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This thesis consists of four papers, referred to in the text as Paper I, II, III and IV, respectively. In Paper I and II, we study the classical Hasse invariant on the geometric special fiber of several Hodge and abelian-type Shimura varieties. We explicitly compute the order of vanishing of the Hasse invariant and describe it in terms of the Bruhat and Ekedahl-Oort stratifications. We compute the conjugate line position (referred to as the a-number by van der Geer and Katsura, respectively Oort) and thus show that it agrees with the order of the Hasse invariant. This equality was obtained by Ogus for certain families of Calabi-Yau varieties mod p, and we thus dub it as Ogus' principle. We give a group-theoretical generalization of this principle via the theory of G-zips, and argue that this theory is a suitable framework within which to understand the principle. Paper III and IV concern (moduli of) G-bundles over smooth projective curves over an algebraically closed field, for G a reductive group. In Paper III we introduce the notion of essentially finite (EF) G-bundles, which generalizes the notion of EF vector bundles, studied initially by Weil and more generally by Nori. We state some elementary characterizations of such bundles, and we prove that they are semistable of torsion degree. Let MGef denote the subset of EF G-bundles in the moduli space of degree 0 semistable G-bundles. We show that in characteristic 0, MGef is not dense if G has semisimple rank 1 and the curve has genus g ≥ 2. This is in contrast to the case of line bundles in arbitrary characteristic, and vector bundles of arbitrary rank in positive characteristic. In Paper IV we first study the moduli stack BunG of G-bundles for nonconnected reductive group schemes G over the curves. We prove that the semistable locus of BunG admits a projective good moduli space. To this end we prove a ``decomposition theorem'' stating that BunG is a finite disjoint union of substacks Xi, each of which admits a

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E, V ), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter a. We study the geometry of the moduli space of coherent systems for different values of a when k ≤ n and the variation of the moduli spaces when we vary a. As a consequence, for sufficiently large , we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case k = n − 1 explicitly, and give the Poincare polynomials for the case k = n − 2. In an appendix, we describe the geometry of the “flips” which take place at critical values of a in the simplest case, and include a proof of the existence of universal families of coherent systems when GCD(n, d, k)= 1., EAGER, EDGE, European Scientific Exchange Programme (Royal Society of London ), Consejo Superior de Investigaciones Científicas, National Science Foundation, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

©2005 Springer Science+Business Media, Inc. One of the authors (L. M. A.) thanks the members of the Physics Department of Lecce University for their warm hospitality. This work was supported in part by the DGCYT (Project No. BFM2002-01607), COFIN (Grant 2002 “Sintesi”), and the National Science Foundation (Grant No. DMS-0404931)., We present a general scheme for determining and studying integrable deformations of algebraic curves, based on the use of Lenard relations. We emphasize the use of several types of dynamical variables: branches, power sums, and potentials., DGCYT, COFIN, National Science Foundation, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

GHIGI, ALESSANDRO CALLISTO, open, This thesis deals with totally geodesic subvarieties of the moduli space A_g of principally polarized abelian varieties and their relation with the Torelli locus. This is the closure in A_g of the image of the moduli space M_g of smooth, complex algebraic curves of genus g via the Torelli map j: M_g-->A_g. The moduli space A_g is a quotient of the Siegel space, which is a Riemannian symmetric space. An algebraic subvariety of A_g is totally geodesic if it is the image, under the natural projection map, of some totally geodesic submanifold of the Siegel space. Geometric considerations lead to the expectation that j(M_g) should contain very few totally geodesic subvarieties of A_g. This expectation also agrees with the Coleman-Oort conjecture. The differential geometry of symmetric spaces is described through Lie theory. In particular, totally geodesic submanifolds can be characterized via Lie algebras. This motivates the discussion carried out in this thesis, in which we use some Lie-theoretic tools to investigate geometric aspects of the inclusion of j(M_g) in A_g. The main results presented are the following. In Chapter 2, we consider the pull-back of the Lie bracket operation on the tangent space of A_g via the Torelli map, and we characterize it in terms of the geometry of the curve. We use the Bergman kernel form associated with the curve. Also, we link the Bergman kernel form to the second fundamental form of the Torelli map. In Chapter 3, we determine which symmetric space uniformizes each of the known counterexamples to the Coleman-Oort conjecture via the computation of the associated Lie algebra decomposition. These known examples were obtained studying families of Galois coverings of curves. Chapter 4 focuses on these families for their own sake, and we describe a new topological construction of families of G-coverings of the line., Oggetto di questa tesi sono le sottovarietà totalmente geodetiche dello spazio dei moduli A_g di varietà abeliane principalmente polarizzate e la loro relazione con il luogo di Torelli. Questo è definito come la chiusura in A_g dell'immagine dello spazio dei moduli M_g di curve algebriche complesse lisce di genere g tramite la mappa di Torelli j: M_g-->A_g. Lo spazio dei moduli A_g è un quoziente dello spazio di Siegel, che è uno spazio simmetrico. Una sottovarietà algebrica di A_g è totalmente geodetica se è l'immagine, tramite la naturale mappa di proiezione, di una qualche sottovarietà totalmente geodetica dello spazio di Siegel. Ci si aspetta che j(M_g) contenga poche sottovarietà totalmente geodetiche di A_g. Questo è anche in accordo con la congettura di Coleman-Oort. La geometria differenziale degli spazi simmetrici si può descrivere attraverso la teoria di gruppi e algebre di Lie. In particolare, le sottovarietà totalmente geodetiche di spazi simmetrici possono essere caratterizzate in termini di algebre di Lie. Queste considerazioni sono alla base della trattazione svolta in questa tesi, in cui utilizziamo alcuni strumenti della teoria di Lie per indagare alcuni aspetti geometrici dell'inclusione di j(M_g) in A_g. I principali risultati presentati sono i seguenti. Nel Capitolo 2, consideriamo il pull-back dell'operazione di Lie-bracket sullo spazio tangente ad A_g tramite la mappa di Torelli e lo caratterizziamo in termini della geometria della curva. Per farlo usiamo il nucleo di Bergman associato alla curva. Inoltre, colleghiamo il nucleo di Bergman alla seconda forma fondamentale della mappa Torelli. Nel Capitolo 3, determiniamo quale spazio simmetrico uniforma ciascuno dei controesempi noti alla congettura di Coleman-Oort attraverso il calcolo della decomposizione dell'algebra di Lie associata. Questi esempi noti erano stati ottenuti studiando famiglie di rivestimenti di Galois. Nel capitolo 4 ci concentriamo sullo studio di queste famiglie e descriviam, 0, open, Tamborini, C

A method for computing limits of quotients of real analytic functions in two variables was developed in [4]. In this article we generalize the results obtained in that paper to the case of quotients q = f(x, y, z)/g(x, y, z) of polynomial functions in three variables with rational coefficients. The main idea consists in examining the behavior of the function q along certain real variety X(q) (the discriminant variety associated to q). The original problem is then solved by reducing to the case of functions of two variables. The inductive step is provided by the key fact that any algebraic curve is birationally equivalent to a plane curve. Our main result is summarized in Theorem 2. In Section 4 we describe an effective method for computing such limits. We provide a high level description of an algorithm that generalizes the one developed in [4], now available in Maple as the limit/multi command.

In this note we present explicit examples of maximal and minimal curves over finite fields in odd characteristic. The curves are of Artin-Schreier type and the construction is closely related to quadratic forms from F-qn to F-q., Ministry for Education, University and Research of Italy (MIUR) (Project PRIN 2012 Geometrie di Galois e strutture di incidenza)Ministry of Education, Universities and Research (MIUR) [2012XZE22K_005]; Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM)Istituto Nazionale di Alta Matematica (INDAM); Science Foundation IrelandScience Foundation IrelandEuropean Commission [13/IA/1914], Research of the first author partially supported by Ministry for Education, University and Research of Italy (MIUR) (Project PRIN 2012 Geometrie di Galois e strutture di incidenza-Prot. N.2012XZE22K_005) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM). Research of the fourth author supported by Science Foundation Ireland Grant 13/IA/1914. The authors would like to thank the anonymous reviewer for his/her useful comments.

In this note we present explicit examples of maximal and minimal curves over finite fields in odd characteristic. The curves are of Artin-Schreier type and the construction is closely related to quadratic forms from F-qn to F-q., Ministry for Education, University and Research of Italy (MIUR) (Project PRIN 2012 Geometrie di Galois e strutture di incidenza)Ministry of Education, Universities and Research (MIUR) [2012XZE22K_005]; Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM)Istituto Nazionale di Alta Matematica (INDAM); Science Foundation IrelandScience Foundation IrelandEuropean Commission [13/IA/1914], Research of the first author partially supported by Ministry for Education, University and Research of Italy (MIUR) (Project PRIN 2012 Geometrie di Galois e strutture di incidenza-Prot. N.2012XZE22K_005) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM). Research of the fourth author supported by Science Foundation Ireland Grant 13/IA/1914. The authors would like to thank the anonymous reviewer for his/her useful comments.

[EN] We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.

In this paper, explicit equations for algebraic curves with genus 4, 5, and 10 already studied in characteristic zero, are analyzed in positive characteristic p. We show that these curves have an interesting behaviour on the number of their rational places. Namely, they are either maximal or minimal over the finite field with p2 elements for infinitely many p's. The key tool is the investigation of their Jacobian decomposition. Lists of small p's for which maximality holds are provided. In some cases we also describe the automorphism group of the curve.

Let X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of odd characteristic p ≥ 0, and let Aut(X) be the group of all automorphisms of X which fix K element-wise. For any a subgroup G of Aut(X) whose order is a power of an odd prime d other than p, the bound proven by Zomorrodian for Riemann surfaces is |G| ≤ 9(g − 1) where the extremal case can only be obtained for d = 3 and g ≥ 10. We prove Zomorrodian’s result for any K. The essential part of our paper is devoted to extremal 3-Zomorrodian curves X. Two cases are distinguished according as the quotient curve X/Z for a central subgroup Z of Aut(X) of order 3 is either elliptic, or not. For elliptic type extremal 3-Zomorrodian curves X, we completely determine the two possibilities for the abstract structure of G using deeper results on finite 3-groups. We also show infinite families of extremal 3-Zomorrodian curves for both types, of elliptic or non-elliptic. Our paper does not adapt methods from the theory of Riemann surfaces, nevertheless it sheds a new light on the connection between Riemann surfaces and their automorphism groups.

Let XX be a (projective, non-singular, geometrically irreducible) curve of even genus g(X)≥2g(X)≥2 defined over an algebraically closed field KK of odd characteristic pp. If the pp-rank γ(X)γ(X) equals g(X)g(X), then XX is \emph{ordinary}. In this paper, we deal with \emph{large} automorphism groups GG of ordinary curves of even genus. We prove that |G|<821.37g(X)7/4|G|<821.37g(X)7/4. The proof of our result is based on the classification of automorphism groups of curves of even genus in positive characteristic, see \cite{giulietti-korchmaros-2017}. According to this classification, for the exceptional cases Aut(X)≅Alt7Aut(X)≅Alt7 and Aut(X)≅M11Aut(X)≅M11 we show that the classical Hurwitz bound |Aut(X)|<84(g(X)−1)|Aut(X)|<84(g(X)−1) holds, unless p=3p=3, g(X)=26g(X)=26 and Aut(X)≅M11Aut(X)≅M11; an example for the latter case being given by the modular curve X(11)X(11) in characteristic 33.

Diese Arbeit widmet sich der Untersuchung von zwei Problemen der abzählenden Geometrie im Zusammenhang mit linearen Systemen auf algebraischen Kurven. Das erste Problem besteht darin, die Frage der Gültigkeit der Jonquières-Formeln zu klären. Diese Formeln berechnen die Anzahl von Divisoren mit vorgeschriebener Multiplizität, genannt de Jonquières-Divisoren, die in einem linearen System auf einer glatten projektiven Kurve enthalten sind. Um dies zu tun, konstruieren wir den Raum der de Jonquières-Divisoren als einen Determinantenzyklus des symmetrischen Produkts der Kurve und beweisen, dass er für eine allgemeine Kurve die erwartete Dimension hat. Dabei beschreiben wir die Degenerationen der Jonquières-Divisoren zu den Knotenkurven sowohl mit linearen Systemen als auch mit kompaktifizierten Picard-Schemata. Das zweite Problem behandelt Zyklen von Untergeordneten-, oder allgemeiner, Sekanten-Divisoren zu einem gegebenen linearen System auf einer Kurve. Wir betrachten den Durchschnitt zweier solcher Zyklen, die Sekanten-Divisoren von zwei verschiedenen linearen Systemen auf der gleichen Kurve entsprechen, und untersuchen die Gültigkeit der enumerativen Formeln, die die Anzahl der Teiler im Durchschnitt zählen. Wir untersuchen einige interessante Fälle mit unerwarteten Transversalitätseigenschaften und etablieren eine allgemeine Methode, um zu überprüfen, wann dieser Durchschnitt leer ist., This thesis is dedicated to the study of two enumerative geometry problems in the context of linear series on algebraic curves. The first problem is that of settling the issue of the validity of the de Jonquières formulas. These formulas compute the number of divisors with prescribed multiplicity, or de Jonquières divisors, contained in a linear series on a smooth projective curve. To do so, we construct the space of de Jonquières divisors as a determinantal cycle of the symmetric product of the curve and prove that, for a general curve with a general linear series, it is of expected dimension. In doing so, we describe the degenerations of de Jonquières divisors to nodal curves using both limit linear series and compactified Picard schemes. The second problem deals with cycles of subordinate or, more generally, secant divisors to a given linear series on a curve. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.

Let chi be a smooth, geometrically irreducible and projective curve over a finite field F-q of odd characteristic. The L-polynomial L-chi(t) of chi determines the number of rational points of chi not only over F-q but also over F-qs for any integer s >= 1. In this paper we determine L-polynomials of a class of such curves over F-q.

Let q be a positive power of a prime number. For arbitrary positive integers h, n, m with n dividing m and arbitrary gamma, alpha is an element of F-qm with gamma not equal 0 the number of F-qm - rational points of the curve y(qn) - y = gamma x(qh+1) - alpha is determined in many cases (Ozbudak and Saygi, in: Larcher et al. (eds.) Applied algebra and number theory, 2014) with odd q. In this paper we complete some of the remaining cases for odd q and we also present analogous results for even q.

In this work we present explicit classes of maximal and minimal Artin-Schreier type curves over finite fields having odd characteristics. Our results include the proof of Conjecture 5.9 given in [1] as a very special subcase. We use some techniques developed in [2], which were not used in [1]. (C) 2016 Elsevier Inc. All rights reserved.

In this thesis we state and give elementary proofs for some fundamental results about intersections of algebraic curves, namely Bezout's, Max Noether's, Pappus's, Pascal's and Chasles' theorems. Our main tools are linear algebra and basic ring theory. We conclude the thesis by applying the results to elliptic curves., I denna kandidatuppsats visas vissa grundläggande satser inom algebraisk geometri. Satserna tillämpas slutligen på elliptiska kurvor, vilka är intressenta, bland annat för deras tillämplighet inom talteori, och speciellt kryptografi. Bezouts, Max Noethers, Pappus, Pascals och Chasles satser, vilka visas här, har varit kända länge. Det äldsta resultatet, Pappus sats, visades nämligen redan på 300-talet e. Kr., visserligen utan de metoder som används här. Cirka 1500 år senare visade Max Noether sin fundamentalsats, vilket är det nyaste resultatet som visas här. Trots åldern är resultaten oumbärliga verktyg för att verkligen förstå de moderna tillämpningarna inom elliptiska kurvor. Även om viktiga resultat, som bland annat Lenstras algoritm för snabb primtalsfaktorisering av stora heltal, och asymmetrisk kryptering med hjälp av elliptiska kurvor, inte visas här, så har dessa tillämpningar gemensamt att de bygger på att man kan definiera addition på elliptiska kurvor, vilket visas rigoröst i denna uppsats. Metoderna som används är elementära och kräver endast förkunskaper inom linjär algebra och grundläggande ringteori. Bevisen kommer från böcker (se referenslistan) i algebraisk geometri, men de som hittas i dessa är mycket mer kortfattade, än de som presenteras här. Detta arbete fyller i detaljerna i resonemangen, för att göra materialet tillgängligt för människor med grundläggande matematisk skolning.

Let p denote the characteristic of , the finite field with q elements. We prove that if q is odd then an arc of size in the projective plane over , which is not contained in a conic, is contained in the intersection of two curves, which do not share a common component, and have degree at most , provided a certain technical condition on t is satisfied. This implies that if q is odd then an arc of size at least is contained in a conic if q is square and an arc of size at least is contained in a conic if q is prime. This is of particular interest in the case that q is an odd square, since then there are examples of arcs, not contained in a conic, of size , and it has long been conjectured that if is an odd square then any larger arc is contained in a conic. These bounds improve on previously known bounds when q is an odd square and for primes less than 1783. The previously known bounds, obtained by Segre [26], Hirschfeld and Korchmáros [17][18], and Voloch [32][33], rely on results on the number of points on algebraic curves over finite fields, in particular the Hasse–Weil theorem and the Stöhr–Voloch theorem, and are based on Segre's idea to associate an algebraic curve in the dual plane containing the tangents to an arc. In this paper we do not rely on such theorems, but use a new approach starting from a scaled coordinate-free version of Segre's lemma of tangents. Arcs in the projective plane over of size q and , q odd, were classified by Segre [25] in 1955. In this article, we complete the classification of arcs of size and . The main theorem also verifies the MDS conjecture for a wider range of dimensions in the case that q is an odd square. The MDS conjecture states that if , a k-dimensional linear MDS code has length at most . Here, we verify the conjecture for , in the case that q is an odd square., Postprint (author's final draft)

For a class of non-hyperelliptic genus 3 curves C which are twofold coverings of elliptic curves E, we give an explicit algebraic description of all birationally nonequivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the Prym varieties associated with such coverings. Our description is based on previous studies of Prym varieties with polarization (1,2) in connection with separation of variables in a series of classical and new algebraic integrable systems linearized on such varieties. We also consider some special cases of the covering C ¿ E, in particular, when the corresponding Prym varieties contain pairs of elliptic curves and the Jacobian of C is isogeneous (but not isomorphic) to the product of three different elliptic curves. Our description is accompanied with explicit algorithms of calculation of periods of the Prym varieties and of absolute invariants of genus 2 curves. They are followed by numerical examples, which experimentally confirm main results of the articl, Peer Reviewed, Postprint (published version)

The Hasse–Weil bound is a deep result in mathematics and has found wide applications in mathematics, theoretical computer science, information theory etc. In general, the bound is tight and cannot be improved. However

Diese Dissertation hat zwei Teile. Im ersten Teil untersuchen wir die Modulräume von Kurven mit multiplen Spinstrukturen. Wir stellen eine neue Kompaktifizierung dieser Räume mit geometrisch sinnvollem Grenzverhalten vor. Die irreduziblen Komponenten dieser Räume werden vollstandig klassifiziert. Die Ergebnisse aus diesem ersten Teil der Dissertation sind fundamental für die Degenerationstechniken im zweiten Teil. Im zweiten Teil untersuchen wir eine Reihe von Problemen, die von der klassischen Geometrie inspiriert werden. Unser Hauptaugenmerk liegt hierbei auf dem Fall von zwei Hyperebenen, die eine kanonische Kurve in jedem Schnittpunkt tangential berühren. Wir fragen, ob eingemensamer Tangentialpunk existieren kann. Unsere Analyse zeigt, dass so ein gemeinsamer Punkt nur in Kodimension 1 im Modulraum existieren kann. Wir berechen dann weiter die Klasse dieses Divisors. Insbesonders zeigen wir, dass diese Klasse eine hinreichend kleine Steigung hat, sodass die kanonischen Klassen von Modulräumen von Kurven mit zwei ungeraden Spinstrukturen gross ist, wenn der Genus grösser ist als neun. Falls die zugehörigen groben Modulräume gutartige Singularitäten haben, dann haben sie in diesem Intervall maximale Kodaria Dimension., This thesis has two parts. In Part I we consider the moduli spaces of curves with multiple spin structures and provide a compactification using geometrically meaningful limiting objects. We later give a complete classification of the irreducible components of these spaces. The moduli spaces built in this part provide the basis for the degeneration techniques required in the second part. In the second part we consider a series of problems inspired by projective geometry. Given two hyperplanes tangential to a canonical curve at every point of intersection, we ask if there can be a common point of tangency. We show that such a common point can appear only in codimension 1 in moduli and proceed to compute the class of this divisor. We then study the general properties of curves in this divisor. Our divisor class has small enough slope to imply that the canonical class of the moduli space of curves with two odd spin structures is big when the genus is greater than 9. If the corresponding coarse moduli spaces have mild enough singularities, then they have maximal Kodaira dimension in this range.

How many rational points are there on a random algebraic curve of large genus g over a given finite field Fq? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q+1+1/(q-1). We prove a weaker version of this statement in which g and q tend to infinity, with q much larger than g.

How many rational points are there on a random algebraic curve of large genus g over a given finite field Fq? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q+1+1/(q-1). We prove a weaker version of this statement in which g and q tend to infinity, with q much larger than g.

The main objects of this thesis are branched coverings obtained as projection from a point in P^2. Our general goal is to understand how a given meromorphic function f: X -> P^1 can be induced from a composition X --> C -> P^1, where C is a plane curve in P^2 which is birationally equivalent to the smooth curve X. In particular, we want to characterize meromorphic functions on plane curves which are obtained in such a way. For instance, we want to describe the relations on branching points of projections of plane projective curves of degree d and enumerate such functions. To this end, in a series of two papers, we show that any degree d meromorphic function on a smooth projective plane curve C of degree d > 4 is isomorphic to a linear projection from a point p belonging to P^2 \ C to P^1. Secondly, we introduce a planarity filtration of the small Hurwitz space using the minimal degree of a plane curve such that a given meromorphic function can be fit into a composition X --> C -> P^1. Finally, we also introduce the notion of plane Hurwitz numbers in this thesis., Boris Shapiro

Diese Dissertation untersucht Brill-Noether-Theorie der algebraischen Kurven, unter besonderer Berücksichtigung von Kurven auf K3-Flächen und Del-Pezzo-Flächen. In Kapitel 2 studieren wir den Gieseker-Petri-Ort GP_g im Modulraum M_g der glatten irreduziblen Kurven vom Geschlecht g. Dieser Ort wird definiert durch Kurven mit einer Brill-Noether-Varietät G^r_d(C), die singulär ist oder deren Dimension größer als erwartet ist. Der Satz von Gieseker-Petri impliziert, dass GP_g mindestens Kodimension 1 hat, und es wurde vermutet, dass er von reiner Kodimension 1 ist. Wir beweisen diese Vermutung für Geschlecht höchstens 13. Dies wird dadurch ermöglicht, dass man für kleine Geschlechter die Dimension der irreduziblen Komponenten von GP_g mittels "ad hoc"-Beweisführungen untersuchen kann. Lazarsfelds Beweis des Gieseker-Petri-Theorems mittels Kurven auf allgemeninen K3-Flächen deutet darauf hin, dass die Brill-Noether-Theorie von K3-Schnitten wichtig ist, um den Gieseker-Petri-Ort besser zu verstehen. Linearscharen von Kurven, die auf K3-Flächen liegen, stehen in tiefgehender Beziehung zu sogenannten Lazarsfeld-Mukai-Vektorbündeln. In Kapitel 3 untersuchen wir die Stabilität der Lazarsfeld-Mukai-Vektorbündel vom Rang 3 auf einer K3-Fläche S, und wir zeigen, dass sie viele Informationen über Netze vom Typ g^2_d auf Kurven in S enthalten. Wenn d größ genug ist, erhalten wir eine obere Schranke für die Dimension der Varietät G^2_d(C). Wenn die Brill-Noether-Zahl negativ ist, beweisen wir, dass jedes g^2_d in einer von einem Geradenbündel induzierten Linearschar enthalten ist, wie von Donagi und Morrison vermutet wurde. Kapitel 4 befasst sich mit Syzygien einer gegebenen Kurve C, die auf einer Del-Pezzo-Fläche liegt. Wir insbesondere, dass C die Greens Vermutung erfüllt, die impliziert, dass die Existenz gewisser spezieller Linearscharen auf C von den Gleichungen ihrer kanonischen Einbettung abgelesen werden kann., We investigate Brill-Noether theory of algebraic curves, with special emphasis on curves lying on $K3$ surfaces and Del Pezzo surfaces. In Chapter 2, we study the Gieseker-Petri locus GP_g inside the moduli space M_g of smooth, irreducible curves of genus g. This consists, by definition, of curves [C] in M_g such that for some r, d the Brill-Noether variety G^r_d(C), which parametrizes linear series of type g^r_d on C, either is singular or has some components exceeding the expected dimension. The Gieseker-Petri Theorem implies that GP_g has codimension at least 1 in M_g and it has been conjectured that it has pure codimension 1. We prove this conjecture up to genus 13; this is possible since, when the genus is low enough, one is able to determine the irreducible components of GP_g and to study their codimension by "ad hoc" arguments. Lazarsfeld''s proof of the Gieseker-Petri-Theorem by specialization to curves lying on general K3 surfaces suggests the importance of the Brill-Noether theory of K3-sections for a better understanding of the Gieseker-Petri locus. Linear series on curves lying on a K3 surface are deeply related to the so-called Lazarsfeld-Mukai bundles. In Chapter 3, we study the stability of rank-3 Lazarsfeld-Mukai bundles on a K3 surface S, and show it encodes much information about nets of type g^2_d on curves C contained in S. When d is large enough and C is general in its linear system, we obtain a dimensional statement for the variety G^2_d(C). If the Brill-Noether number is negative, we prove that any g^2_d is contained in a linear series which is induced from a line bundle on S, as conjectured by Donagi and Morrison. Chapter 4 concerns syzygies of any given curve C lying on a Del Pezzo surface S. In particular, we prove that C satisfies Green''s Conjecture, which implies that the existence of some special linear series on C can be read off the equations of its canonical embedding.

Let $C$ be an algebraic curve of genus $g\ge2$. A coherent system on $C$ consists of a pair $(E,V)$, where $E$ is an algebraic vector bundle over $C$ of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of the space of sections of $E$. The stability of the coherent system depends on a parameter $\alpha$. We study the geometry of the moduli space of coherent systems for $0

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.

Let X be a curve of genus g. A coherent system on X consists of a pair (E; V ), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter a. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyze the cases k = 1; 2; 3 and n = 2 explicitly. For small values of , the moduli spaces of coherent systems are related to the Brill-Noether loci, the subschemes of the moduli spaces of stable bundles consisting of those bundles with at least a prescribed number of independent sections. The study of coherent systems is applied to nd the dimension, prove the irreducibility, and in some cases calculate the Picard groups of the Brill{Noether loci with k < 3., EAGER, EDGE, Acciones Integradas Programme, National Science Foundation, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We define Picard-Einstein metrics on complex algebraic surfaces as Kähler-Einstein metrics with negative constant sectional curvature pushed down from the unit ball via Picard modular groups 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). We give a complete proof for the fact that it belongs to the congruence subgroup of level 1+i of the full Picard modular group of Gauß numbers together with precise octahedral- symmetric interpretation as moduli space of an explicit Shimura family of curves of genus 3. Proofs are based only on the Proportionality Theorem and classification results for hermitian lattices and algebraic surfaces.

We define Picard_Einstein metrics on complex algebraic surfaces as Kähler-Einstein metrics with negative constant sectional curvature pushed down from the unit ball via Picard modular groups 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). We give a complete proof for the fact that it belongs to the congruence subgroup of level 1 + i of the full Picard modular group of Gauß numbers together with precise octahedral- symmetric interpretation as moduli space of an explicit Shimura family of curves of genus 3. Proofs are based only on the Proportionality Theorem and classification results for hermitian lattices and algebraic surfaces.

Jury : Michel Coste, Ilia Itenberg, Louis Mahé, Stepan Orevkov, Adam Parusinski (Président), Jean Jacques Risler, This thesis is motivated by the study of real algebraic curves in the real projective plane and in rational geometrically ruled surfaces equipped with their standard real structure. We were especially interested in two particular problems. The ovals of a nonsingular curves in the projective plane of even degree are naturally divided in two disjoint sets : even ovals, contained in an even number of ovals, and odd ovals. Combining the Harnack and Petrovsky inequalities, one obtains an upper bound on the number of even ovals, and on the number of odd ovals with respect to the degree of the curve. We generalize here a previous construction of I. Itenberg, and show that this upper bound is asymptotically sharp. Almost all known prohibitions on the topology of real algebraic curves are still valid for a wider class of objects, real pseudoholomorphic curves. An open problem is the existence of a real scheme realizable by a nonsingular real pseudoholomorphic curve but not realizable by a nonsingular real algebraic curve of the same degree. In this thesis, we study real nonsingular algebraic and pseudoholomorphic symmetric curves of degree 7 in the real projective plane. In particular, we give several classifications and exhibit two real schemes realizable by dividing nonsingular real symmetric pseudoholomorphic curves of degree 7, but not realizable by such algebraic curves. Some results of this thesis are based on the techniques of dessins d'enfants. In real algebraic geometry these objects were first used by S. Yu. Orevkov. In particular, they allow one to answer the following question : does there exist two real polynomials P and Q of degree n such that the real roots of P, Q, and P+Q realize a given arrangement? Following Orevkov, we give a necessary and sufficient condition for the existence of two such polynomials, formulated in terms of dessins d'enfants. We also give an algorithm which gives a possibility to check whether a given L-scheme is realizable by a real tri, Cette thèse est motivée par l'étude des courbes algébriques réelles dans le plan projectif réel et dans les surfaces rationnelles géométriquement réglées, munis de leur structure réelle standard. Deux problèmes ont particulièrement retenus notre attention. Les ovales d'une courbe non singulière dans dans le plan projectif réel de degré pair sont naturellement divisés en deux ensembles disjoints : les ovales pairs, contenus dans un nombre pair d'ovales, et les ovales impairs. La combinaison des inégalités de Harnack et de Petrovsky permet d'obtenir une borne supérieure pour le nombre d'ovales pairs et le nombre d'ovales impairs en fonction du degré de la courbe. Généralisant une construction antérieure d'I. Itenberg, nous montrons que cette borne est asymptotiquement optimale. La majorité des restrictions connues sur la topologie des courbes algébriques réelles sont aussi valables pour une classe plus vaste d'objets, les courbes pseudoholomorphes réelles. Un problème ouvert est celui de l'existence d'un schéma réel réalisable par une courbe pseudoholomorphe réelle non singulière, mais pas par une courbe algébrique réelle non singulière de même degré. Nous étudions dans cette thèse les courbes réelles non singulières symétriques de degré 7 dans le plan projectif réel, algébriques et pseudoholomorphes. Nous obtenons en particulier plusieurs classifications, et exhibons deux schémas réels réalisables par des courbes pseudoholomorphes réelles séparantes symétriques non singulières de degré 7 mais pas par de telles courbes algébriques. Certains des résultats de cette thèse sont basés sur l'utilisation des dessins d'enfants. En géométrie algébrique réelle, ces objets ont été utilisés la première fois par S. Yu. Orevkov. Ils permettent en particulier de répondre à la question suivante : Existe-t-il deux polynômes réels P et Q de degré n tels que les racines réelles de P, Q et P+Q réalisent un arrangement donné? Suivant Orevkov, nous donnons une condition nécessaire et suffi