DOCSLIB.ORG

On Smooth Projective D-Affine Varieties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Smooth Projective D-Affine Varieties On smooth projective D-affine varieties Adrian Langer December 17, 2019 ADDRESS: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland e-mail: [email protected] Abstract We show various properties of smooth projective D-affine varieties. In particular, any smooth pro- jective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine varieties are also uniruled. We also show that (apart from a few small characteristics) a smooth projective surface is D-affine if and only if it is isomorphic to either P2 or P1 P1. In positive characteristic, a basic tool in the proof is a new generalization of Miyaoka’s generic semipositivity× theorem. Introduction Let X be a scheme defined over some algebraicallyclosed field k. Let DX be thesheafof k-linear differential operators on X. A DX -module is a left DX -module, which is quasi-coherent as an OX -module. X is called D-quasi-affine if every DX -module M is generated over DX by its global sections. X is called D-affine if it i is D-quasi-affine and for every DX -module M we have H (X,M)= 0 for all i > 0. In [2] Beilinson and Bernstein proved that every flag variety (i.e., a quotient of a reductive group by some parabolic subgroup) in characteristic 0 is D-affine. This fails in positive characteristic (see [17]), although some flag varieties are still D-affine (see, e.g., [12], [22] and [36]). However, there are no known examples of smooth projective varieties that are D-affine and that are not flag varieties. In [39] Thomsen proved that any smooth projective toric variety that is D-affine is a product of projective spaces. arXiv:1906.00227v2 [math.AG] 16 Dec 2019 Note that OX has a canonical structure of a DX -module coming from the inclusion DX End kOX . In i ⊂ particular, if X is a D-affine variety then H (X,OX )= 0 for all i > 0. This shows that a smooth projective curve is D-affine if and only if it is isomorphic to P1. However, in higher dimensions this restriction is essentially the only known condition that must be satisfied by D-affine varieties. In the first part of this note we show some other properties of smooth projective D-affine varieties. In particular, we prove the following theorem: THEOREM 0.1. Let X be a smooth projective variety defined over an algebraically closed field k. Let us assume that X is D-affine. Then the following conditions are satisfied: ´et 1. π1 (X)= 0. 2. All left DX -modules, which are coherent as OX -modules, are direct sums of finitely many DX -modules isomorphic to the canonical DX -module OX . 3. X does not admit any dominant rational map to a curve of genus 1. ≥ 4. If chark = 0 then X is uniruled. 1 2 Proof of parts 1 and 2 of Theorem 0.1 is divided into two cases depending on the characteristic of the base field. In case of characteristic zero the theorem follows from Theorem 3.2. The proof depends on reducing to the study of unitary representations of the topological fundamental group of X. In positive char- acteristic Theorem 0.1 follows from Theorem 3.5. Here we use interpretation of DX -modules as stratified bundles. Part 3 follows from part 1 and Proposition 1.7. The last part of the theorem is an application of Miyaoka’s theorem [31, Corollary 8.6] on generic semipositivity of the cotangent bundle of a non-uniruled variety (see Proposition 3.1 and Remark 4.4). In fact, this part of Theorem 0.1 suggests that X should be rationally connected. This problem is studied in Subsection 3.3, but here we obtain only a partial result on the maximal rationally connected fibration of a D-affine variety. Our next aim is the study of morphisms from smooth D-affine varieties. Here we obtain the following results: THEOREM 0.2. Let X be a smooth complete variety defined over an algebraically closed field k. Let us assume that X is D-affine and let f : X Y be a surjective morphism to some projective variety Y . Then the following conditions are satisfied: → 1. If f is a fibration thenY does not admit any divisorial contractions. i 2. For any DX -module M we have R f M = 0 for i > 0. ∗ i 3. IfY issmoothand f OX is locally free then for any DY -module N we have L f ∗N = 0 for i > 0. ∗ 4. If f is a fibrationandY is smooth thenY is D-affine. Part 1 follows from the fact that for any effective divisor D the module OX ( D) carries a natural DX - module structure (see Lemma 4.1). In case Y is smooth and k has characteristic∗ zero the second part follows from [14, 2.14 Proposition] . In general, we use a similar proof following [16, Theorem 1.4.1] (see Proposition 4.3). Part 3 follows from part 2 and some calculation in derived categories (see Corollary 4.14). The last part follows from parts 2 and 3 and again can be found in [14, 2.14 Proposition] in case k has characteristic zero. Part 3 of Theorem 0.1 says that the only smooth projective curve, which is dominated by a smooth projective D-affine variety, is P1. In the next part of the paper we prove that, except possibly for some small characteristics, all smooth projective surfaces that are images of smooth projective D-affine varieties are flag varieties: THEOREM 0.3. Let X be a smooth projective variety defined over an algebraically closed field k and let f : X Y be a fibration over a smooth projective surface. If chark = 0 or chark > 7 and X is D-affine then f is flatandY→ = P2 or Y = P1 P1. × Let us recall that products of projective spaces are D-affine in any characteristic (see [12, Korollar 3.2] or [39]). In particular, a smooth projective surface defined over an algebraically closed field of characteristic 0 or > 7 is D-affine if and only if it is isomorphic to either P2 or P1 P1. We prove a slightly more precise result in Section 6. Let us mention× that recently D. Rumynin in [35] proved that the only D-affine rational surfaces are flag varieties. In characteristic 0 the above classification result follows from Theorem 0.1 and Theorem 0.2. However, the positive characteristic case is more delicate and we need the following positive characteristic version of Miyaoka’s generic semipositivity theorem. THEOREM 0.4. Let X be a smooth projective surface defined over an algebraically closed field k of char- acteristic p. Let us fix an ample divisor H on X and assume that p > KX H + 1. If X is not uniruled then ΩX is generically H-semipositive. For the definition and basic properites of generically semipositive sheaves in positive characteristic see Subsection 2.2. The most important property is their good behaviour under various tensor operations like symmetric or divided powers. The only known result on generic semipositivity of cotangent bundle for non-uniruled varieties in posi- tive characteristic concerns varieties with trivial canonical divisor (see [24, Theorem 0.1]). However, it does 3 not say anything about the most interesting case of varieties of general type. This is covered by the above theorem but only in the surface case. The higher dimensional version seems to require different techniques. A more precise version of Theorem 0.4 is contained in Theorem 5.3. We show that this generalization is optimal (see Subsection 5.1 and Remark 5.2). The structure of the paper is as follows. In Section 1 we recall some auxiliary results. In Section 2 we prove several facts about tensor operations and generic semipositivity of sheaves in positive characteristic. In Section 3 we prove Theorem 0.1. In Section 4 we prove Theorem 0.2. In Section 5 we study uniruled- ness of surfaces in positive characteristic proving a generalization of Theorem 0.4. Finally, in Section 6 we use these results to study smooth projective surfaces that are images of D-affine varieties, proving a generalization of Theorem 0.3. We also make some remarks on the case of D-affine 3-folds. Notation Let X and Y be algebraic varieties defined over an algebraically closed field. A divisorial contraction is a proper birational morphism f : X Y, which contracts some divisor D to a subscheme of codimension 2 and that is an isomorphism outside→ of D. ≥ A fibration is a morphism f : X Y such that f OX = OY (in particular, we allow f to be the identity or a birational morphism). → ∗ Let us assume that X is a smooth projective variety and let us fix an ample divisor H on X. If E is a torsion free coherent OX -module then by µmax,H (E) we denote the slope of the maximal destabilizing sub- sheaf of E (with respect to H). Similarly, we use µmin,H (E) to denote the slope of the minimal destabilizing quotient of E. When it is clear from the context which polarization is used, we omit H in the notation and write µmax(E) and µmin(E) instead of µmax,H (E) and µmin,H (E), respectively. 1 Preliminaries 1.1 A -affine varieties Let X be a smooth variety defined over an algebraically closed field k. Let A be any sheaf of rings on X with a ring homomorphism OX A such that the image of k OX A is contained in the center of A .
Load more

Recommended publications
  • 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.
    [Show full text]
  • Notes on Grassmannians
    NOTES ON GRASSMANNIANS ANDERS SKOVSTED BUCH This is class notes under construction. We have not attempted to account for the history of the results covered here. 1. Construction of Grassmannians 1.1. The set of points. Let k = k be an algebraically closed field, and let kn be the vector space of column vectors with n coordinates. Given a non-negative integer m ≤ n, the Grassmann variety Gr(m, n) is defined as a set by Gr(m, n)= {Σ ⊂ kn | Σ is a vector subspace with dim(Σ) = m} . Our first goal is to show that Gr(m, n) has a structure of algebraic variety. 1.2. Space with functions. Let FR(n, m) = {A ∈ Mat(n × m) | rank(A) = m} be the set of all n × m matrices of full rank, and let π : FR(n, m) → Gr(m, n) be the map defined by π(A) = Span(A), the column span of A. We define a topology on Gr(m, n) be declaring the a subset U ⊂ Gr(m, n) is open if and only if π−1(U) is open in FR(n, m), and further declare that a function f : U → k is regular if and only if f ◦ π is a regular function on π−1(U). This gives Gr(m, n) the structure of a space with functions. ex:morphism Exercise 1.1. (1) The map π : FR(n, m) → Gr(m, n) is a morphism of spaces with functions. (2) Let X be a space with functions and φ : Gr(m, n) → X a map.
    [Show full text]
  • Utah/Fall 2020 3. Abstract Varieties. the Categories of Affine and Quasi
    Algebraic Geometry I (Math 6130) Utah/Fall 2020 3. Abstract Varieties. The categories of affine and quasi-affine varieties over k are (by definition) full subcategories of the category of sheaved spaces (X; OX ) over k. We now enlarge the category just enough within the category of sheaved spaces to define a category of abstract varieties analogous to the categories of differentiable or analytic manifolds. Varieties are Noetherian and locally affine and separated (analogous to Hausdorff), the latter defined via products, which are also used to define the notion of a proper (analogous to compact or complete) variety. Definition 3.1. A topology on a set X is Noetherian if every descending chain: X ⊇ X1 ⊇ X2 ⊇ · · · of closed sets eventually stabilizes (or every ascending chain of open sets stabilizes). Remarks. (a) A Noetherian topological space X is quasi-compact, i.e. every open cover of X has a finite subcover, but a quasi-compact space need not be Noetherian. (b) The Zariski topology on a quasi-affine variety is Noetherian. (c) It is possible for a topological space X to fail to be Noetherian even if it has a cover by open sets, each of which is Noetherian with the induced topology. If the open cover is finite, however, then X is Noetherian. Definition 3.2. A Noetherian topological space X is reducible if X = X1 [ X2 for closed sets X1;X2 properly contained in X. Otherwise it is irreducible. Remarks. (a) As we saw in x1, every Noetherian topological space X is a finite union of irreducible closed subsets, accounting for the irreducible components of X.
    [Show full text]
  • Grothendieck Ring of Varieties
    Neeraja Sahasrabudhe Grothendieck Ring of Varieties Thesis advisor: J. Sebag Université Bordeaux 1 1 Contents 1. Introduction 3 2. Grothendieck Ring of Varieties 5 2.1. Classical denition 5 2.2. Classical properties 6 2.3. Bittner's denition 9 3. Stable Birational Geometry 14 4. Application 18 4.1. Grothendieck Ring is not a Domain 18 4.2. Grothendieck Ring of motives 19 5. APPENDIX : Tools for Birational geometry 20 Blowing Up 21 Resolution of Singularities 23 6. Glossary 26 References 30 1. Introduction First appeared in a letter of Grothendieck in Serre-Grothendieck cor- respondence (letter of 16/8/64), the Grothendieck ring of varieties is an interesting object lying at the heart of the theory of motivic inte- gration. A class of variety in this ring contains a lot of geometric information about the variety. For example, the topological euler characteristic, Hodge polynomials, Stably-birational properties, number of points if the variety is dened on a nite eld etc. Besides, the question of equality of these classes has given some important new results in bira- tional geometry (for example, Batyrev-Kontsevich's theorem). The Grothendieck Ring K0(V ark) is the quotient of the free abelian group generated by isomorphism classes of k-varieties, by the relation [XnY ] = [X]−[Y ], where Y is a closed subscheme of X; the ber prod- 0 0 uct over k induces a ring structure dened by [X]·[X ] = [(X ×k X )red]. Many geometric objects verify this kind of relations. It gives many re- alization maps, called additive invariants, containing some geometric information about the varieties.
    [Show full text]
  • Completeness in Partial Differential Fields
    Completeness in Partial Differential Fields James Freitag [email protected] Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago 322 Science and Engineering Offices (M/C 249) 851 S. Morgan Street Chicago, IL 60607-7045 Abstract We study completeness in partial differential varieties. We generalize many of the results of Pong to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and use it to give a new class of examples of complete partial differential varieties. In addition to completeness we prove some new embedding theorems for differential algebraic varieties. We use methods from both differential algebra and model theory. Completeness is a fundamental notion in algebraic geometry. In this paper, we examine an analogue in differential algebraic geometry. The paper builds on Pong’s [16] and Kolchin’s [7] work on differential completeness in the case of differential va- rieties over ordinary differential fields and generalizes to the case differential varieties over partial differential fields with finitely many commuting derivations. Many of the proofs generalize in the straightforward manner, given that one sets up the correct definitions and attempts to prove the correct analogues of Pong’s or Kolchin’s results. arXiv:1106.0703v2 [math.LO] 3 Feb 2012 Of course, some of the results are harder to prove because our varieties may be infinite differential transcendence degree. Nevertheless, Proposition 3.3 generalizes a theorem of [17] even when we restrict to the ordinary case. The proposition also generalizes a known result projective varieties. Many of the results are model-theoretic in nature or use model-theoretic tools.
    [Show full text]
  • Compactification of Tame Deligne–Mumford Stacks
    COMPACTIFICATION OF TAME DELIGNE–MUMFORD STACKS DAVID RYDH Abstract. The main result of this paper is that separated Deligne– Mumford stacks in characteristic zero can be compactified. In arbitrary characteristic, we give a necessary and sufficient condition for a tame Deligne–Mumford stack to have a tame Deligne–Mumford compactifi- cation. The main tool is a new class of stacky modifications — tame stacky blow-ups — and a new ´etalification result which in it simplest form asserts that a tamely ramified finite flat cover becomes ´etale after a tame stacky blow-up. This should be compared to Raynaud–Gruson’s flatification theorem which we extend to stacks. Preliminary draft! Introduction A fundamental result in algebraic geometry is Nagata’s compactification theorem which asserts that any variety can be embedded into a complete variety [Nag62]. More generally, if f : X → Y is a separated morphism of finite type between noetherian schemes, then there exists a compactification of f, that is, there is a proper morphism f : X → Y such that f is the restriction of f to an open subset X ⊆ X [Nag63]. The main theorem of this paper is an extension of Nagata’s result to sep- arated morphisms of finite type between quasi-compact Deligne–Mumford stacks of characteristic zero and more generally to tame Deligne–Mumford stacks. The compactification result relies on several different techniques which all are of independent interest. • Tame stacky blow-ups — We introduce a class of stacky modifica- tions, i.e., proper birational morphisms of stacks, and show that this class has similar properties as the class of blow-ups.
    [Show full text]
  • Algebraic Geometry of Curves
    Algebraic Geometry of Curves Andrew Kobin Fall 2016 Contents Contents Contents 0 Introduction 1 0.1 Geometry and Number Theory . .2 0.2 Rational Curves . .4 0.3 The p-adic Numbers . .6 1 Algebraic Geometry 9 1.1 Affine and Projective Space . .9 1.2 Morphisms of Affine Varieties . 15 1.3 Morphisms of Projective Varieties . 19 1.4 Products of Varieties . 21 1.5 Blowing Up . 22 1.6 Dimension of Varieties . 24 1.7 Complete Varieties . 26 1.8 Tangent Space . 27 1.9 Local Parameters . 32 2 Curves 33 2.1 Divisors . 34 2.2 Morphisms Between Curves . 36 2.3 Linear Equivalence . 38 2.4 Differentials . 40 2.5 The Riemann-Hurwitz Formula . 42 2.6 The Riemann-Roch Theorem . 43 2.7 The Canonical Map . 45 2.8 B´ezout'sTheorem . 45 2.9 Rational Points of Conics . 47 3 Elliptic Curves 52 3.1 Weierstrass Equations . 53 3.2 Moduli Spaces . 55 3.3 The Group Law . 56 3.4 The Jacobian . 58 4 Rational Points on Elliptic Curves 61 4.1 Isogenies . 62 4.2 The Dual Isogeny . 65 4.3 The Weil Conjectures . 67 4.4 Elliptic Curves over Local Fields . 69 4.5 Jacobians of Hyperelliptic Curves . 75 5 The Mordell-Weil Theorem 77 5.1 Some Galois Cohomology . 77 5.2 Selmer and Tate-Shafarevich Groups . 80 5.3 Twists, Covers and Principal Homogeneous Spaces . 84 i Contents Contents 5.4 Descent . 89 5.5 Heights . 97 6 Elliptic Curves and Complex Analysis 99 6.1 Elliptic Functions . 99 6.2 Elliptic Curves .
    [Show full text]
  • Collected Papers of C. S. Seshadri. Volume 1. Vector Bundles and Invariant Theory, Vikraman Balaji, V
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 51, Number 2, April 2014, Pages 367–372 S 0273-0979(2013)01429-0 Article electronically published on September 23, 2013 Collected papers of C. S. Seshadri. Volume 1. Vector bundles and invariant theory, Vikraman Balaji, V. Lakshmibai, M. Pavaman Murthy, and Madhav V. Nori (Editors), Hindustan Book Agency, New Delhi, India, 2012, xxiv+1008 pp., ISBN 978-93-80250-17-5 (2 volume set) Collected papers of C. S. Seshadri. Volume 2. Schubert geometry and representation Theory, Vikraman Balaji, V. Lakshmibai, M. Pavaman Murthy, and Madhav V. Nori (Editors), Hindustan Book Agency, New Delhi, India, 2012, xxvi+633 pp., ISBN 978-93-80250-17-5 (2 volume set) 1. Introduction It would probably take several volumes to describe the enormous impact of Se- shadri’s work contained in these two volumes of his collected papers. His impressive achievements in the last several decades span a large variety of diverse topics, includ- ing moduli of vector bundles on curves, geometric invariant theory, representation theory, and Schubert calculus. We briefly address here the main results of his work presented in the volumes under review. 2. The early work In the nineteenth century the theory of divisors on a compact Riemann surface was developed, mainly by Abel and Riemann. In modern algebraic geometry, di- visors modulo linear equivalence are identified with line bundles. So, this theory is the same as the theory of line bundles. It can also be generalized to higher di- mensions. Seshadri’s early contributions are a sequence of important expos´es on divisors in algebraic geometry written in the Chevalley seminar.
    [Show full text]
  • Some Lectures on Algebraic Geometry∗
    Some lectures on Algebraic Geometry∗ V. Srinivas 1 Affine varieties Let k be an algebraically closed field. We define the affine n-space over k n n to be just the set k of ordered n-tuples in k, and we denote it by Ak (by 0 convention we define Ak to be a point). If x1; : : : ; xn are the n coordinate n functions on Ak , any polynomial f(x1; : : : ; xn) in x1; : : : ; xn with coefficients n in k yields a k-valued function Ak ! k, which we also denote by f. Let n A(Ak ) denote the ring of such functions; we call it the coordinate ring of n Ak . Since k is algebraically closed, and in particular is infinite, we easily see n that x1; : : : ; xn are algebraically independent over k, and A(Ak ) is hence a 0 polynomial ring over k in n variables. The coordinate ring of Ak is just k. n Let S ⊂ A(Ak ) be a collection of polynomials. We can associate to it n an algebraic set (or affine variety) in Ak defined by n V (S) := fx 2 Ak j f(x) = 0 8 f 2 Sg: This is called the variety defined by the collection S. If I =< S >:= ideal in k[x1; : : : ; xn] generated by elements of S, then we clearly have V (S) = V (I). Also, if f is a polynomial such that some power f m vanishes at a point P , then so does f itself; hence if p m I := ff 2 k[x1; : : : ; xn] j f 2 I for some m > 0g p is the radical of I, then V (I) = V ( I).
    [Show full text]
  • (Linear) Algebraic Groups 1 Basic Definitions and Main Examples
    Study Group: (Linear) Algebraic Groups 1 Basic Denitions and Main Examples (Matt) Denition 1.1. Let (where is some eld) then n is an ane algebraic I C K[x] K VI = fP 2 A : f(P ) = 08f 2 Ig set. If I is prime, then VI is an ane algebraic variety. Denition 1.2. A linear algebraic group, G, is a variety V=K with a group structure such that the group operations are morphisms and V is ane. K-rational point e 2 G and K-morphisms µ : G × G ! G (µ(x; y) = xy) and i : G ! G (i(x) = x−1) K[G] is the K algebra of regular functions in V . ∆ : K[G] ! K[G] ⊗K K[G] the comultiplication, ι : K[G] ! K[G] is the coinverse We get the following axioms: (∆ ⊗ 1) ◦ ∆ = (1 ⊗ ∆) ◦ ∆ Example. : where , 1. , ∼ . n2 ∼ . Ga µ(x; y) = x+y a; y 2 A K[x] ∆ : x 7! x⊗1+1⊗y 2 K[x]⊗K[y] = K[x; y] Ga = Mn(K) : ∗ 2 dened by . Then . So Gm K ! V = (x; y) 2 A jxy = 1 t 7! (t; 1=t) (x1; y1) · (x2; y2) = (x1x2; y1y2) ∆ : x 7! x ⊗ x; y 7! y ⊗ y. Note that =∼ GL (K) p p Gm p 1 p p p Let , , −1 x1−y1 2 . Let K = Q( 2)=Q (x1 + y1 2)(x2 + y2 2) = (x1x2 + 2y1y2) + (x1y2 + x2y1) 2 (x1 + y1 2) = 2 2 x1−2y1 2 2, then we can use . , . g = x1−2y1 V (g 6= 0) ∆ : x1 7! x1⊗x2+2y1⊗y2 y1 7! x1⊗y2+x2⊗y1 ι : x1 7! x1=g; y1 7! −y1=g n .
    [Show full text]
  • Complete Homogeneous Varieties: Structure and Classification
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 9, Pages 3651{3667 S 0002-9947(03)03280-X Article electronically published on March 17, 2003 COMPLETE HOMOGENEOUS VARIETIES: STRUCTURE AND CLASSIFICATION CARLOS SANCHO DE SALAS Abstract. Homogeneous varieties are those whose group of automorphisms acts transitively on them. In this paper we prove that any complete homo- geneous variety splits in a unique way as a product of an abelian variety and a parabolic variety. This is obtained by proving a rigidity theorem for the parabolic subgroups of a linear group. Finally, using the results of Wenzel on the classification of parabolic subgroups of a linear group and the results of Demazure on the automorphisms of a flag variety, we obtain the classifi- cation of the parabolic varieties (in characteristic different from 2; 3). This, together with the moduli of abelian varieties, concludes the classification of the complete homogeneous varieties. 0. Introduction Let X be a variety over an algebraically closed field k.LetG be the functor of automorphisms of X that acts naturally on X.WesaythatX is homogeneous if G acts transitively on X, that is, if for each pair of points x; x0 2 Hom(S; X), there exists an automorphism τ 2 AutS(X × S) (after a faithfully flat base change on S) transforming one into the other: τ(x)=x0. It is known that the group of automorphisms of a complete variety exists, i.e., the functor G is representable, and it is locally of finite type (see [7]). Hence, if X is smooth, connected and homogeneous, then the reduced connected component through the origin of G acts transitively on X.
    [Show full text]
  • Algebraic Geometry
    Algebraic Geometry J.S. Milne Version 6.01 August 23, 2015 These notes are an introduction to the theory of algebraic varieties emphasizing the simi- larities to the theory of manifolds. In contrast to most such accounts they study abstract algebraic varieties, and not just subvarieties of affine and projective space. This approach leads more naturally into scheme theory. BibTeX information @misc{milneAG, author={Milne, James S.}, title={Algebraic Geometry (v6.01)}, year={2015}, note={Available at www.jmilne.org/math/}, pages={226} } v2.01 (August 24, 1996). First version on the web. v3.01 (June 13, 1998). v4.00 (October 30, 2003). Fixed errors; many minor revisions; added exercises; added two sections/chapters; 206 pages. v5.00 (February 20, 2005). Heavily revised; most numbering changed; 227 pages. v5.10 (March 19, 2008). Minor fixes; TEXstyle changed, so page numbers changed; 241 pages. v5.20 (September 14, 2009). Minor corrections; revised Chapters 1, 11, 16; 245 pages. v5.22 (January 13, 2012). Minor fixes; 260 pages. v6.00 (August 24, 2014). Major revision; 223 pages. v6.01 (August 23, 2015). Minor fixes; 226 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The curves are a tacnode, a ramphoid cusp, and an ordinary triple point. Copyright c 1996, 1998, 2003, 2005, 2008, 2009, 2011, 2012, 2013, 2014 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Contents3 1 Preliminaries from commutative algebra 11 a. Rings and ideals, 11 ; b.
    [Show full text]