χΚ at 2 or 3, respectively. The proof is based on detailed knowledge about classification of Picard modular surfaces. It combines algebraic geometric methods (Riemann-Roch, Vanishing- and Proportionality Theorem, curvature, structure of algebraic groups) with modern and classical number theoretic ones (representation densities, Tamagawa measure, strong approximation, functional equation for L-series).
Keywords Some Examples Combinatorics Divisor Complement Ball Quotients Logarithmic Forms Hypergeometric Integrals See also References
A geometric 4-configuration is a collection of points and straight lines with the property that every point lies on exactly four lines in the collection and every line passes through exactly four points in the collection. This paper describes a method for constructing a large number of new infinite families of rotationally symmetric geometric 4-configurations which are movable; that is, there is at least one continuous parameter which preserves the symmetry of the configuration. In fact, the configurations in this paper have 2q continuous parameters for any integer (Formula presented.); previously the known classes of movable 4-configurations had only one or two degrees of freedom. The construction is extended to produce movable 4-configurations with dihedral symmetry. The paper ends with a number of open questions.
Let be an algebraically closed field of characteristic ,
and let C be a nonsingular projective curve over . We prove that
for any real number , there are minimal surfaces of general type X
over such that
a) ,
b) ,
c) and is arbitrarily close to x.
In particular, we show density of Chern slopes in the pathological
Bogomolov-Miyaoka-Yau interval for any given p. Moreover, we
prove that for there exist surfaces X as above with
, this is, with Picard scheme equal to a reduced point.
In this way, we show that even surfaces with reduced Picard scheme are densely
persistent in for any given p.
Using the Miyaoka-Sakai inequality for curves with normal crossings, we find
lower bounds for H-constants of curves involving their geometric genus and
the number of their singularities. Similarly to the linear H-constant
introduced recently, we study the elliptic H-constants of
and Abelian surfaces and we find some lower bounds. We exhibit configurations
of smooth plane cubic curves whose H-constant is arbitrarily close to .
As a Corollary, we obtain that the global H-constant of any surface X is
less or equal to . Related to these problems, we moreover give a new
inequality for the number and type of singularities of elliptic curves
arrangements on Abelian surfaces, inequality which has a close similarity to
the one of Hirzebruch for arrangements of lines in the plane.
Algebraic surfaces with many nondegenerate singularities can be constructed with the help of a class of bivariate polynomials with complex coefficients, associated with the affine Weyl group of the root system A 2. Real variants of the polynomials are related to certain simple arrangements of real lines in the plane. In the study of its critical points, additional arrangements appear, which can be used to generate other singular surfaces. The existence of a high number of singularities in the associated surfaces is due to the fact that the polynomials based on the arrangements have many critical points with few critical values. Surfaces with singularities of types A 2, A 3n + 1, and D 4 are constructed.
We call a complex (quasiprojective) surface of hyperbolic type, fiff-after removing finitely many points and/or curves-the universal cover is the complex two-dimensional unit ball. We characterize abelian surfaces which have a birational transform of hyperbolic type by the existence of a reduced divisor with only elliptic curve components and maximal sin-gularity rate (equal to 4). We discover a Picard modular surface of Gauß numbers of bielliptic type connected with the rational cuboid problem.
ResearchGate has not been able to resolve any references for this publication.