Exploiting Torus Actions: Immaculate Line Bundles on Toric Varieties and Parametrizations of Gröbner Cells

This dissertation contains two chapters on the use of torus actions in algebraic geometry. In chapter 2 we study ”immaculate line bundles” on projective toric varieties. The cohomology groups of those line bundles vanish in all degrees, including the 0-th degree. Immaculate line bundles can be seen as building blocks of full exceptional sequences of line bundles of the variety. All the immaculate line bundles of a toric variety X = TV(Σ) can be identified in two steps. First identify those subsets of the rays Σ(1) whose geometric realization is not k-acyclic, they will be called tempting. Those subsets of the rays give ”maculate sets/regions” in the class group of the variety. A line bundle is immaculate, if it is not in any of those maculate sets. So the first step in finding immaculate line bundles is to find all tempting subsets. When X is projective, the main result for this is that primitive collections – subsets of the rays that do not span a cone, but each proper subset spans a cone – are always tempting. And a subset of rays can only be tempting if it is the union of primitive collections. The same has to hold for the complement, too. We give descriptions of the immaculate line bundles for different examples. In particular, we describe the immaculate locus for projective toric varieties of Picard rank 3. Most of the results have been published in [ABKW20]. In chapter 3 we study the Hilbert scheme of n points in affine plane. It describes all ideals in the polynomial ring of two variables whose quotient is an n-dimensional vector space. The Hilbert scheme can be decomposed into so called Gröbner cells. They consist of those ideals that have a prescribed leading term ideal with respect to a given term order. The Gröbner cells for the lexicographic and the degree-lexicographic order are parametrized in [CV08] and [Con11], respectively, by canonical Hilbert-Burch matrices. A Hilbert-Burch matrix of an ideal is a matrix generating the syzygies of the ideal. Its maximal minors also generate the ideal. These results are generalized in two directions. Firstly, we consider the ring of formal power series. Here we give a parametrization of the cells that respects the Hilbert function stratification of the punctual Hilbert scheme. In particular, this cellular decomposition restricts to a cellular decomposition of the subscheme consisting of ideals with a prescribed Hilbert function. We use the parametrization to describe subsets of the Gröbner cells associated to lex-segment ideals with a given minimal number of generators. These subsets are quasi-affine varieties inside the cell. Most of these results have been published in [HW21] and [HW23]. The second way of changing the setting is to consider a general term order on the polynomial ring. We give a surjection to the Gröbner cell with respect to this ordering and parametrizations of subsets of the cell, as well as a conjecture how the parametrization of the whole cell should look like. We also study intersections of Gröbner cells with respect to different term orders.Die vorliegende Dissertation besteht aus zwei Kapiteln zu zwei unterschiedlichen Anwendungen von Toruswirkungen in der algebraischen Geometrie. Die wichtigsten Objekte des Kapitels 2 sind unbefleckte Geradenbündel auf projektiven torischen Varietäten X = TV(Σ), Geradenbündel, deren Kohomologiegruppen alle verschwinden. Unbefleckte Geradenbündel können als Bausteine für exzeptionelle Sequenzen aus Geradenbündeln dienen und somit die derivierte Kategorie der Varietät beschreiben. Die Bestimmung von unbefleckten Geradenbündeln lässt sich in zwei Schritte aufteilen. Es lassen sich Teilmengen der Strahlen Σ(1) des die torische Varietät beschreibenden Fächers Σ identifizieren, deren geometrische Realisierungen nicht k-azyklisch sind. Diese verlockenden Teilmengen der Strahlen definieren befleckte Teilmengen der Klassengruppe Cl(X). Ein Geradenbündel ist genau dann unbefleckt, wenn es in keiner befleckten Teilmenge von Cl(X) liegt. Die Bestimmung aller unbefleckten Geradenbündel lässt sich also in zwei Schritte aufteilen. Das Bestimmen der verlockenden Teilmengen der Strahlen und das Bestimmen der zugehörigen befleckten Regionen. Primitive Kollektionen – Teilmengen der Strahlen, die selbst keinen Kegel des Fächers aufspannen, aber jede ihrer Teilmenge spannt einen Kegel des Fächers auf – sind verlockend und außerdem ist eine Teilmenge nur dann verlockend, wenn sie eine Vereinigung von primitiven Kollektionen ist. Dies muss auch für das Komplement gelten. Wir geben die Beschreibung für die unbefleckten Geradenbündel für verschiedene Beispielklassen von projektiven torischen Varietäten. Insbesondere beschreiben wir die unbefleckten Geradenbündel für projektive torische Varietäten von Picardrang 3. Die meisten dieser Ergebnisse sind in [ABKW20] erschienen. In Kapitel 3 geht es um das Hilbertschema von n Punkten in der affinen Ebene. Seine Punkte sind Ideale im Polynomenring k[x, y], deren Quotient ein n-dimensionaler k-Vektorraum ist. Das Hilbertschema kann in sogenannte Gröbnerzellen unterteilt werden. Sie umfassen Ideale, die bezüglich einer Termordnung τ ein festgelegtes Leitideal haben. In [CV08] und [Con11] werden für die lexikographische und gradlexikographische Termordnung Parametrisierung der Gröbnerzellen durch kanonische Hilbert-Burch Matrizen angegeben. Hilbert-Burch Matrizen beschreiben die Syzygien des Ideals und ihre maximalen Minoren erzeugen das Ideal. Die Ergebnisse werden in zwei Richtungen verallgemeinert. Zunächst betrachten wir Ideale im Ring der formalen Potenzreihen. Wir geben eine Parametrisierung der Zellen, bei der die lokale Struktur der Ideale berücksichtigt wird. Insbesondere lässt sich diese zelluläre Unterteilung des lokalen Hilbertschemas auf eine zelluläre Unterteilung des Unterschemas einschränken, das nur Ideale mit einer gegebenen Hilbertfunktion beinhaltet. Durch diese Parametrisierung lassen sich für Ideale in diesen Zellen kanonische Hilbert-Burch Matrizen definieren. Diese benutzen wir um Teilmengen der Gröbnerzellen mit einer vorgegebenen minimalen Anzahl von Erzeugern zu beschreiben. Diese Teilmengen sind quasi-affine Varietäten in der Gröbnerzelle. Die meisten der Resultate sind in [HW21] und [HW23] erschienen. Die zweite Möglichkeit das Setting zu ändern, ist beliebige Termordnungen auf dem Polynomenring zu betrachten. Im zweiten Teil von Kapitel 3 geben wir eine Surjektion auf diese Gröbnerzellen, sowie Parametrisierungen von Teilmengen und geben eine Vermutung, wie eine Parametrisierung der ganzen Zelle aussieht. Außerdem untersuchen wir Schnitte von Gröbnerzellen bezüglich verschiedener Termordnungen

Tropical Derivation of Cohomology Ring of Heavy/Light Hassett Spaces

The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as \calm_{g, w} for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for these heavy/light Hassett spaces. For $g = 0$, we want to find the tropicalization of \calm_{0, w}, a polyhedral complex parametrizing leaf-labeled metric trees that can be thought of as Bergman fan, which furthermore creates a toric variety $X_{\Sigma}$. We use the presentation of \overline{\calm}_{0,w} as a tropical compactification associated to an explicit Bergman fan, to give a concrete presentation of the cohomology

Applications of monodromy in solving polynomial systems

Polynomial systems of equations that occur in applications frequently have a special structure. Part of that structure can be captured by an associated Galois/monodromy group. This makes numerical homotopy continuation methods that exploit this monodromy action an attractive choice for solving these systems; by contrast, other symbolic-numeric techniques do not generally see this structure. Naturally, there are trade-offs when monodromy is chosen over other methods. Nevertheless, there is a growing literature demonstrating that the trade can be worthwhile in practice. In this thesis, we consider a framework for efficient monodromy computation which rivals the state-of-the-art in homotopy continuation methods. We show how its implementation in the package MonodromySolver can be used to efficiently solve challenging systems of polynomial equations. Among many applications, we apply monodromy to computer vision---specifically, the study and classification of minimal problems used in RANSAC-based 3D reconstruction pipelines. As a byproduct of numerically computing their Galois/monodromy groups, we observe that several of these problems have a decomposition into algebraic subproblems. Although precise knowledge of such a decomposition is hard to obtain in general, we determine it in some novel cases.Ph.D

Computing Intersection and Self-intersection Loci of Parametrized Surfaces Using Regular Systems and Gröbner Bases

International audienceThis paper presents two general and efficient methods for determining intersection and self-intersection loci of rationally parametrized surfaces. One of the methods, based on regular systems, is capable of computing the exact parametric loci of intersection and self-intersection. The other, based on Gröbner bases, can compute the minimal varieties passing through the exact parametric loci. The relation between the results computed by the two methods is established and algorithms for computing parametric loci of intersection and self-intersection are described. Experimental results and comparisons with some existing methods show that our algorithms have a good performance

Computing Self-intersection Loci of Parametrized Surfaces Using Regular Systems and {Gröbner} Bases

International audienceThe computation of self-intersection loci of parametrized surfaces is needed for constructing trimmed parametrizations and describing the topology of the considered surfaces in real settings. This paper presents two general and efficient methods for determining self-intersection loci of rationally parametrized surfaces. One of the methods, based on regular systems, is capable of computing the exact parametric locus of self-intersection of a given surface and the other, based on Grobner bases, can compute the minimal variety passing through the exact parametric locus. The relation between the results computed by the two methods is established and two algorithms for computing parametric loci of self-intersection are described. Experimental results and comparisons with some existing methods show that our algorithms have a good performance for parametrized surfaces

Advances in Robot Kinematics : Proceedings of the 15th international conference on Advances in Robot Kinematics

International audienceThe motion of mechanisms, kinematics, is one of the most fundamental aspect of robot design, analysis and control but is also relevant to other scientific domains such as biome- chanics, molecular biology, . . . . The series of books on Advances in Robot Kinematics (ARK) report the latest achievement in this field. ARK has a long history as the first book was published in 1991 and since then new issues have been published every 2 years. Each book is the follow-up of a single-track symposium in which the participants exchange their results and opinions in a meeting that bring together the best of world’s researchers and scientists together with young students. Since 1992 the ARK symposia have come under the patronage of the International Federation for the Promotion of Machine Science-IFToMM.This book is the 13th in the series and is the result of peer-review process intended to select the newest and most original achievements in this field. For the first time the articles of this symposium will be published in a green open-access archive to favor free dissemination of the results. However the book will also be o↵ered as a on-demand printed book.The papers proposed in this book show that robot kinematics is an exciting domain with an immense number of research challenges that go well beyond the field of robotics.The last symposium related with this book was organized by the French National Re- search Institute in Computer Science and Control Theory (INRIA) in Grasse, France

College of Arts and Sciences

Cornell University Courses of Study Vol. 101 2009/201