Lecture Notes for the Algebraic Geometry Course Held by Rahul Pandharipande
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1 Affine Varieties
1 Affine Varieties We will begin following Kempf's Algebraic Varieties, and eventually will do things more like in Hartshorne. We will also use various sources for commutative algebra. What is algebraic geometry? Classically, it is the study of the zero sets of polynomials. We will now fix some notation. k will be some fixed algebraically closed field, any ring is commutative with identity, ring homomorphisms preserve identity, and a k-algebra is a ring R which contains k (i.e., we have a ring homomorphism ι : k ! R). P ⊆ R an ideal is prime iff R=P is an integral domain. Algebraic Sets n n We define affine n-space, A = k = f(a1; : : : ; an): ai 2 kg. n Any f = f(x1; : : : ; xn) 2 k[x1; : : : ; xn] defines a function f : A ! k : (a1; : : : ; an) 7! f(a1; : : : ; an). Exercise If f; g 2 k[x1; : : : ; xn] define the same function then f = g as polynomials. Definition 1.1 (Algebraic Sets). Let S ⊆ k[x1; : : : ; xn] be any subset. Then V (S) = fa 2 An : f(a) = 0 for all f 2 Sg. A subset of An is called algebraic if it is of this form. e.g., a point f(a1; : : : ; an)g = V (x1 − a1; : : : ; xn − an). Exercises 1. I = (S) is the ideal generated by S. Then V (S) = V (I). 2. I ⊆ J ) V (J) ⊆ V (I). P 3. V ([αIα) = V ( Iα) = \V (Iα). 4. V (I \ J) = V (I · J) = V (I) [ V (J). Definition 1.2 (Zariski Topology). We can define a topology on An by defining the closed subsets to be the algebraic subsets. -
Algebraic Geometry Part III Catch-Up Workshop 2015
Algebraic Geometry Part III Catch-up Workshop 2015 Jack Smith June 6, 2016 1 Abstract Algebraic Geometry This workshop will give a basic introduction to affine algebraic geometry, assuming no prior exposure to the subject. In particular, we will cover: • Affine space and algebraic sets • The Hilbert basis theorem and applications • The Zariski topology on affine space • Irreducibility and affine varieties • The Nullstellensatz • Morphisms of affine varieties. If there's time we may also touch on projective varieties. What we expect you to know • Elementary point-set topology: topological spaces, continuity, closure of a subset etc • Commutative algebra, at roughly the level covered in the Rings and Modules workshop: rings, ideals (including prime and maximal) and quotients, algebras over fields (in particular, some familiarity with polynomial rings over fields). Useful for Part III courses Algebraic Geometry, Commutative Algebra, Elliptic Curves 2 Talk 2.1 Preliminaries Useful resources: • Hartshorne `Algebraic Geometry' (classic textbook, on which I think this year's course is based, although it's quite dense; I'll mainly try to match terminology and notation with Chapter 1 of this book). • Ravi Vakil's online notes `Math 216: Foundations of Algebraic Geometry'. • Eisenbud `Commutative Algebra with a view toward algebraic geometry' (covers all the algebra you might need, with a geometric flavour|it has pictures). 1 • Pelham Wilson's online notes for the `Preliminary Chapter 0' of his Part III Algebraic Geometry course from last year cover much of this catch-up material but are pretty brief (warning: this year's course has a different lecturer so will be different). -
Segre Embedding and Related Maps and Immersions in Differential Geometry
SEGRE EMBEDDING AND RELATED MAPS AND IMMERSIONS IN DIFFERENTIAL GEOMETRY BANG-YEN CHEN Abstract. Segre embedding was introduced by C. Segre (1863–1924) in his famous 1891 article [50]. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathemat- ical physics, and coding theory. In this article, we survey main results on Segre embedding in differential geometry. Moreover, we also present recent differential geometric results on maps and immersions which are constructed in ways similar to Segre embedding. Table of Contents.1 1. Introduction. 2. Basic formulas and definitions. 3. Differential geometric characterizations of Segre embedding. 4. Degree of K¨ahlerian immersions and homogeneous K¨ahlerian subman- ifolds via Segre embedding. 5. CR-products and Segre embedding. 6. CR-warped products and partial Segre CR-immersions. 7. Real hypersurfaces as partial Segre embeddings. 8. Complex extensors, Lagrangian submanifolds and Segre embedding. 9. Partial Segre CR-immersions in complex projective space. 10. Convolution of Riemannian manifolds. Arab. J. Math. Sci. 8 (2002), 1–39. 2000 Mathematics Subject Classification. Primary 53-02, 53C42, 53D12; Secondary arXiv:1307.0459v1 [math.AG] 1 Jul 2013 53B25, 53C40, 14E25. Key words and phrases. Segre embedding, totally real submanifold, Lagrangian subman- ifold, CR-submanifold, warped product, CR-warped product, complex projective space, geo- metric inequality, convolution, Euclidean Segre map, partial Segre map, tensor product im- mersion, skew Segre embedding. 1 The author would like to express his thanks to the editors for the invitation to publish this article in this Journal 1 2 BANG-YEN CHEN 11. Convolutions and Euclidean Segre maps. -
Chapter 2 Affine Algebraic Geometry
Chapter 2 Affine Algebraic Geometry 2.1 The Algebraic-Geometric Dictionary The correspondence between algebra and geometry is closest in affine algebraic geom- etry, where the basic objects are solutions to systems of polynomial equations. For many applications, it suffices to work over the real R, or the complex numbers C. Since important applications such as coding theory or symbolic computation require finite fields, Fq , or the rational numbers, Q, we shall develop algebraic geometry over an arbitrary field, F, and keep in mind the important cases of R and C. For algebraically closed fields, there is an exact and easily motivated correspondence be- tween algebraic and geometric concepts. When the field is not algebraically closed, this correspondence weakens considerably. When that occurs, we will use the case of algebraically closed fields as our guide and base our definitions on algebra. Similarly, the strongest and most elegant results in algebraic geometry hold only for algebraically closed fields. We will invoke the hypothesis that F is algebraically closed to obtain these results, and then discuss what holds for arbitrary fields, par- ticularly the real numbers. Since many important varieties have structures which are independent of the field of definition, we feel this approach is justified—and it keeps our presentation elementary and motivated. Lastly, for the most part it will suffice to let F be R or C; not only are these the most important cases, but they are also the sources of our geometric intuitions. n Let A denote affine n-space over F. This is the set of all n-tuples (t1,...,tn) of elements of F. -
1. Affine Varieties
6 Andreas Gathmann 1. Affine Varieties As explained in the introduction, the goal of algebraic geometry is to study solutions of polynomial equations in several variables over a fixed ground field, so we will start by making the corresponding definitions. In order to make the correspondence between geometry and algebra as clear as possible (in particular to allow for Hilbert’s Nullstellensatz in Proposition 1.10) we will restrict to the case of algebraically closed ground fields first. In particular, if we draw pictures over R they should always be interpreted as the real points of an underlying complex situation. Convention 1.1. (a) In the following, until we introduce schemes in Chapter 12, K will always denote a fixed algebraically closed ground field. (b) Rings are always assumed to be commutative with a multiplicative unit 1. If J is an ideal in a ring R we will write this as J E R, and denote the radical of J by p k J := f f 2 R : f 2 J for some k 2 Ng: The ideal generated by a subset S of a ring will be written as hSi. (c) As usual, we will denote the polynomial ring in n variables x1;:::;xn over K by K[x1;:::;xn], n and the value of a polynomial f 2 K[x1;:::;xn] at a point a = (a1;:::;an) 2 K by f (a). If there is no risk of confusion we will often denote a point in Kn by the same letter x as we used for the formal variables, writing f 2 K[x1;:::;xn] for the polynomial and f (x) for its value at a point x 2 Kn. -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
A Course in Commutative Algebra
Graduate Texts in Mathematics 256 A Course in Commutative Algebra Bearbeitet von Gregor Kemper 1. Auflage 2010. Buch. xii, 248 S. Hardcover ISBN 978 3 642 03544 9 Format (B x L): 0 x 0 cm Gewicht: 1200 g Weitere Fachgebiete > Mathematik > Algebra Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Chapter 1 Hilbert’s Nullstellensatz Hilbert’s Nullstellensatz may be seen as the starting point of algebraic geom- etry. It provides a bijective correspondence between affine varieties, which are geometric objects, and radical ideals in a polynomial ring, which are algebraic objects. In this chapter, we give proofs of two versions of the Nullstellensatz. We exhibit some further correspondences between geometric and algebraic objects. Most notably, the coordinate ring is an affine algebra assigned to an affine variety, and points of the variety correspond to maximal ideals in the coordinate ring. Before we get started, let us fix some conventions and notation that will be used throughout the book. By a ring we will always mean a commutative ring with an identity element 1. In particular, there is a ring R = {0},the zero ring,inwhich1=0.AringR is called an integral domain if R has no zero divisors (other than 0 itself) and R ={0}.Asubring of a ring R must contain the identity element of R, and a homomorphism R → S of rings must send the identity element of R to the identity element of S. -
A Linear Algebraic Group? Skip Garibaldi
WHATIS... ? a Linear Algebraic Group? Skip Garibaldi From a marketing perspective, algebraic groups group over F, a Lie group, or a topological group is are poorly named.They are not the groups you met a “group object” in the category of affine varieties as a student in abstract algebra, which I will call over F, smooth manifolds, or topological spaces, concrete groups for clarity. Rather, an algebraic respectively. For algebraic groups, this implies group is the analogue in algebra of a topological that the set G(K) is a concrete group for each field group (fromtopology) or a Liegroup (from analysis K containing F. and geometry). Algebraic groups provide a unifying language Examples for apparently different results in algebra and The basic example is the general linear group number theory. This unification can not only sim- GLn for which GLn(K) is the concrete group of plify proofs, it can also suggest generalizations invertible n-by-n matrices with entries in K. It is the and bring new tools to bear, such as Galois co- collection of solutions (t, X)—with t ∈ K and X an homology, Steenrod operations in Chow theory, n-by-n matrix over K—to the polynomial equation etc. t ·det X = 1. Similar reasoning shows that familiar matrix groups such as SLn, orthogonal groups, Definitions and symplectic groups can be viewed as linear A linear algebraic group over a field F is a smooth algebraic groups. The main difference here is that affine variety over F that is also a group, much instead of viewing them as collections of matrices like a topological group is a topological space over F, we view them as collections of matrices that is also a group and a Lie group is a smooth over every extension K of F. -
Tensor Categorical Foundations of Algebraic Geometry
Tensor categorical foundations of algebraic geometry Martin Brandenburg { 2014 { Abstract Tannaka duality and its extensions by Lurie, Sch¨appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete tensor categories (resp. cocontinuous tensor functors) which usually correspond to constructions of schemes (resp. their morphisms) in the case of quasi-coherent sheaves. This means to globalize the usual local-global algebraic geometry. For this we first have to develop basic commutative algebra in an arbitrary cocom- plete tensor category. We then discuss tensor categorical globalizations of affine morphisms, projective morphisms, immersions, classical projective embeddings (Segre, Pl¨ucker, Veronese), blow-ups, fiber products, classifying stacks and finally tangent bundles. It turns out that the universal properties of several moduli spaces or stacks translate to the corresponding tensor categories. arXiv:1410.1716v1 [math.AG] 7 Oct 2014 This is a slightly expanded version of the author's PhD thesis. Contents 1 Introduction 1 1.1 Background . .1 1.2 Results . .3 1.3 Acknowledgements . 13 2 Preliminaries 14 2.1 Category theory . 14 2.2 Algebraic geometry . 17 2.3 Local Presentability . 21 2.4 Density and Adams stacks . 22 2.5 Extension result . 27 3 Introduction to cocomplete tensor categories 36 3.1 Definitions and examples . 36 3.2 Categorification . 43 3.3 Element notation . 46 3.4 Adjunction between stacks and cocomplete tensor categories . 49 4 Commutative algebra in a cocomplete tensor category 53 4.1 Algebras and modules . 53 4.2 Ideals and affine schemes . -
Characterization of Quantum Entanglement Via a Hypercube of Segre Embeddings
Characterization of quantum entanglement via a hypercube of Segre embeddings Joana Cirici∗ Departament de Matemàtiques i Informàtica Universitat de Barcelona Gran Via 585, 08007 Barcelona Jordi Salvadó† and Josep Taron‡ Departament de Fisíca Quàntica i Astrofísica and Institut de Ciències del Cosmos Universitat de Barcelona Martí i Franquès 1, 08028 Barcelona A particularly simple description of separability of quantum states arises naturally in the setting of complex algebraic geometry, via the Segre embedding. This is a map describing how to take products of projective Hilbert spaces. In this paper, we show that for pure states of n particles, the corresponding Segre embedding may be described by means of a directed hypercube of dimension (n − 1), where all edges are bipartite-type Segre maps. Moreover, we describe the image of the original Segre map via the intersections of images of the (n − 1) edges whose target is the last vertex of the hypercube. This purely algebraic result is then transferred to physics. For each of the last edges of the Segre hypercube, we introduce an observable which measures geometric separability and is related to the trace of the squared reduced density matrix. As a consequence, the hypercube approach gives a novel viewpoint on measuring entanglement, naturally relating bipartitions with q-partitions for any q ≥ 1. We test our observables against well-known states, showing that these provide well-behaved and fine measures of entanglement. arXiv:2008.09583v1 [quant-ph] 21 Aug 2020 ∗ [email protected] † [email protected] ‡ [email protected] 2 I. INTRODUCTION Quantum entanglement is at the heart of quantum physics, with crucial roles in quantum information theory, superdense coding and quantum teleportation among others. -
Tropical Algebraic Geometry Keyvan Yaghmayi University of Utah Advisor: Prof
Tropical Algebraic Geometry Keyvan Yaghmayi University of Utah Advisor: Prof. Tommaso De Fernex Field of interest and the problem There are different approaches to tropical algebraic geometry. In the following I describe briefly three approaches: the geometry over max-plus semifield, the Maslov dequantization of classical algebraic geometry, and the valuation approach. We are interested to study the valuation approach and also understand the relation with Berkovich spaces. Tropical algebraic geometry Example 1 (integer linear programming): Let be a matrix of non-negative The most straightforward way to tropical algebraic geometry is doing algebraic geometry on integers, let be a row vector with real entries, and let be a column tropical semifield where the tropical sum is maximum and the tropical vector with non-negative integer entries. Our task is to find a non-negative integer column vector product is the usual addition: . which solves the following optimization problem: Ideally, one may hope that every construction in algebraic geometry should have a tropical Minimize subject to and . (1) counterpart and thus, obtain results in algebraic geometry by looking at the tropical picture and Let us further assume that all columns of the matrix sum to the same number and that then trying to transfer the results back to the original setting. This assumption is convenient because it ensures that all feasible solutions of (1) satisfy . We can solve the integer programming problem (1) using tropical arithmetic We can look at as the Maslov dequantization of the semifield . Let for . as follows. Let be indeterminates and consider the expression The bijection given by induces a semifield structure on such that (2) is an isomorphism. -
An Algebraic Structure Theory Applicable to Tropical Mathematics Shirshov 100 Conference
Semialgebra systems: An algebraic structure theory applicable to tropical mathematics Shirshov 100 Conference. Louis Rowen, Bar-Ilan University 16 August, 2021 Ode to Shirshov I first heard of Shirshov from Amitsur (also born the same year!) who discovered the amazing Shirshov Height Theorem, which revolutionized PI-theory. I did not have the opportunity to meet Shirshov, but have the privilege of knowing several of his students, including Bokut (who has been a frequent guest at Bar-Ilan University and introduced me to many members of the important algebra group in Novosibirsk), Zelmanov, Kemer, (both joint students of Bokut and Shirshov), and Shestakov. The main original idea was to take the limit of the logarithm of the absolute values of the coordinates of an affine variety as the base of the logarithm goes to 1. Tropical background in 3 slides Tropical geometry has assumed a prominent position in mathematics because of its ability to simplify algebraic geometry while not changing some invariants (involving intersection numbers of varieties), thereby simplifying difficult computations. Tropical background in 3 slides Tropical geometry has assumed a prominent position in mathematics because of its ability to simplify algebraic geometry while not changing some invariants (involving intersection numbers of varieties), thereby simplifying difficult computations. The main original idea was to take the limit of the logarithm of the absolute values of the coordinates of an affine variety as the base of the logarithm goes to 1. Tropicalization For any complex affine variety W = f(z1;:::; zn): zi 2 Cg; and any t > 0, its amoeba A(W ) is defined as f(logt jz1j;:::; logt jznj) :(z1;:::; zn) 2 W g (n) ⊂ (R [ {−∞}) ; graphed according to the (rescaled) coordinates logt jz1j;:::; logt jznj.