Averaging Residue Currents and the Stückrad–Vogel Algorithm

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant KΘ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H1 and H2 have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) KΘ(H1) and KΘ(H2) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ0 be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ0) for different varieties Θ and also the groups of autoequivalences of Θ0. The problem is to describe these groups for different Θ.

On the Hilbert function of Artinian local complete intersections of codimension three

Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground elds to arbitrary non-archimedean valued elds. To achieve this, we develop a theory of toric schemes over valuation rings of rank 1. As a basic tool, we use techniques from non-archimedean analysis. MSC2010: 14T05, 14M25, 32P05 X) of polyhedra in R n . This process is called tropicaliza- tion and it can be used to transform a problem from algebraic geometry into a corresponding problem in convex geometry which is usually easier. If the toric co- ordinates are well suited to the problem, it is sometimes possible to use a solution of the convex problem to solve the original algebraic problem. Another strategy is to vary the ambient torus to compensate the loss of information due to the tropi- calization process. Tropicalization originates from a paper of Bergman (Berg) on logarithmic limit sets. The convex structure of the tropical variety Trop(X) was worked out by Bieri{Groves (BG) with applications to geometric group theory in mind. Sturmfels (Stu) pointed out that Trop(X) is a subcomplex of the Grobner complex. In fact, the polyhedral complex Trop(X) has some natural weights satisfying a balancing condition which appears rst in Speyer's thesis (Spe). This relies on the description

The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well. This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.

This Appendix is a somewhat expanded version of the “Appendix on Complete Intersections” that appeared in the original (1978) version of this book. The goal here is to offer some comments on the problem of complete intersections in algebraic geometry, and to explain some of its connections to Serre’s Conjecture in the period 1955–1976 (when the Conjecture stood open). Since such discussions are still essential today in coming to a full understanding of Serre’s Conjecture, I have chosen to preserve the Appendix in this version of the book, in spite of the fact that the main text of it was written in 1977.

This article discusses the geometric application of the method of multiplier ideal sheaves. It first briefly describes its application to effective problems in algebraic geometry and then presents and explains its application to the deformational invariance of plurigenera for general compact algebraic manifolds. Finally its application to the conjecture of the finite generation of the canonical ring is explored and the use of complex algebraic geometry in complex Neumann estimates is discussed.

One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kahler metric leads to all sorts of Hodge theoretical restrictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it. In spite of the differential-geometric nature of the questions and methods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main over-arching problem in algebraic geometry is to understand the classification of algebro-geometric objects. The topology of the usual complex-valued points of a variety plays an important role, because the topological type is a locally constant function on any classifying space. Thus the partitioning of the classification problem according to topological type provides a coarse, and more calculable, alternative to the partition by connected components. Furthermore, the topology of a variety strongly influences its geometric properties. A sociological observation is that the quest for understanding the topology of algebraic varieties has led to a rich set of techniques which found applications elsewhere, even in physics. Perhaps a word about the choice of ground field is appropriate. One could also look at the shapes of the real points of real algebraic varieties. However, by Weierstrass approximation, pretty much anything can arise if you let the degree get big enough. Thus the classical question in this case is “which shapes can occur for a given degree?” This is so much more difficult

In a series of works on uniruled projective manifolds starting in the late 1990’s, Jun-Muk Hwang and the author have developed the basics of a geometric theory of uniruled projective manifolds arising from the study of varieties of minimal rational tangents (VMRTs), i.e., the collection at a general point of tangents to minimal rational curves passing through the point. From its onset, our theory is a cross-over between algebraic geometry and differential geometry. While we deal with problems in algebraic geometry, the heart of our perspective is differential-geometric in nature, revolving around foliations, G-structures, differential systems, etc. and dealing with various issues relating to connections, curvature and integrability.The current article is written with the aim of highlighting certain aspects in the geometric theory of VMRTs revolving around the theme of analytic continuation of geometric structures and substructures. For the parts of the article where adequate exposition already exists, we recall fundamental elements and results in the theory essential for the understanding of more recent development and provide occasional examples for illustration. The presentation will be more systematic on sub-VMRT structures since the latter topic is relatively new. We will discuss various perspectives concerning sub-VMRT structures, and indicate how the subject has intimate links with other areas of mathematics including several complex variables, local differential geometry and Kahler geometry.

Geography of projective varieties is one of the fundamental problems in algebraic geometry. There are many researches toward the characteristics of Chern number of some projective spaces, for example Noethers inequalities, the theorem of Chang-Lopez, and the Miyaoka-Yau inequality. In this paper, we compute the Chern numbers of any smooth complete intersection threefold in the product of projective spaces via the standard exact sequences of cotangent bundles. Then we obtain linear Chern number inequalities for (c_1 (X)c_2 (X))/(c_1^3 (X)) and (c_3 (X))/(c_1^3 (X)) on such threefolds under conditions of d_ij4 and d_ij6 respectively. They can be considered as a generalization of the Miyaoka-Yau inequality and an improvement of Yaus inequality for such threefolds.

This article is devoted to an empirical study of metacognitive decision-making strategies and their relationship with students’ academic performance. The aim of the study is to monitor the impact of metacognitive strategies in subject-specific mathematics training on increasing students’ metacognitive engagement in educational activities. To achieve this goal, an intervention program based on metacognitive strategies was developed for first-year students studying linear algebra and analytic geometry. Two approaches were used: strategy training and a supportive social environment. Students from Saratov National Research State University and Omsk State Technical University participated in the experiment. Statistical processing of the obtained data was performed using the Python programming language. Descriptive statistics were used, and sample comparisons were carried out using the Mann-Whitney U-test. The empirical results showed that the integration of metacognitive strategies transforms the study of linear algebra and analytic geometry from passive information acquisition into an active, conscious dialogue with one’s own thinking. This isn’t simply developing the ability to “solve more problems,” but to connect new knowledge with existing ones and “solve problems with constant self-reflection.” The experiment proved quite successful in expanding knowledge about cognition, which includes three distinct but interrelated characteristics of knowledge (declarative, procedural, and conditional knowledge). Furthermore, it was found that the developed metacognitive skills help students with low abilities compensate for their deficiencies in solving mathematical problems, develop emotional intelligence, and a desire for personal growth.

In this paper we formulate an abstract quotient theorem for convergent maps of PO-spaces (4.3), being inspired by the work [6] of I. F. Donin. This theorem, which is proved in detail in [5], gives a criterion for certain quotient groupoids over the category of germs of analytic spaces to have a semiuniversal deformation. With this result in hands, one can solve many local moduli problems in analytic geometry in a unified manner, see [3, 4] and [5]. Let us finally remark that we use the notions and notations introduced in [2] without any further comment.

Empirical evidence in literature identified significant association between spatial ability and educational performance particularly in science, technology, engineering and mathematics (STEM). The purpose of this study was to explore pre-service teachers’ spatial skills in solving mathematics problems, in the context of coordinate geometry. It is envisaged that spatial skills allow for the perception of visual information and, therefore, spatial cognition has been considered as a key skill in teaching mathematics. However, literature asserts that teachers are ill prepared to teach mathematics, hence there is limited use and misuse of spatial skills in teaching the subject. This study, therefore, examines the spatial orientation of pre-service teachers in teaching coordinate geometry. This is a mixed methods study in which pre-service teachers answered a coordinate geometry test to explore their content knowledge and their ability to interpret, analyse and apply visual spatial models to solve mathematical problems in coordinate geometry. The study established that the spatial orientation skills of pre-service teachers determine their performance in mathematics, especially coordinate geometry.

In this survey, we discuss some basic problems in representation theory of finite groups, including some long-standing conjectures of Alperin, Brauer, and others. A possible approach to some of these problems is to use the classification of finite simple groups to reduce the problem in consideration to some, more specific, questions about simple groups. We will describe recent progress on reduction theorems in this direction. We will also outline applications of these results to various problems in group theory, number theory, and algebraic geometry.

Methods of computer algebra become more familiar to a wide audience of theoretical mathematicians and physicists. The environment of computer algebra system leads to a greater acceptance of computer instruments in the mathematical research. Methods of symbolic manipulation provided by computer algebra systems in combination with high-power number crunching abilities of traditional hardware and software open the way to truly large scale computations often needed by mathematicians and physicists. This chapter describes new low complexity (both operational and logical) methods and algorithms for computations of solutions of differential equations and their efficient evaluation, and of solution of algebraic equations. These computations are often basic in applied problems, but even more so in a variety of problems of number theory and algebraic geometry. Power series manipulations over various rings and fields are one of the most important features of advanced computer algebra systems.

The Stacks Project Expository Collection (SPEC) compiles expository articles in advanced algebraic geometry, intended to bring graduate students and researchers up to speed on recent developments in the geometry of algebraic spaces and algebraic stacks. The articles in the text make explicit in modern language many results, proofs, and examples that were previously only implicit, incomplete, or expressed in classical terms in the literature. Where applicable this is done by explicitly referring to the Stacks project for preliminary results. Topics include the construction and properties of important moduli problems in algebraic geometry (such as the Deligne–Mumford compactification of the moduli of curves, the Picard functor, or moduli of semistable vector bundles and sheaves), and arithmetic questions for fields and algebraic spaces.