Le point de départ de la thèse fut l'étude d'une variété holomorphiquement symplectique irréductible (VHSI) à singularités orbifold, de dimension 4, construite en 2007 par Markushevich—Tikhomirov comme une compactification d'une famille lagrangienne de surfaces de Prym de polarisation (1,2). La famille des surfaces de Prym en question est associée au système linéaire de courbes de genre 3 sur une surface K3 quartique, munie d'une involution anti-symplectique. Dans la première partie de la thèse, on calcule la forme de Beauville—Bogomolov (BB) sur la seconde cohomologie entière de cette VHSI. L'existence d'une forme BB sur les VHSI singulières aux singularités en codimension 4 était démontrée par Namikawa, mais aucun exemple explicite d'une telle forme n'était connu, et la thèse présente les premiers exemples explicites de formes BB de VHSI singulières. Le calcul de ces formes BB a nécessité de développer des outils permettant de déterminer la cohomologie entière de variétés quotientées par un groupe d'automorphismes d'ordre premier. Dans la deuxième partie de la thèse, la famille miroir de la VHSI de Markushevich—Tikhomirov, formée des surfaces abéliennes duales, est déterminée. Il se trouve qu'elle est aussi une famille de prymiennes, associée à une quartique K3 avec involution anti-symplectique, donc admet une compactification qui est la symétrique miroir de la VHSI d'origine. Une description géométrique très précise de cette correspondance est donnée, basée sur la construction bigonale de Pantazis. De plus, on montre que la symétrie miroir ainsi construite représente une involution birationnelle non-triviale sur l'espace de modules de VHSI de ce type. / The starting point of the thesis was the study of a singular irreducible holomorphically symplectic variety (IHSV) of dimension 4 with orbifold singularities which was constructed by Markushevich—Tikhomirov in 2007 as a compactification of a Lagrangian family of (1,2)-polarized Prym surfaces. This family of Prym surfaces is associated to a linear system of genus-3 curves on a quartic K3 surface endowed with an anti-symplectic involution. In the fist part of the thesis, the Beauville—Bogomolov form (BB) on the second integer cohomology group of this IHSV is computed. The existence of the BB form for an IHSV with singular locus of codimension 4 was proved by Namikawa, but no explicit example of such a form was known. The thesis provides the first concrete examples of BB forms on singular IHSV. The calculation of these BB forms required the development of some tools for computing the integer cohomology of varieties quotiented by automorphism groups of prime order. In the second part of the thesis, the mirror family of dual abelian surfaces for the Markushevich—Tikhomirov IHSV is determined. As it turns out, it is also a family of Prym surfaces associated to a quartic K3 surface with an anti-symplectic involution and hence admits a compactification, which is the mirror of the original IHSV. A very precise geometric description of this duality is given, using Pantazis's bigonal construction. Moreover, it is proved that the mirror symmetry constructed in this way represents a non-trivial birational involution on the moduli space of Markushevich—Tikhomirov IHSV.

Diese Dissertation beschäftigt sich mit asymptotischen Syzygien und Gleichungen Abelscher Varietäten, sowie mit deren Anwendung auf zyklische Überdeckungen von Kurven von Geschlecht zwei. Was asymptotischen Syzygien angeht, zeigen wir für beliebige Geradenbündel auf projektiven Schemata: Wenn die asymptotischen Syzygien von Grad p eines Geradenbündels verschwinden, dann ist das Geradenbündel p-sehr ampel. Darüber hinaus verwenden wir die Bridgeland-King-Reid-Haiman Korrespondenz, um zu zeigen, dass dieses Ergebnis auch umgekehrt wahr ist, wenn es um eine glatte Fläche und kleine p geht. Dies dehnt Ergebnisse von Ein-Lazarsfeld und Ein-Lazarsfeld-Yang aus. Wir verwenden unsere Ergebnisse, um zu untersuchen, wie Syzygien verwendet werden können, um den Grad der Irrationalität einer Varietät zu begrenzen. Ferner, beweisen wir eine Vermutung von Gross and Popescu über Abelsche Flächen, deren Ideal durch Quadriken und Kubiken erzeugt wird. Außerdem verwenden wir die projektive Normalität einer Abelschen Fläche, um die Prym Abbildung, die mit zyklischen Überdeckungen von Geschlecht zwei Kurven assoziert ist, zu untersuchen. Wir zeigen, dass das Differential der Abbildung generisch injektiv ist, wenn der Grad der Überdeckung mindestens sieben ist. Wir dehnen damit Ergebnisse von Lange und Ortega aus. Abschließend zeigen wir, dass das Differential genau für bielliptische Überdeckungen nicht injectiv ist. / In this thesis we study asymptotic syzygies of algebraic varieties and equations of abelian surfaces, with applications to cyclic covers of genus two curves. First, we show that vanishing of asymptotic p-th syzygies implies p-very ampleness for line bundles on arbitrary projective schemes. For smooth surfaces we prove that the converse holds, when p is small, by studying the Bridgeland-King-Reid-Haiman correspondence for the Hilbert scheme of points. This extends previous results of Ein-Lazarsfeld and Ein-Lazarsfeld-Yang. As an application of our results, we show how to use syzygies to bound the irrationality of a variety. Furthermore, we confirm a conjecture of Gross and Popescu about abelian surfaces whose ideal is generated by quadrics and cubics. In addition, we use projective normality of abelian surfaces to study the Prym map associated to cyclic covers of genus two curves. We show that the differential of the map is generically injective as soon as the degree of the cover is at least seven, extending a previous result of Lange and Ortega. Moreover, we show that the differentials fails to be injective precisely at bielliptic covers.

On étudie dans cette thèse les cycles algébriques sur les variétés Jacobiennes de courbes complexes projectives lisses qui admettent des automorphismes non triviaux. Il s'agit plus précisément d'étudier de nouveaux anneaux tautologiques associés à des groupes d’automorphismes de la courbe. On montre que ces Q-algèbres naturelles de cycles algébriques sur les Jacobiennes se restreignent en des familles de cycles sur certaines sous-variétés spéciales de la Jacobienne et que celles-ci méritent encore le nom d'anneaux tautologiques sur ces sous-variétés. On étudie en détail le cas des courbes hyperelliptiques; situation dans laquelle les algèbres introduites admettent un nombre fini de générateurs, et en particulier sont de dimension finie. On peut alors être très précis dans l'étude des relations entre ces générateurs. Enfin, on montre que ces anneaux tautologiques apparaissent naturellement dans un autre contexte : celui des systèmes linéaires complets sans point de base. / In this thesis we study algebraic cycles on Jacobian varieties of smooth projective complex curves with non trivial automorphisms. More precisely, we introduce new tautological rings associated to groups of automorphisms of the curve. We show that these natural Q-algebras of algebraic cycles on Jacobians induce a good notion of tautological rings on some particular subvarieties of the Jacobian. We then study in detail the case of hyperelliptic curves. In this case, the tautological rings admit a finite number of generators, and in particular are of finite dimension. We can then be very precise when studying the relations between these generators. Finally, we present another situation in which these tautological rings appear: when we consider complete linear series without base point.

Master of Science / Department of Mathematics / Ilia Zharkov / When considering an unramified double cover π: C’→ C of nonsingular algebraic curves, the Prym variety (P; θ) of the cover arises from the sheet exchange involution of C’ via extension to the Jacobian J(C’). The Prym is defined to be the anti-invariant (odd) part of this induced map on J(C’), and it carries twice a principal polarization of J(C’). The pair (P; θ), where θ is a representative of a theta divisor of J(C’) on P, makes the Prym a candidate for the Jacobian of another curve. In 1974, David Mumford proved that for an unramified double cover π : C’η →C of a plane quintic curve, where η is a point of order two in J(C), then the Prym (P; θ) is not a Jacobian if the theta characteristic L(η) is odd, L the hyperplane section. We sought to find an analog of Mumford's result in the tropical geometry setting. We consider the Prym variety of certain unramified double covers of three types of tropical plane quintics. Applying the theory of lattice dicings, which give affine invariants of the Prym lattice, we found that when the parity α(H3) is even, H3 the cycle associated to the hyperplane section and the analog to η in the classical setting, then the Prym is not a Jacobian, and is a Jacobian when the parity is odd.

In dieser Doktorarbeit untersuchen wir einige Modulräume der Prym-Paaren, Prym-Varietäten und Spin-Kurven. Nachdem der passende theoretische Rahmen eingeführt wird, erhalten wir neue Ergebnisse zu zwei verschiedenen Aspekten ihrer Geometrie, die wir in zwei entsprechenden Kapiteln beschreiben. In Kapitel 1 betrachten wir die universelle Prym-Varietät über dem Modulraum R_g der Prym-Paaren vom Geschlecht g und bestimmen ihre Unirationalität für g=3. Dazu bilden wir eine explizite rationale Parametrisierung der universellen 2-fachen Prym-Kurve über R_3, die die universelle Prym-Varietät durch die globale Version der Abel-Prym-Abbildung dominiert. Darüber hinaus passen wir den Beweis an den Rahmen von Nikulin-Flächen an und zeigen, dass die universelle doppelte Nikulin-Fläche ebenfalls unirational ist. In Kapitel 2 untersuchen wir die Wechselwirkung zwischen R_g und dem Modulraum S_g der (stabilen) Spin-Kurven vom Geschlecht g. Wenn man den Divisor der Kurven, die mit einem verschwindenden Thetanull ausgestattet sind, von S_g^+ nach R_g versetzt, erhält man zwei geometrische Divisoren der (stabilen) Prym-Kurven mit einem verschwindenden Thetanull. Wir verwenden Testkurventechniken, um die Klassen dieser (Prym-Null-)Divisoren für g>=5 zu berechnen, und werten die Prymnull-Klassen auf einigen weiteren Familien von Kurven aus, um ihre verschwindenden Thetanulls zu analysieren. Darüber hinaus diskutieren wir am Ende von Kapitel 2 eine mögliche Kompaktifizierung des Modulraums der Kurven, die eine doppelte Quadratwurzel tragen. Anschließend untersuchen wir den Rand des Modulraums RS_g der (stabilen) Prym-Spin-Kurven vom Geschlecht g und überprüfen die Prymnull-Klassen anhand des Diagramms R_g<--RS_g-->S_g. Zum Schluss schlagen wir eine Erweiterung des Produkts von Wurzeln, das über glatten Kurven durch das Tensorprodukt definiert ist, zu einer Operation auf stabilen Doppelwurzeln vor. / In this thesis, we study several moduli spaces of Prym pairs, Prym varieties, and spin curves. After the appropriate theoretical framework is introduced, we obtain new results concerning two different aspects of their geometry, which we describe across two corresponding chapters. In Chapter 1, we consider the universal Prym variety over the moduli space R_g of Prym pairs of genus g, and determine its unirationality for g=3. To do this, we build an explicit rational parametrization of the universal 2-fold Prym curve over R_3, which dominates the universal Prym variety through the global version of the Abel-Prym map. Furthermore, we adapt the proof to the setting of Nikulin surfaces and show that the universal double Nikulin surface is also unirational. In Chapter 2, we explore the interaction between R_g and the moduli space S_g of (stable) spin curves of genus g. When the divisor of curves equipped with a vanishing theta-null is moved from S_g^+ to R_g, it yields two geometric divisors of (stable) Prym curves with a vanishing theta-null. We use test curve techniques to compute the classes of these (Prym-null) divisors for g>=5, and evaluate the Prym-null classes on some more families of curves in order to analyse their vanishing theta-nulls. In addition, at the end of Chapter 2 we discuss a potential compactification of the moduli space of curves carrying a double square root. We then examine the boundary of the moduli space RS_g of (stable) Prym-spin curves of genus g and check the Prym-null classes against the diagram R_g<--RS_g-->S_g. Finally, we propose an extension of the product of roots, defined over smooth curves by the tensor product, to an operation on stable double roots.

Literature shows that serious concerns are being raised about the wellbeing of young people in South Africa, however somehow youth manage to sustain their health and wellbeing despite the risks they face. This phenomenon is called resilience; youth are coping well in the face of adversity, nevertheless little is known about the relationship between resilience and service usage. Resources such as empathy, religious leaders and personal faith, supportive family relationships and bonding with a parent empower youth against risks they might face, which might counteract various risks which impair youth from becoming resilient. This study focuses on the correlation between services and resilience through a quantitative cross-sectional survey of the Pathways to Resilience Youth Measure (PRYM). 1209 participants between the ages of 12 and 19, from QwaQwa and Bethlehem in the Free State, South Africa were involved. Statistical analysis found that been questioned by the Police, not as a witness (-0.203), foster home (-0.200), gone to court, not as a witness (-0.190), been put into jail (-0.227), been on probation (-0.222) and substance abuse or addiction services (-0.222) scored statistically practically significantly. These results might indicate that participants in this study do not necessarily use services which are identified in the PRYM; moreover low resilience youth use services due to their involvement in activities which might get them into trouble or have them witness such activities. Findings might also indicate that those at-risk youth whose family cannot care for them sufficiently might have low resilience levels and have to make use of services such as placements in Foster homes. Finally the limited use of services by high resilience youth might correspond with reports that youth make positive meaning of live events and circumstances. / Thesis (MEd (Educational Psychology))--North-West University, Potchefstroom Campus, 2012.

Literature shows that serious concerns are being raised about the wellbeing of young people in South Africa, however somehow youth manage to sustain their health and wellbeing despite the risks they face. This phenomenon is called resilience; youth are coping well in the face of adversity, nevertheless little is known about the relationship between resilience and service usage. Resources such as empathy, religious leaders and personal faith, supportive family relationships and bonding with a parent empower youth against risks they might face, which might counteract various risks which impair youth from becoming resilient. This study focuses on the correlation between services and resilience through a quantitative cross-sectional survey of the Pathways to Resilience Youth Measure (PRYM). 1209 participants between the ages of 12 and 19, from QwaQwa and Bethlehem in the Free State, South Africa were involved. Statistical analysis found that been questioned by the Police, not as a witness (-0.203), foster home (-0.200), gone to court, not as a witness (-0.190), been put into jail (-0.227), been on probation (-0.222) and substance abuse or addiction services (-0.222) scored statistically practically significantly. These results might indicate that participants in this study do not necessarily use services which are identified in the PRYM; moreover low resilience youth use services due to their involvement in activities which might get them into trouble or have them witness such activities. Findings might also indicate that those at-risk youth whose family cannot care for them sufficiently might have low resilience levels and have to make use of services such as placements in Foster homes. Finally the limited use of services by high resilience youth might correspond with reports that youth make positive meaning of live events and circumstances. / Thesis (MEd (Educational Psychology))--North-West University, Potchefstroom Campus, 2012.

The first goal of this Thesis is to contribute to the study of principally polarized abelian varieties (ppav), especially to the Schottky and the Torelli problems. Ppav admit a duality theory analogous to that of projective spaces, where the role played by hyperplanes in projective spaces is played by divisors representing the principal polarization. Thus, given a subvariety Y of a ppav, we can define its thetadual T(Y) as the set of divisors representing the principal polarization that contain this subvariety. This set admits a natural schematic structure (as defined by Pareschi and Popa). Jacobian and Prym varieties are classical examples of ppav constructed from curves. Besides, they are interesting because some properties of the curves involved in their construction are reflected in their geometry or in the geometry of some special subvarieties. For example, in the case of Jacobians we have the BrillNoether loci Wd ( W1 corresponds to the AbelJacobi curve) and in the case of Pryms we have the AbelPrym curve C. In chapter III, we study the schematic structure of the thetadual of the BrillNoether loci Wd and the AbelPrym curve. In the first case, we obtain with different methods, the result of Pareschi and Popa T(Wd)= Wgd1. In the case of the AbelPrym curve C, we get that T(C)=V², where V² is the second PrymBrillNoether locus with the schematic structure defined by Welters. Pareschi and Popa have proved a result for ppavs analogous to the Castelnuovo Lemma for projective spaces. That is, if (A,Θ) is a ppav of dimension g, then g+2 distinct points in general position with respect to Θ, but in special position with respect to 2Θ, have to be contained in a curve of minimal degree in A, i.e. an AbelJacobi curve. In particular, they obtain a Schottky result because A has to be a Jacobian variety and a Torelli result, because the curve is the intersection of all the divisors in |2Θ| that contain the g+2 points. In chapter IV, as Eisenbud and Harris have done in the projective Castelnuovo Lemma, we extend this result to possibly nonreduced finite schemes. The second goal of this thesis is the study of varieties of general type. Almost by definition, pluricanonical maps are the essential tool to study them. One of the main problems in this area is to find geometric or numerical conditions to guarantee that the mth pluricanonical map (for low m) induces a birational equivalence with its image. The classification of surfaces whose bicanonical map is nonbirational has attracted considerable interest among algebraic geometers. In chapter V, we give a sufficient numerical condition for the birationality of the bicanonical map of irregular varieties of arbitrary dimension. We also prove that, if X is a primitive variety, then it only admits very special fibrations to other irregular varieties. For primitive varieties we get that the following are equivalent: X is birational to a divisor Θ in an indecomposable ppav, the irregularity q(X) > dim X and the bicanonical map is nonbirational. When X is a primitive variety of general type and q(X) = dim X we prove, under certain conditions over the Stein factorization of the Albanese map, that the only possibility for the bicanonical map being nonbirational is that X is a double cover branched along a divisor in |2Θ|. These results extend to arbitrary dimension, wellknown theorems in the case of surfaces and curves. / El primer objectiu d'aquesta tesi és contribuir a l'estudi de les varietats abelianes principalment polaritzades (vapp), especialment als problemes de Schottky i Torelli. Les vapp admeten una teoria de dualitat anàloga a la dualitat dels espais projectius, on el paper que juguen els hiperplans de l'espai projectiu és substituït pels divisors que representen la polarització principal. Així doncs, donada una subvarietat Y d'una vapp, podem definir el seu thetadual T(Y) com el conjunt dels divisors que representen la polarització principal i contenen aquesta subvarietat. Aquest conjunt admet una estructura esquemàtica natural (tal i com la defineixen Pareschi i Popa). Les varietats Jacobianes i de Prym són exemples clàssics de vapp construïdes a partir de corbes. A més, són interessants perquè certes propietats de les corbes involucrades es veuen reflectides en elles o en algunes subvarietats especials. Per exemple, en el cas de les Jacobianes tenim els llocs de BrillNoether Wd ( W1 correspon a la corba d'AbelJacobi) i en el cas de les Pryms tenim la corba d'AbelPrym C. Al capítol III de la tesi s'estudia l'estructura esquemàtica del thetadual dels llocs de BrillNoether Wd i de la corba d'AbelPrym. En el primer cas, es reobté amb uns altres mètodes, el resultat de Pareschi i Popa T(Wd)= Wgd1. En el cas de la corba d'AbelPrym C, s'obté que T(C)=V², onV² és el segon lloc de PrymBrillNoether amb l'estructura esquemàtica definida per Welters. Pareschi i Popa han demostrat un resultat anàleg per les vapp al Lemma de Castelnuovo pels espais projectius. És a dir, si (A,Θ) és una vapp de dimensió g, aleshores g+2 punts en posició general respecte Θ, però en posició especial respecte 2Θ, han d'estar continguts en una corba de grau minimal a A, i.e. una corba d'AbelJacobi. En particular, s'obté un resultat de Schottky ja que A ha de ser una Jacobiana i un resultat de Torelli, ja que la corba és la intersecció de tots els divisors de |2Θ| que contenen els g+2 punts. Al capítol IV, tal i com Eisenbud i Harris van fer en el cas projectiu, s'estén aquest resultat a esquemes finits possiblement no reduïts. El segon objectiu d'aquesta tesi és contribuir a l'estudi de les varietats de tipus general. Pràcticament per definició, les aplicacions pluricanòniques són essencials pel seu estudi. Un dels problemes principals de l'àrea és donar condicions geomètriques o numèriques per assegurar que la mèsima aplicació pluricanònica (per m baix) indueix una equivalència biracional amb la imatge. La classificació de les superfícies que tenen l'aplicació bicanònica no biracional ha atret l'atenció de molts geòmetres algebraics. Al capítol V, es dóna un criteri numèric suficient per assegurar la biracionalitat de l'aplicació bicanònica de les varietats irregulars de dimensió arbitrària. També es demostra que si X és una varietat primitiva, aleshores només admet fibracions molt especials a altres varietats irregulars. Per aquestes varietats s'obté que és equivalent que X sigui biracional a un divisor Θ en una vapp indescomponible, a què la irregularitat q(X) > dim X i l'aplicació bicanònica sigui no biracional. Quan X és una varietat primitiva de tipus general i q(X) = dim X es demostra sota certes condicions de la descomposició de Stein del morfisme d'Albanese, que l'única possibilitat per tal que l'aplicació bicanònica sigui no biracional és que X sigui un recobriment doble sobre una vapp ramificat al llarg d'un divisor a |2Θ|. Aquest resultats estenen a dimensió arbitrària, teoremes ben coneguts en el cas de superfícies i corbes.