Relevant bibliographies by topics / Birational geometry of surfaces

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Birational geometry of surfaces'

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Published: 8 March 2025

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Journal articles on the topic "Birational geometry of surfaces"

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Ciliberto, Ciro, Thomas Dedieu, Flaminio Flamini, and Rita Pardini. "Birational geometry of surfaces." Bollettino dell'Unione Matematica Italiana 11, no. 1 (March 2018): 1–3. http://dx.doi.org/10.1007/s40574-018-0157-1.

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Mella, Massimiliano. "Birational geometry of rational quartic surfaces." Journal de Mathématiques Pures et Appliquées 141 (September 2020): 89–98. http://dx.doi.org/10.1016/j.matpur.2020.07.007.

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Toda, Yukinobu. "Stability conditions and birational geometry of projective surfaces." Compositio Mathematica 150, no. 10 (July 17, 2014): 1755–88. http://dx.doi.org/10.1112/s0010437x14007337.

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AbstractWe show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.
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Chi, Quo-Shin, Luis Fernández, and Hongyou Wu. "Normalized potentials of minimal surfaces in spheres." Nagoya Mathematical Journal 156 (1999): 187–214. http://dx.doi.org/10.1017/s0027763000007133.

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We determine explicitly the normalized potential, a Weierstrass-type representation, of a superconformal surface in an even-dimensional sphere S2n in terms of certain normal curvatures of the surface. When the Hopf differential is zero the potential embodies a system of first order equations governing the directrix curve of a superminimal surface in the twistor space of the sphere. We construct a birational map from the twistor space of S2n into ℂPn(n+1)/2. In general, birational geometry does not preserve the degree of an algebraic curve. However, we prove that the birational map preserves the degree, up to a factor 2, of the twistor lift of a superminimal surface in S6 as long as the surface does not pass through the north pole. Our approach, which is algebro-geometric in nature, accounts in a rather simple way for the aforementioned first order equations, and as a consequence for the particularly interesting class of superminimal almost complex curves in S6. It also yields, in a constructive way, that a generic superminimal surface in S6 is not almost complex and can achieve, by the above degree property, arbitrarily large area.
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Blanc, Jérémy, and Frédéric Mangolte. "Geometrically rational real conic bundles and very transitive actions." Compositio Mathematica 147, no. 1 (September 13, 2010): 161–87. http://dx.doi.org/10.1112/s0010437x10004835.

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AbstractIn this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.
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Laza, Radu, and Kieran O’Grady. "Birational geometry of the moduli space of quartic surfaces." Compositio Mathematica 155, no. 9 (August 2, 2019): 1655–710. http://dx.doi.org/10.1112/s0010437x19007516.

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By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-$2$ $K3$ surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-$4$ $K3$ surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to $D$-lattices. In particular, for the $19$-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-$4$ $K3$ surfaces. The analogous $18$-dimensional case, which corresponds to hyperelliptic degree-$4$ $K3$ surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients.
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Morrison, David R. "The birational geometry of surfaces with rational double points." Mathematische Annalen 271, no. 3 (September 1985): 415–38. http://dx.doi.org/10.1007/bf01456077.

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Ryan, Tim, and Ruijie Yang. "Nef Cones of Nested Hilbert Schemes of Points on Surfaces." International Mathematics Research Notices 2020, no. 11 (May 28, 2018): 3260–94. http://dx.doi.org/10.1093/imrn/rny088.

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Abstract Let X be the projective plane, a Hirzebruch surface, or a general K3 surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on X. We calculate the nef cone for two types of nested Hilbert schemes. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces.
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Tanaka, Hiromu. "Minimal Models and Abundance for Positive Characteristic Log Surfaces." Nagoya Mathematical Journal 216 (2014): 1–70. http://dx.doi.org/10.1215/00277630-2801646.

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AbstractWe discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for ℚ-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.
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Tanaka, Hiromu. "Minimal Models and Abundance for Positive Characteristic Log Surfaces." Nagoya Mathematical Journal 216 (2014): 1–70. http://dx.doi.org/10.1017/s0027763000022431.

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AbstractWe discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for ℚ-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.
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Dissertations / Theses on the topic "Birational geometry of surfaces"

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Barros, Ignacio. "K3 surfaces and moduli of holomorphic differentials." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/19290.

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In dieser Arbeit behandeln wir die birationale Geometrie verschiedener Modulräume; die Modulräume von Kurven mit einem k-Differential mit vorgeschierbenen Nullen, besser bekannt als Strata von Differenzialen, Moduln von K3 Flächen mit markierten Punkten und Moduln von Kurven. Für bestimmte Geschlechter nennen wir Abschätzungen der Kodaira-Dimension, konstruieren unirationale Parametrisierungen, rationale deckende Kurven und unterschiedliche birationale Modelle. In Kapitel 1 führen wir die zu untersuchenden Objekte ein und geben einen kurzen Überblick ihrer wichtigsten Eigenschaften und offenen Problemen. In Kapitel 2 konstruieren wir einen Hilfsmodulraum, der als Brücke zwischen bestimmten finiten Quotienten von Mgn für kleines g und den Moduln der polarisierten K3 Flächen vom Geschlecht 11 dient. Wir entwickeln die Deformationstheorie, die nötig ist, um die Eigenschaften und die oben genannten Modulräume zu erforschen. In Kapitel 3 bedienen wir uns dieser Werkzeuge, um birationale Modelle für Moduln polarisierter K3 Flächen vom Geschlecht 11 mit markierten Punkten zu konstruieren. Diese nutzen wir, um Resultate über die Kodaira-Dimension herzuleiten. Wir beweisen, dass der Modulraum von polarisierten K3 Flächen vom Geschlecht 11 mit n markierten Punkten unirational ist, falls n<=6, und uniruled, falls n<=7. Wir beweisen auch, dass die Kodaira-Dimension von Modulraum von polarisierten K3 Flächen vom Geschlecht 11 mit n markierten Punkten nicht-negativ ist für n>= 9. Im letzten Kapitel gehen wir noch auf die fehlenden Fälle der Kodaira-Klassifizierung von Mgnbar ein. Schliesslich behandeln wir in Kapitel 4 die birationale Geometrie mit Blick auf die Strata von holomorphen und quadratischen Differentialen. Wir zeigen, dass die Strata holomorpher und quadratischer Differentiale von niedrigem Geschlecht uniruled sind, indem wir rationale Kurven mit pencils auf K3 und del Pezzo Flächen konstruieren. Durch das Beschränken des Geschlechts 3<= g<=6 bilden wir projektive Bündel über rationale Varietäten, die die holomorphe Strata mit maximaler Länge g-1 dominieren. Also zeigen wir auch, dass diese Strata unirational sind.
In this thesis we investigate the birational geometry of various moduli spaces; moduli spaces of curves together with a k-differential of prescribed vanishing, best known as strata of differentials, moduli spaces of K3 surfaces with marked points, and moduli spaces of curves. For particular genera, we give estimates for the Kodaira dimension, construct unirational parameterizations, rational covering curves, and different birational models. In Chapter 1 we introduce the objects of study and give a broad brush stroke about their most important known features and open problems. In Chapter 2 we construct an auxiliary moduli space that serves as a bridge between certain finite quotients of Mgn for small g and the moduli space of polarized K3 surfaces of genus eleven. We develop the deformation theory necessary to study properties of the mentioned moduli space. In Chapter 3 we use this machinery to construct birational models for the moduli spaces of polarized K3 surfaces of genus eleven with marked points and we use this to conclude results about the Kodaira dimension. We prove that the moduli space of polarized K3 surfaces of genus eleven with n marked points is unirational when n<= 6 and uniruled when n<=7. We also prove that the moduli space of polarized K3 surfaces of genus eleven with n marked points has non-negative Kodaira dimension for n>= 9. In the final section, we make a connection with some of the missing cases in the Kodaira classification of Mgnbar. Finally, in Chapter 4 we address the question concerning the birational geometry of strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to genus 3<= g<=6 we construct projective bundles over rational varieties that dominate the holomorphic strata with length at most g-1, hence showing in addition, these strata are unirational.
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Beri, Pietro. "On birational transformations and automorphisms of some hyperkähler manifolds." Thesis, Poitiers, 2020. http://www.theses.fr/2020POIT2267.

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Mon travail de thèse porte sur les doubles EPW sextiques, une famille de variétés hyperkähleriennes qui, dans le cas général, sont équivalentes par déformation au schéma de Hilbert de deux points sur une surface K3. Notamment j'ai utilisé le lien que ces variétés ont avec les variétés de Gushel-Mukai, qui sont des variétés de Fano dans une Grassmannienne si leur dimension est plus grande que deux, des surface K3 si la dimension est deux.Le premier chapitre contient quelques rappels de théorie des équations de Pell et des réseaux, qui sont fondamentals pour l’étude des variétés hyperkähleriennes. Ensuite je rappelle la construction qui associe un revêtement double à un faisceau sur une variété normale.Dans le deuxième chapitre j’aborde les variétés hyperkähleriennes et je décris leurs premières propriétés ; j’introduis aussi le premier cas de variété hyperkählerienne qui a été étudiée, les surfaces K3. Cette famille de surfaces correspond aux variétés hyperkähleriennes en dimension deux.Je présente ensuite brièvement certains des derniers résultats dans ce domaine, notamment je définis différents espaces de modules de variétés hyperkähleriennes et je décris l’action d’un automorphisme sur le deuxième groupe de cohomologie d’une variété hyperkähleriennes.Les outils introduits dans le chapitre précédent ne fournissent pas de description géométrique de l'action de l'automorphisme sur la variété, dans le cas où la variété est un schéma de Hilbert de points sur une surface K3. Dans le troisième chapitre, j’introduis donc une description géométrique à une certaine déformation près. Cette déformation prend en compte la structure du schéma de la variété de Hilbert. Pour ce faire, j'introduis un isomorphisme entre une composante connexe de l'espace de modules des variétés de type K3[n] avec une polarization, et l'espace de modules des variétés de même type avec une involution dont le rang de l'invariant est un. Il s’agit d’une généralisation d’un résultat obtenu par Boissière, An. Cattaneo, Markushevich et Sarti en dimension deux. Les deux premières parties de ce chapitre sont un travail en collaboration avec Alberto Cattaneo.Dans le quatrième chapitre, je définis les EPW sextiques, en présentant l'argument de O'Grady, qui montre qu'un double revêtement d'un EPW sextique dans le cas général est une variété de type K3[2]. Ensuite, je présente les variétés Gushel-Mukai, en mettant l'accent sur leur lien avec les EPW sextiques ; cette approche a été introduite par O'Grady, poursuivie par Iliev et Manivel et systématisée par Kuznetsov et Debarre.Dans le cinquième chapitre, j’utilise les outils introduits dans le quatrième chapitre dans le cas où on peut associer une surface K3 à une EPW sextique X. Dans ce cas je donne des conditions explicites sur le groupe de Picard de la surface pour que X soit une variété hyperkählerienne. Cela permet d'utiliser le théorème de Torelli pour une surface K3 pour démontrer l'existence de quelques automorphismes sur X. Je donne des bornes sur la structure d'un sous-groupe d'automorphismes d'une EPW sextique sous conditions d'existence d'un point fixe pour l'action du groupe.Toujours dans le cas d'existence d'une surface K3 associée à une EPW sextique X, j’améliore la borne obtenue précédemment sur les automorphismes de X, en donnant un lien explicite avec le nombre de coniques sur la surface K3. Je montre que la symplecticité d'un automorphisme sur X dépend de la symplecticité d'un automorphisme correspondant sur la surface K3.Le sixième chapitre est un travail en collaboration avec Alberto Cattaneo. J'étudie le groupe d'automorphismes birationels sur le schéma de Hilbert des points sur une surface projective K3, dans le cas générique. Cela généralise le résultat obtenu en dimension deux par Debarre et Macrì. Ensuite j’étudie les cas où il existe un modèle birationel où ces automorphismes sont réguliers. Je décris de façon géométrique quelques involutions dont on avait prouvé l'existence auparavant
My thesis work focuses on double EPW sextics, a family of hyperkähler manifolds which, in the general case, are equivalent by deformation to Hilbert's scheme of two points on a K3 surface. In particular I used the link that these manifolds have with Gushel-Mukai varieties, which are Fano varieties in a Grassmannian if their dimension is greater than two, K3 surfaces if their dimension is two.The first chapter contains some reminders of the theory of Pell's equations and lattices, which are fundamental for the study of hyperkähler manifolds. Then I recall the construction which associates a double covering to a sheaf on a normal variety.In the second chapter I discuss hyperkähler manifolds and describe their first properties; I also introduce the first case of hyperkähler manifold that has been studied, the K3 surfaces. This family of surfaces corresponds to the hyperkähler manifolds in dimension two.Furthermore, I briefly present some of the latest results in this field, in particular I define different module spaces of hyperkähler manifolds, and I describe the action of automorphism on the second cohomology group of a hyperkähler manifold.The tools introduced in the previous chapter do not provide a geometrical description of the action of automorphism on the manifold for the case of the Hilbert scheme of points on a general K3 surface. In the third chapter, I therefore introduce a geometrical description up to a certain deformation. This deformation takes into account the structure of Hilbert scheme. To do so, I introduce an isomorphism between a connected component of the module space of manifolds of type K3[n] with a polarization, and the module space of manifolds of the same type with an involution of which the rank of the invariant is one. This is a generalization of a result obtained by Boissière, An. Cattaneo, Markushevich and Sarti in dimension two. The first two parts of this chapter are a joint work with Alberto Cattaneo.In the fourth chapter, I define EPW sextics, using O'Grady's argument, which shows that a double covering of a EPW sextic in the general case is deformation equivalent to the Hilbert square of a K3 surface. Next, I present the Gushel-Mukai varieties, with emphasis on their connection with EPW sextics; this approach was introduced by O'Grady, continued by Iliev and Manivel and systematized by Kuznetsov and Debarre.In the fifth chapter, I use the tools introduced in the fourth chapter in the case where a K3 surface can be associated to a EPW sextic X. In this case I give explicit conditions on the Picard group of the surface for X to be a hyperkähler manifold. This allows to use Torelli's theorem for a K3 surface to demonstrate the existence of some automorphisms on X. I give some bounds on the structure of a subgroup of automorphisms of a sextic EPW under conditions of existence of a fixed point for the action of the group.Still in the case of the existence of a K3 surface associated with a EPW sextic X, I improve the bound obtained previously on the automorphisms of X, by giving an explicit link with the number of conics on the K3 surface. I show that the symplecticity of an automorphism on X depends on the symplecticity of a corresponding automorphism on the surface K3.The sixth chapter is a work in collaboration with Alberto Cattaneo. I study the group of birational automorphisms on Hilbert's scheme of points on a projective surface K3, in the generic case. This generalizes the result obtained in dimension two by Debarre and Macrì. Then I study the cases where there is a birational model where these automorphisms are regular. I describe in a geometrical way some involutions, whose existence has been proved before
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Fanelli, Andrea. "Two structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristic." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/26285.

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The aim of this thesis is to investigate two questions which naturally arise in the context of the classification of algebraic varieties. The first project concerns the structure of Mori fibre spaces: these objects naturally appear in the birational classification of higher dimensional varieties and the minimal model program. We ask which Fano varieties can appear as a fibre of a Mori fibre space and introduce the notion of fibre-likeness to study this property. This turns out to be a rather restrictive condition: in order to detect this property, we obtain two criteria (one sufficient and one necessary), which turn into a characterisation in the rigid case. Many applications are discussed and the basis for the classification of fibre-like Fano varieties is presented. In the second part of the thesis, an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic is provided: this gives an effective bound on the multiple which makes an ample line bundle D very ample. A careful study of pathological surfaces is presented here in order to bypass the classical cohomological approach. As a consequence, we obtain a Kawamata-Viehweg-type vanishing theorem for arbitrary smooth algebraic surfaces in positive characteristic.
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Benzerga, Mohamed. "Structures réelles sur les surfaces rationnelles." Thesis, Angers, 2016. http://www.theses.fr/2016ANGE0081.

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Le but de cette thèse est d’apporter des éléments de réponse au problème de la finitude du nombre de classes de R-isomorphisme de formes réelles d’une surface rationnelle projective complexe lisse X quelconque, i.e. du nombre de classes d’isomorphisme de R-schémas dont le complexifié est isomorphe à X. Nous étudions ce problème en termes de structures réelles (ou involutions antiholomorphes, généralisant la conjugaison complexe) sur X : l’intérêt de cette approche est qu’elle permet une réécriture du problème faisant intervenir les groupes d’automorphismes de surfaces rationnelles, à travers la cohomologie galoisienne. Grâce à des résultats récents concernant ces groupes et en nous appuyant sur de la géométrie hyperbolique et aussi dans une moindre mesure sur de la dynamique holomorphe et de la géométrie métrique, nous prouvons plusieurs résultats généraux de finitude qui dépassent largement le seul cadre des surfaces de Del Pezzo et peuvent s’appliquer à certaines surfaces rationnelles à grands groupes d’automorphismes
The aim of this PhD thesis is to give a partial answer to the finiteness problem for R-isomorphism classes of real forms of any smooth projective complex rational surface X, i.e. for the isomorphism classes of R-schemes whose complexification is isomorphic to X. We study this problem in terms of real structures (or antiholomorphic involutions, which generalize complex conjugation) on X: the advantage of this approach is that it helps us rephrasing our problem with automorphism groups of rational surfaces, via Galois cohomology. Thanks to recent results on these automorphism groups, using hyperbolic geometry and, to a lesser extent, holomorphic dynamics and metric geometry, we prove several finiteness results which go further than Del Pezzo surfaces and can apply to some rational surfaces with large automorphism groups
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Boitrel, Aurore. "Groupes d'automorphismes des surfaces del Pezzo sur un corps parfait." Electronic Thesis or Diss., université Paris-Saclay, 2025. http://www.theses.fr/2025UPASM002.

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Les surfaces del Pezzo sont des surfaces algébriques dotées de propriétés particulières, et qui jouent un rôle important dans la classification des surfaces algébriques projectives à transformations birationnelles près.La classification des surfaces del Pezzo rationnelles et lisses de degré d sur un corps parfait arbitraire est classique pour d = 7, 8, 9 et nouvelle pour d = 6. Il en va de même pour ladescription de leurs groupes d'automorphismes. Leur classification et la description de leursgroupes d'automorphismes sont beaucoup plus difficiles pour d ≤ 5, comme on peut déjà le voir si le corps de base est le corps des nombres réels, et la classification est ouverte sur un corps parfait général. Des classifications partielles existent sur des corps finis. Par conséquent, nous ne connaissons pas leurs groupes d'automorphismes en général.L'objectif de la thèse est de classifier les surfaces del Pezzo rationnelles lisses de degréd = 5 et d = 4 sur un corps parfait arbitraire et de décrire leurs groupes d'automorphismes.En raison de la difficulté du projet, le cas d = 4 ne sera étudié que sur le corps des nombres réels
Del Pezzo surfaces are algebraic surfaces with quite special properties, that play an importantpart in the classification of projective algebraic surfaces up to birational transformations.The classification of smooth rational del Pezzo surfaces of degree d over an arbitraryperfect field is classical for d = 7, 8, 9 and new for d = 6. The same is the case for thedescription of their groups of automorphisms. Their classification and the description of theirautomorphism groups is much more difficult for d ≤ 5, as one can see already if the groundfield is the field of real numbers, and the classification is open over a general perfect field.Partial classifications exist over finite fields. Accordingly, we do not know their automorphismgroups in general.The objective of the thesis is to classify the smooth rational del Pezzo surfaces of degreed = 5 and d = 4 over an arbitrary perfect field and describe their automorphism groups.Due to the difficulty of the project, the case d = 4 will only be studied over the field ofreal numbers
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Krylov, Igor. "Birational geometry of Fano fibrations." Thesis, University of Edinburgh, 2017. http://hdl.handle.net/1842/28857.

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An algebraic variety is called rationally connected if two generic points can be connected by a curve isomorphic to the projective line. The output of the minimal model program applied to rationally connected variety is variety admitting Mori fiber spaces over a rationally connected base. In this thesis we study the birational geometry of a particular class of rationally connected Mori fiber spaces: Fano fibrations over the projective line. We construct examples of Fano fibrations with a unique Mori fiber space in their birational classes. We prove that these examples are not birational to varieties of Fano type, thus answering the question of Cascini and Gongyo. That is we prove that the classes of rationally connected varieties and varieties of Fano type are not birationally equivalent. To construct the examples we use the techniques of birational rigidity. A Mori fiber space is called birationally rigid if there is a unique Mori fiber space structure in its birational class. The birational rigidity of smooth varieties admitting a del Pezzo fibration of degrees 1 and 2 is a well studied question. Unfortunately it is not enough to study smooth del Pezzo fibrations as there are fibrations which do not have smooth or even smoothable minimal models. In the case of fibrations of degree 2 we know that there is a minimal model with 2-Gorenstein singularities. These singularities are degenerations of the simplest terminal quotient singularity: singular points of the type 1/2(1,1,1). We give first examples of birationally rigid del Pezzo fibrations with 2-Gorenstein singularities. We then apply this result to study finite subgroups of the Cremona group of rank three. We then study the birational geometry of Fano fibrations from a different side. Using the reduction to characteristic 2 method we prove that double covers of Pn-bundles over Pm branched over a divisor of sufficiently high degree are not stably rational. For a del Pezzo fibration Y→P1 of degree 2 such that X is smooth there is a double cover Y→ X, where X is a P2-bundle over P1. In this case a stronger result holds: a very general Y with Pic(Y)≅Z⊕Z is not stably rational. We discuss the proof of this statement.
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DURIGHETTO, Sara. "Classical and Derived Birational Geometry." Doctoral thesis, Università degli studi di Ferrara, 2019. http://hdl.handle.net/11392/2488324.

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In the field of algebraic geometry, the study of birational transforma- tions and their properties plays a primary role. In this, there are two different approaches: the classical one due to the Italian school who focuses on the Cremona group and a modern one which utilizes instruments like derived categories and semiorthogonal decompositions. About the Cremona group, that is the group of birational self-morphisms of P^n, we do not know much in general and we focus on the complex case. We know a set of generators only in dimension n = 2. Moreover, we do not have a classication of curves and linear systems in P^2 up to Cremona transformations. Among the known results there are: irreducible curves and curves with two irreducible components. In this thesis we approach tha case of a conguration of lines in the projective plane. The last theorem lists the known contractible configurations. From a categorical point of view, the semiorthogonal decompositions of the derived category of a variety provide some useful invariants in the study of the variety. Following the work of Clemens-Griffiths about the complex cubic threefold, we want to characterize the obstructions to the rationality of a variety X of dimension n. The idea is to collect the component of a semiorthogonal decomposition which are not equivalent to the derived category of a variety of dimension at least n-1. In this way we defined the so called Griffiths-Kuznetsov component of X. In this thesis we study the case of surfaces on an arbitrary field, we define that component and show that it is a birational invariant. It appears clearly that the Griffiths-Kuznetsov component vanishes only if the surface is rational.
Nell'ambito della geometria algebrica, lo studio delle trasformazioni birazionali e delle loro proprietà riveste un ruolo di importanza primaria. In questo, si affiancano l'approccio classico della scuola italiana che si concentra sul gruppo di Cremona e quello più moderno che utilizza strumenti come categorie derivate e decomposizioni semiortogonali. Del gruppo di Cremona Cr_n, cioé il gruppo degli automorfismi birazionali di P^n, in generale non si conosce molto e ci si concentra sul caso complesso. Si conosce un insieme di generatori solo nel caso di dimensione 2. Inoltre non é ancora nota una classicazione tramite trasformazioni di Cremona delle curve e dei sistemi lineari di P^2. Tra i casi noti ci sono: le curve irriducibili e quelle formate da due componenti irriducibili. In questa tesi ci si approccia al caso di una configurazione di d rette nel piano proiettivo. Il teorema finale fornisce condizioni necessarie o sufficienti alla contraibilità. Da un punto di vista categoriale invece, le decomposizioni semiortogonali della cat- egoria derivata di una varietà ci forniscono degli invarianti utili nello studio della varietà. Seguendo l'approccio di Clemens-Griffiths riguardante la cubica complessa di dimensione 3, si vuole caratterizzare le ostruzioni alla razionalità di una varietà X di dimensione n. L'idea è di raccogliere le componenti di una decomposizione ortog- onale che non sono equivalenti a categorie derivate di varietà di dimensione almeno n-1 e in questo modo definire quella che chiamiamo componente di Griffiths- Kuznetsov di X. In questa tesi si studia il caso delle superci geometricamante razionali su un campo arbitrario, si definisce tale componente e si mostra che essa è un invariante birazionale. Si vede anche che la componente di Griffiths-Kuznetsov è nulla solo se la supercie è razionale.
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8

Zong, Hong R. "Topics in birational geometry of algebraic varieties." Thesis, Princeton University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3665359.

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Various questions related to birational properties of algebraic varieties are concerned.

Rationally connected varieties are recognized as the buildings blocks of all varieties by the Minimal Model theory. We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. As a consequence, we solve a question of Professor Burt Totaro on integral Hodge classes on rationally connected 3-folds. And by a result of Professor Claire Voisin, the general case will be a consequence of the Tate conjecture for surfaces over finite fields.

Using the same philosophy looking for degenerated rational components through forgetful maps between moduli spaces of curves, we prove Weak Approximation conjecture to Prof. Hassett and Prof. Tschinkel for isotrivial families of rationally connected varieties. Theory of Twisted Stable maps is essentially used, with an alternative proof where some notion from Derived Algebraic Geometry is applied. It is remarkable that technics and ideas developed in this part, shed light upon and essentially led to the final solution to weak approximation of Cubic Surfaces, which is a problem concerned by Number Theorists for many years, and this is currently the best known result in this subject.

Then we turn to Minimal Model theory in both zero and positive characteristics. Firstly, projective globally F-regular threefolds of characteristic p ≥ 11, are shown to be rationally chain connected, and back to characteristic zero, we use hard-core technics of Minimal Model program, esp. finite generate of canonical rings due to Professor Hacon, Professor McKernan et al. to characterize Toric varieties and geometric rational varieties as log canonical log-Calabi Yau varieties with "large" boundary, where the specific meanings of "large" are originated from some notion of "charges" from String theory, and hence is related to Mirror Symmetry. This part of works also answered a Conjecture due to Prof. Shokurov.

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9

Rulla, William Frederick. "The birational geometry of M₃ and M₂, ₁ /." Full text (PDF) from UMI/Dissertation Abstracts International, 2001. http://wwwlib.umi.com/cr/utexas/fullcit?p3008434.

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Massarenti, Alex. "Biregular and Birational Geometry of Algebraic Varieties." Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4679.

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Every area of mathematics is characterized by a guiding problem. In algebraic geometry such problem is the classification of algebraic varieties. In its strongest form it means to classify varieties up to biregular morphisms. However, birationally equivalent varieties share many interesting properties. Therefore for any birational equivalence class it is natural to work out a variety, which is the simplest in a suitable sense, and then study these varieties. This is the aim of birational geometry. In the first part of this thesis we deal with the biregular geometry of moduli spaces of curves, and in particular with their biregular automorphisms. However, in doing this we will consider some aspects of their birational geometry. The second part is devoted to the birational geometry of varieties of sums of powers and to some related problems which will lead us to computational geometry and geometric complexity theory.
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Books on the topic "Birational geometry of surfaces"

1

János, Kollár. Birational geometry of algebraic varieties. Cambridge: Cambridge University Press, 1998.

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Matsuki, Kenji. Weyl groups and birational transformations among minimal models. Providence, RI: American Mathematical Society, 1995.

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Kawamata, Yujiro, and Vyacheslav V. Shokurov, eds. Birational Algebraic Geometry. Providence, Rhode Island: American Mathematical Society, 1997. http://dx.doi.org/10.1090/conm/207.

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Brunella, Marco. Birational Geometry of Foliations. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-14310-1.

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Hochenegger, Andreas, Manfred Lehn, and Paolo Stellari, eds. Birational Geometry of Hypersurfaces. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-18638-8.

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Shigefumi, Mori, Miyaoka Yoichi, and Kyōto Daigaku. Sūri Kaiseki Kenkyūjo., eds. Higher dimensional birational geometry. Tokyo, Japan: Mathematical Society of Japan, 2002.

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Colombo, Elisabetta, Barbara Fantechi, Paola Frediani, Donatella Iacono, and Rita Pardini, eds. Birational Geometry and Moduli Spaces. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-37114-2.

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Berenstein, Arkady, and Vladimir Retakh, eds. Noncommutative Birational Geometry, Representations and Combinatorics. Providence, Rhode Island: American Mathematical Society, 2013. http://dx.doi.org/10.1090/conm/592.

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Cheltsov, Ivan, Ciro Ciliberto, Hubert Flenner, James McKernan, Yuri G. Prokhorov, and Mikhail Zaidenberg, eds. Automorphisms in Birational and Affine Geometry. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05681-4.

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Bogomolov, Fedor, Brendan Hassett, and Yuri Tschinkel, eds. Birational Geometry, Rational Curves, and Arithmetic. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-6482-2.

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Book chapters on the topic "Birational geometry of surfaces"

1

Matsuki, Kenji. "Birational Geometry of Surfaces." In Introduction to the Mori Program, 9–108. New York, NY: Springer New York, 2002. http://dx.doi.org/10.1007/978-1-4757-5602-9_2.

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Mumford, David. "The Birational Geometry of Surfaces." In Algebraic Geometry I Complex Projective Varieties, 156–80. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/978-3-642-61833-8_8.

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Liedtke, Christian. "Algebraic Surfaces in Positive Characteristic." In Birational Geometry, Rational Curves, and Arithmetic, 229–92. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-6482-2_11.

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Várilly-Alvarado, Anthony. "Arithmetic of Del Pezzo surfaces." In Birational Geometry, Rational Curves, and Arithmetic, 293–319. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-6482-2_12.

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Ciliberto, Ciro, Thomas Dedieu, Concettina Galati, and Andreas Leopold Knutsen. "A Note on Severi Varieties of Nodal Curves on Enriques Surfaces." In Birational Geometry and Moduli Spaces, 29–36. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-37114-2_3.

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Kovács, Sándor J. "The Cone of Curves of K3 Surfaces Revisited." In Birational Geometry, Rational Curves, and Arithmetic, 163–69. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-6482-2_8.

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Bertram, Aaron, and Izzet Coskun. "The Birational Geometry of the Hilbert Scheme of Points on Surfaces." In Birational Geometry, Rational Curves, and Arithmetic, 15–55. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-6482-2_2.

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Friedman, Robert. "Birational Geometry." In Universitext, 59–83. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-1688-9_4.

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Smith, Karen E., Lauri Kahanpää, Pekka Kekäläinen, and William Traves. "Birational Geometry." In An Invitation to Algebraic Geometry, 105–22. New York, NY: Springer New York, 2000. http://dx.doi.org/10.1007/978-1-4757-4497-2_7.

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Li, Tian-Jun, and Yongbin Ruan. "Symplectic birational geometry." In CRM Proceedings and Lecture Notes, 307–26. Providence, Rhode Island: American Mathematical Society, 2009. http://dx.doi.org/10.1090/crmp/049/12.

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Conference papers on the topic "Birational geometry of surfaces"

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BIRKAR, CAUCHER. "BIRATIONAL GEOMETRY OF ALGEBRAIC VARIETIES." In International Congress of Mathematicians 2018. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789813272880_0068.

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POPA, MIHNEA. "𝒟-MODULES IN BIRATIONAL GEOMETRY." In International Congress of Mathematicians 2018. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789813272880_0077.

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Hacon, Christopher D., and James McKernan. "Boundedness Results in Birational Geometry." In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010). Published by Hindustan Book Agency (HBA), India. WSPC Distribute for All Markets Except in India, 2011. http://dx.doi.org/10.1142/9789814324359_0058.

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Katzarkov, Ludmil. "Birational geometry and homological mirror symmetry." In Proceedings of the Australian-Japanese Workshop. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812706898_0008.

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HACON, CHRISTOPHER D. "CAUCHER BIRKAR’S WORK IN BIRATIONAL ALGEBRAIC GEOMETRY." In International Congress of Mathematicians 2018. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789813272880_0003.

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Koiller, Jair, Stefanella Boatto, Fernando Etayo, Mario Fioravanti, and Rafael Santamaría. "Vortex pairs on surfaces." In GEOMETRY AND PHYSICS: XVII International Fall Workshop on Geometry and Physics. AIP, 2009. http://dx.doi.org/10.1063/1.3146241.

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Takeuchi, Nobuko. "Surfaces which contain many circles." In Geometry and topology of caustics. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc82-0-14.

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Peng, Chia-Kuei, and Liang Xiao. "Willmore Surfaces and Minimal Surfaces with Flat Ends." In Differential Geometry in Honor of Professor S S Chern. WORLD SCIENTIFIC, 2000. http://dx.doi.org/10.1142/9789812792051_0021.

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Babiker, Hassan. "Projections of surfaces with singular boundary." In Geometry and topology of caustics. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc82-0-1.

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Joets, Alain. "Singularities in drawings of singular surfaces." In Geometry and topology of caustics. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc82-0-10.

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Reports on the topic "Birational geometry of surfaces"

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Munteanu, Marian I., and Ana I. Nistor. New Results on the Geometry of the Translation Surfaces. GIQ, 2012. http://dx.doi.org/10.7546/giq-11-2010-157-169.

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Munteanu, Marian I. New Results on the Geometry of the Translation Surfaces. Journal of Geometry and Symmetry in Physics, 2012. http://dx.doi.org/10.7546/jgsp-18-2010-49-62.

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Ludu, Andrei. Differential Geometry of Moving Surfaces and its Relation to Solitons. GIQ, 2012. http://dx.doi.org/10.7546/giq-12-2011-43-69.

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Brosnahan and DeVries. PR-317-10702-R01 Testing for the Dilation Strength of Salt. Chantilly, Virginia: Pipeline Research Council International, Inc. (PRCI), December 2011. http://dx.doi.org/10.55274/r0010026.

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A laboratory testing program on rock salt specimens was performed using test conditions that are consistent with the stresses that are experienced near the surfaces of salt caverns during storage operation. The proposed work effort focuses on improving the methodology for defining the onset of dilation for rock salt. Geomechanical studies use dilation criteria to assess the potential for salt damage that can lead to spalling in the cavern roof and/or walls and subsequent damage to the cavern or hanging string. This constraint is often the one that limits the minimum gas pressure in a natural gas storage cavern. This report documents the PRCI funded follow-on activities to the recently completed Gas Storage Technology Consortium project [DeVries, 2010]. The work activities completed include the following: Laboratory dilation strength testing of eight specimens having preconditioning durations longer than 10 days. Numerical modeling to identify and optimize an appropriate specimen shape for dilation testing in triaxial extension states of stress. Laboratory constant mean stress extension testing on the optimized specimen shape. DeVries [2010] documented the effects of the preconditioning durations on the dilation strength of salt specimens. Preconditioning of specimens is the process whereby specimens are subject to a relatively high hydrostatic stress for a specified period of time. It is believed that preconditioning mitigates some of the damage to the specimens induced by coring, transporting, and specimen preparation. The study documented by DeVries [2010] suggests that increasing the preconditioning duration increases the dilation strength of salt, with the maximum precondition duration limited to 10 days. This project expands upon these findings through additional testing to determine if preconditioning durations longer than 10 days has any additional benefit. In addition to the preconditioning task, this study will also investigate the variability issues observed during dilation strength tests performed under triaxial extension states of stress. It is hypothesized that the high variability seen in extensional test results might be attributed to end effects caused by (1) the friction at the specimen-platen interface and (2) specimens breaking outside the range measured by gages. To help reduce frictional effects and breakage location issues, numerical models of alternate specimen shapes were created to provide a basis for testing a new specimen geometry. Laboratory tests were performed on the new specimen geometry to validate any of its possible benefits.
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Chen, Weixing. PR378-173601-Z01 Effect of Pressure Fluctuations on the Growth Rate of Near-Neutral pH SCC. Chantilly, Virginia: Pipeline Research Council International, Inc. (PRCI), July 2021. http://dx.doi.org/10.55274/r0012112.

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This report summarizes the work completed in PRCI SCC-2-12A project: The Effect of Pressure Fluctuations on the Growth Rate of Near-Neutral pH SCC, which is Phase 3 of the work on the same subject of investigation. The following insights from the current phase of the PRCI SCC-2-12A project are thought to be the most important: - Near neutral pH crack initiation is pressure-fluctuation dependent. Severe pressure fluctuations accelerate the fracture and spallation of mill scale on the pipeline steel surfaces, making it harder to initiate SCC cracks from the bottom of pits that are developed at flawed mill scale sites. On the other hand, the presence of a primer layer before application of the protective coating preserves the mill scale on the pipe steel surface and promotes crack initiation. - The early-stage crack growth primarily features crack length extension on the pipe surface but limited crack growth in the depth direction. Three different mechanisms of crack length extension have been identified, including that determined by the geometry of coating disbondment, a chaotic process of crack coalescence, and the ability of existing cracks to induce further crack initiation and growth. This latter process is pressure-fluctuation sensitive. - A complete set of equations governing crack growth in Stage 2 has been established based on experimental specimens with surface cracks under mechanical loading conditions realistic to pressure fluctuations during the operation of oil and gas pipelines. - The contribution to crack growth by direct dissolution of the steel at the crack tip has been determined, which has been found to be crack depth-dependent and pressure-fluctuation-sensitive. Gas pipelines operated under high mean pressure show higher rates of dissolution. - The severity of crack growth and the accuracy of the predictive model can be significantly affected by crack tip morphology, either sharp or blunt, and this would yield different threshold values for Stage 2 crack growth and therefore different lengths of remaining life. - Full scale testing was performed and has validated the crack growth models contained herein. - The PipeOnline software has been revised to incorporate the new experimental results obtained from the current PRCI SCC 2-12A project. This PipeOnline software was previously developed from the two earlier phases of the PRCI project.
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Snyder, Victor A., Dani Or, Amos Hadas, and S. Assouline. Characterization of Post-Tillage Soil Fragmentation and Rejoining Affecting Soil Pore Space Evolution and Transport Properties. United States Department of Agriculture, April 2002. http://dx.doi.org/10.32747/2002.7580670.bard.

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Tillage modifies soil structure, altering conditions for plant growth and transport processes through the soil. However, the resulting loose structure is unstable and susceptible to collapse due to aggregate fragmentation during wetting and drying cycles, and coalescense of moist aggregates by internal capillary forces and external compactive stresses. Presently, limited understanding of these complex processes often leads to consideration of the soil plow layer as a static porous medium. With the purpose of filling some of this knowledge gap, the objectives of this Project were to: 1) Identify and quantify the major factors causing breakdown of primary soil fragments produced by tillage into smaller secondary fragments; 2) Identify and quantify the. physical processes involved in the coalescence of primary and secondary fragments and surfaces of weakness; 3) Measure temporal changes in pore-size distributions and hydraulic properties of reconstructed aggregate beds as a function of specified initial conditions and wetting/drying events; and 4) Construct a process-based model of post-tillage changes in soil structural and hydraulic properties of the plow layer and validate it against field experiments. A dynamic theory of capillary-driven plastic deformation of adjoining aggregates was developed, where instantaneous rate of change in geometry of aggregates and inter-aggregate pores was related to current geometry of the solid-gas-liquid system and measured soil rheological functions. The theory and supporting data showed that consolidation of aggregate beds is largely an event-driven process, restricted to a fairly narrow range of soil water contents where capillary suction is great enough to generate coalescence but where soil mechanical strength is still low enough to allow plastic deforn1ation of aggregates. The theory was also used to explain effects of transient external loading on compaction of aggregate beds. A stochastic forInalism was developed for modeling soil pore space evolution, based on the Fokker Planck equation (FPE). Analytical solutions for the FPE were developed, with parameters which can be measured empirically or related to the mechanistic aggregate deformation model. Pre-existing results from field experiments were used to illustrate how the FPE formalism can be applied to field data. Fragmentation of soil clods after tillage was observed to be an event-driven (as opposed to continuous) process that occurred only during wetting, and only as clods approached the saturation point. The major mechanism of fragmentation of large aggregates seemed to be differential soil swelling behind the wetting front. Aggregate "explosion" due to air entrapment seemed limited to small aggregates wetted simultaneously over their entire surface. Breakdown of large aggregates from 11 clay soils during successive wetting and drying cycles produced fragment size distributions which differed primarily by a scale factor l (essentially equivalent to the Van Bavel mean weight diameter), so that evolution of fragment size distributions could be modeled in terms of changes in l. For a given number of wetting and drying cycles, l decreased systematically with increasing plasticity index. When air-dry soil clods were slightly weakened by a single wetting event, and then allowed to "age" for six weeks at constant high water content, drop-shatter resistance in aged relative to non-aged clods was found to increase in proportion to plasticity index. This seemed consistent with the rheological model, which predicts faster plastic coalescence around small voids and sharp cracks (with resulting soil strengthening) in soils with low resistance to plastic yield and flow. A new theory of crack growth in "idealized" elastoplastic materials was formulated, with potential application to soil fracture phenomena. The theory was preliminarily (and successfully) tested using carbon steel, a ductile material which closely approximates ideal elastoplastic behavior, and for which the necessary fracture data existed in the literature.
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