Algebraic Geometry
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The Many Lives of the Twisted Cubic
Mathematical Assoc. of America American Mathematical Monthly 121:1 February 10, 2018 11:32 a.m. TwistedMonthly.tex page 1 The many lives of the twisted cubic Abstract. We trace some of the history of the twisted cubic curves in three-dimensional affine and projective spaces. These curves have reappeared many times in many different guises and at different times they have served as primary objects of study, as motivating examples, and as hidden underlying structures for objects considered in mathematics and its applications. 1. INTRODUCTION Because of its simple coordinate functions, the humble (affine) 3 twisted cubic curve in R , given in the parametric form ~x(t) = (t; t2; t3) (1.1) is staple of computational problems in many multivariable calculus courses. (See Fig- ure 1.) We and our students compute its intersections with planes, find its tangent vectors and lines, approximate the arclength of segments of the curve, derive its cur- vature and torsion functions, and so forth. But in many calculus books, the curve is not even identified by name and no indication of its protean nature and rich history is given. This essay is (semi-humorously) in part an attempt to remedy that situation, and (more seriously) in part a meditation on the ways that the things mathematicians study often seem to have existences of their own. Basic structures can reappear at many times in many different contexts and under different names. According to the current consensus view of mathematical historiography, historians of mathematics do well to keep these different avatars of the underlying structure separate because it is almost never correct to attribute our understanding of the sorts of connections we will be con- sidering to the thinkers of the past. -
A Three-Dimensional Laguerre Geometry and Its Visualization
A Three-Dimensional Laguerre Geometry and Its Visualization Hans Havlicek and Klaus List, Institut fur¨ Geometrie, TU Wien 3 We describe and visualize the chains of the 3-dimensional chain geometry over the ring R("), " = 0. MSC 2000: 51C05, 53A20. Keywords: chain geometry, Laguerre geometry, affine space, twisted cubic. 1 Introduction The aim of the present paper is to discuss in some detail the Laguerre geometry (cf. [1], [6]) which arises from the 3-dimensional real algebra L := R("), where "3 = 0. This algebra generalizes the algebra of real dual numbers D = R("), where "2 = 0. The Laguerre geometry over D is the geometry on the so-called Blaschke cylinder (Figure 1); the non-degenerate conics on this cylinder are called chains (or cycles, circles). If one generator of the cylinder is removed then the remaining points of the cylinder are in one-one correspon- dence (via a stereographic projection) with the points of the plane of dual numbers, which is an isotropic plane; the chains go over R" to circles and non-isotropic lines. So the point space of the chain geometry over the real dual numbers can be considered as an affine plane with an extra “improper line”. The Laguerre geometry based on L has as point set the projective line P(L) over L. It can be seen as the real affine 3-space on L together with an “improper affine plane”. There is a point model R for this geometry, like the Blaschke cylinder, but it is more compli- cated, and belongs to a 7-dimensional projective space ([6, p. -
Real Rank Two Geometry Arxiv:1609.09245V3 [Math.AG] 5
Real Rank Two Geometry Anna Seigal and Bernd Sturmfels Abstract The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two. 1 Introduction Low-rank approximation of tensors is a fundamental problem in applied mathematics [3, 6]. We here approach this problem from the perspective of real algebraic geometry. Our goal is to give an exact semi-algebraic description of the set of tensors of real rank two and to characterize its boundary. This complements the results on tensors of non-negative rank two presented in [1], and it offers a generalization to the setting of arbitrary varieties, following [2]. A familiar example is that of 2 × 2 × 2-tensors (xijk) with real entries. Such a tensor lies in the closure of the real rank two tensors if and only if the hyperdeterminant is non-negative: 2 2 2 2 2 2 2 2 x000x111 + x001x110 + x010x101 + x011x100 + 4x000x011x101x110 + 4x001x010x100x111 −2x000x001x110x111 − 2x000x010x101x111 − 2x000x011x100x111 (1) −2x001x010x101x110 − 2x001x011x100x110 − 2x010x011x100x101 ≥ 0: If this inequality does not hold then the tensor has rank two over C but rank three over R. To understand this example geometrically, consider the Segre variety X = Seg(P1 × P1 × P1), i.e. the set of rank one tensors, regarded as points in the projective space P7 = 2 2 2 7 arXiv:1609.09245v3 [math.AG] 5 Apr 2017 P(C ⊗ C ⊗ C ). -
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. -
Hyperbolic Monopoles and Rational Normal Curves
HYPERBOLIC MONOPOLES AND RATIONAL NORMAL CURVES Nigel Hitchin (Oxford) Edinburgh April 20th 2009 204 Research Notes A NOTE ON THE TANGENTS OF A TWISTED CUBIC B Y M. F. ATIYAH Communicated by J. A. TODD Received 8 May 1951 1. Consider a rational normal cubic C3. In the Klein representation of the lines of $3 by points of a quadric Q in Ss, the tangents of C3 are represented by the points of a rational normal quartic O4. It is the object of this note to examine some of the consequences of this correspondence, in terms of the geometry associated with the two curves. 2. C4 lies on a Veronese surface V, which represents the congruence of chords of O3(l). Also C4 determines a 4-space 2 meeting D. in Qx, say; and since the surface of tangents of O3 is a developable, consecutive tangents intersect, and therefore the tangents to C4 lie on Q, and so on £lv Hence Qx, containing the sextic surface of tangents to C4, must be the quadric threefold / associated with C4, i.e. the quadric determining the same polarity as C4 (2). We note also that the tangents to C4 correspond in #3 to the plane pencils with vertices on O3, and lying in the corresponding osculating planes. 3. We shall prove that the surface U, which is the locus of points of intersection of pairs of osculating planes of C4, is the projection of the Veronese surface V from L, the pole of 2, on to 2. Let P denote a point of C3, and t, n the tangent line and osculating plane at P, and let T, T, w denote the same for the corresponding point of C4. -
Asymptotic Behavior of the Length of Local Cohomology
ASYMPTOTIC BEHAVIOR OF THE LENGTH OF LOCAL COHOMOLOGY STEVEN DALE CUTKOSKY, HUY TAI` HA,` HEMA SRINIVASAN, AND EMANOIL THEODORESCU Abstract. Let k be a field of characteristic 0, R = k[x1, . , xd] be a polynomial ring, and m its maximal homogeneous ideal. Let I ⊂ R be a homogeneous ideal in R. In this paper, we show that λ(H0 (R/In)) λ(Extd (R/In,R(−d))) lim m = lim R n→∞ nd n→∞ nd e(I) always exists. This limit has been shown to be for m-primary ideals I in a local Cohen Macaulay d! ring [Ki, Th, Th2], where e(I) denotes the multiplicity of I. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have polynomial growth. Introduction Let R = k[x1, . , xd] be a polynomial ring over a field k, with graded maximal ideal m, and I ⊂ R d n a proper homogeneous ideal. We investigate the asymptotic growth of λ(ExtR(R/I ,R)) as a function of n. When R is a local Gorenstein ring and I is an m-primary ideal, then this is easily seen to be equal to λ(R/In) and hence is a polynomial in n. A theorem of Theodorescu and Kirby [Ki, Th, Th2] extends this to m-primary ideals in local Cohen Macaulay rings R. We consider homogeneous ideals in a polynomial ring which are not m-primary and show that a limit exists asymptotically although it can be irrational. -
Arxiv:1703.06832V3 [Math.AC]
REGULARITY OF FI-MODULES AND LOCAL COHOMOLOGY ROHIT NAGPAL, STEVEN V SAM, AND ANDREW SNOWDEN Abstract. We resolve a conjecture of Ramos and Li that relates the regularity of an FI- module to its local cohomology groups. This is an analogue of the familiar relationship between regularity and local cohomology in commutative algebra. 1. Introduction Let S be a standard-graded polynomial ring in finitely many variables over a field k, and let M be a non-zero finitely generated graded S-module. It is a classical fact in commutative algebra that the following two quantities are equal (see [Ei, §4B]): S • The minimum integer α such that Tori (M, k) is supported in degrees ≤ α + i for all i. i • The minimum integer β such that Hm(M) is supported in degrees ≤ β − i for all i. i Here Hm is local cohomology at the irrelevant ideal m. The quantity α = β is called the (Castelnuovo–Mumford) regularity of M, and is one of the most important numerical invariants of M. In this paper, we establish the analog of the α = β identity for FI-modules. To state our result precisely, we must recall some definitions. Let FI be the category of finite sets and injections. Fix a commutative noetherian ring k. An FI-module over k is a functor from FI to the category of k-modules. We write ModFI for the category of FI-modules. We refer to [CEF] for a general introduction to FI-modules. Let M be an FI-module. Define Tor0(M) to be the FI-module that assigns to S the quotient of M(S) by the sum of the images of the M(T ), as T varies over all proper subsets of S. -
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). -
ULRICH BUNDLES on CUBIC FOURFOLDS Daniele Faenzi, Yeongrak Kim
ULRICH BUNDLES ON CUBIC FOURFOLDS Daniele Faenzi, Yeongrak Kim To cite this version: Daniele Faenzi, Yeongrak Kim. ULRICH BUNDLES ON CUBIC FOURFOLDS. 2020. hal- 03023101v2 HAL Id: hal-03023101 https://hal.archives-ouvertes.fr/hal-03023101v2 Preprint submitted on 25 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ULRICH BUNDLES ON CUBIC FOURFOLDS DANIELE FAENZI AND YEONGRAK KIM Abstract. We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6. Introduction An Ulrich sheaf on a closed subscheme X of PN of dimension n and degree d is a non-zero coherent sheaf F on X satisfying H∗(X, F(−j)) = 0 for 1 ≤ j ≤ n. In particular, the cohomology table {hi(X, F(j))} of F is a multiple of the cohomology table of Pn. -
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. -
TWO FAMILIES of STABLE BUNDLES with the SAME SPECTRUM Nonreduced [M]
proceedings of the american mathematical society Volume 109, Number 3, July 1990 TWO FAMILIES OF STABLE BUNDLES WITH THE SAME SPECTRUM A. P. RAO (Communicated by Louis J. Ratliff, Jr.) Abstract. We study stable rank two algebraic vector bundles on P and show that the family of bundles with fixed Chern classes and spectrum may have more than one irreducible component. We also produce a component where the generic bundle has a monad with ghost terms which cannot be deformed away. In the last century, Halphen and M. Noether attempted to classify smooth algebraic curves in P . They proceeded with the invariants d , g , the degree and genus, and worked their way up to d = 20. As the invariants grew larger, the number of components of the parameter space grew as well. Recent work of Gruson and Peskine has settled the question of which invariant pairs (d, g) are admissible. For a fixed pair (d, g), the question of determining the number of components of the parameter space is still intractable. For some values of (d, g) (for example [E-l], if d > g + 3) there is only one component. For other values, one may find many components including components which are nonreduced [M]. More recently, similar questions have been asked about algebraic vector bun- dles of rank two on P . We will restrict our attention to stable bundles a? with first Chern class c, = 0 (and second Chern class c2 = n, so that « is a positive integer and i? has no global sections). To study these, we have the invariants n, the a-invariant (which is equal to dim//"'(P , f(-2)) mod 2) and the spectrum. -
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.