Cohomological Descent for Logarithmic Differential Forms in the Log Etale
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Rational Curves on Hypersurfaces
Rational curves on hypersurfaces ||||||{ Lecture notes for the II Latin American School of Algebraic Geometry and Applications 1{12 June 2015 Cabo Frio, Brazil ||||||{ Olivier Debarre January 4, 2016 Contents 1 Projective spaces and Grassmannians 3 1.1 Projective spaces . 3 1.2 The Euler sequence . 4 1.3 Grassmannians . 4 1.4 Linear spaces contained in a subscheme of P(V )................ 5 2 Projective lines contained in a hypersurface 9 2.1 Local study of F (X) ............................... 9 2.2 Schubert calculus . 10 2.3 Projective lines contained in a hypersurface . 12 2.3.1 Existence of lines in a hypersurface . 12 2.3.2 Lines through a point . 16 2.3.3 Free lines . 17 2.4 Projective lines contained in a quadric hypersurface . 18 2.5 Projective lines contained in a cubic hypersurface . 19 2.5.1 Lines on a smooth cubic surface . 20 2.5.2 Lines on a smooth cubic fourfold . 22 2.6 Cubic hypersurfaces over finite fields . 25 3 Conics and curves of higher degrees 29 3.1 Conics in the projective space . 29 3.2 Conics contained in a hypersurface . 30 3.2.1 Conics contained in a quadric hypersurface . 32 3.2.2 Conics contained in a cubic hypersurface . 33 3.3 Rational curves of higher degrees . 35 4 Varieties covered by rational curves 39 4.1 Uniruled and separably uniruled varieties . 39 4.2 Free rational curves and separably uniruled varieties . 41 4.3 Minimal free rational curves . 44 Abstract In the past few decades, it has become more and more obvious that rational curves are the primary player in the study of the birational geometry of projective varieties. -
The Jouanolou-Thomason Homotopy Lemma
The Jouanolou-Thomason homotopy lemma Aravind Asok February 9, 2009 1 Introduction The goal of this note is to prove what is now known as the Jouanolou-Thomason homotopy lemma or simply \Jouanolou's trick." Our main reason for discussing this here is that i) most statements (that I have seen) assume unncessary quasi-projectivity hypotheses, and ii) most applications of the result that I know (e.g., in homotopy K-theory) appeal to the result as merely a \black box," while the proof indicates that the construction is quite geometric and relatively explicit. For simplicity, throughout the word scheme means separated Noetherian scheme. Theorem 1.1 (Jouanolou-Thomason homotopy lemma). Given a smooth scheme X over a regular Noetherian base ring k, there exists a pair (X;~ π), where X~ is an affine scheme, smooth over k, and π : X~ ! X is a Zariski locally trivial smooth morphism with fibers isomorphic to affine spaces. 1 Remark 1.2. In terms of an A -homotopy category of smooth schemes over k (e.g., H(k) or H´et(k); see [MV99, x3]), the map π is an A1-weak equivalence (use [MV99, x3 Example 2.4]. Thus, up to A1-weak equivalence, any smooth k-scheme is an affine scheme smooth over k. 2 An explicit algebraic form Let An denote affine space over Spec Z. Let An n 0 denote the scheme quasi-affine and smooth over 2m Spec Z obtained by removing the fiber over 0. Let Q2m−1 denote the closed subscheme of A (with coordinates x1; : : : ; x2m) defined by the equation X xixm+i = 1: i Consider the following simple situation. -
Vector Bundles As Generators on Schemes and Stacks
Vector Bundles as Generators on Schemes and Stacks Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨atD¨usseldorf vorgelegt von Philipp Gross aus D¨usseldorf D¨usseldorf,Mai 2010 Aus dem Mathematischen Institut der Heinrich-Heine-Universit¨atD¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨atder Heinrich-Heine-Universit¨atD¨usseldorf Referent: Prof. Dr. Stefan Schr¨oer Koreferent: Prof. Dr. Holger Reich Acknowledgments The work on this dissertation has been one of the most significant academic challenges I have ever had to face. This study would not have been completed without the support, patience and guidance of the following people. It is to them that I owe my deepest gratitude. I am indebted to my advisor Stefan Schr¨oerfor his encouragement to pursue this project. He taught me algebraic geometry and how to write academic papers, made me a better mathematician, brought out the good ideas in me, and gave me the opportunity to attend many conferences and schools in Europe. I also thank Holger Reich, not only for agreeing to review the dissertation and to sit on my committee, but also for showing an interest in my research. Next, I thank the members of the local algebraic geometry research group for their time, energy and for the many inspiring discussions: Christian Liedtke, Sasa Novakovic, Holger Partsch and Felix Sch¨uller.I have had the pleasure of learning from them in many other ways as well. A special thanks goes to Holger for being a friend, helping me complete the writing of this dissertation as well as the challenging research that lies behind it. -
NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation. -
Bertini's Theorem on Generic Smoothness
U.F.R. Mathematiques´ et Informatique Universite´ Bordeaux 1 351, Cours de la Liberation´ Master Thesis in Mathematics BERTINI1S THEOREM ON GENERIC SMOOTHNESS Academic year 2011/2012 Supervisor: Candidate: Prof.Qing Liu Andrea Ricolfi ii Introduction Bertini was an Italian mathematician, who lived and worked in the second half of the nineteenth century. The present disser- tation concerns his most celebrated theorem, which appeared for the first time in 1882 in the paper [5], and whose proof can also be found in Introduzione alla Geometria Proiettiva degli Iperspazi (E. Bertini, 1907, or 1923 for the latest edition). The present introduction aims to informally introduce Bertini’s Theorem on generic smoothness, with special attention to its re- cent improvements and its relationships with other kind of re- sults. Just to set the following discussion in an historical perspec- tive, recall that at Bertini’s time the situation was more or less the following: ¥ there were no schemes, ¥ almost all varieties were defined over the complex numbers, ¥ all varieties were embedded in some projective space, that is, they were not intrinsic. On the contrary, this dissertation will cope with Bertini’s the- orem by exploiting the powerful tools of modern algebraic ge- ometry, by working with schemes defined over any field (mostly, but not necessarily, algebraically closed). In addition, our vari- eties will be thought of as abstract varieties (at least when over a field of characteristic zero). This fact does not mean that we are neglecting Bertini’s original work, containing already all the rele- vant ideas: the proof we shall present in this exposition, over the complex numbers, is quite close to the one he gave. -
On the De Rham Cohomology of Algebraic Varieties
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. ROBIN HARTSHORNE On the de Rham cohomology of algebraic varieties Publications mathématiques de l’I.H.É.S., tome 45 (1975), p. 5-99 <http://www.numdam.org/item?id=PMIHES_1975__45__5_0> © Publications mathématiques de l’I.H.É.S., 1975, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES by ROBIN HARTSHORNE CONTENTS INTRODUCTION .............................................................................. 6 CHAPTER I. — Preliminaries .............................................................. n § i. Cohomology theories ............................................................... 9 § 2. Direct and inverse images .......................................................... 10 § 3. Hypercohomology ................................................................. 11 § 4. Inverse limits...................................................................... 12 § 5. Completions ...................................................................... 18 § -
The Spectral Sequence Relating Algebraic K-Theory to Motivic Cohomology
Ann. Scient. Éc. Norm. Sup., 4e série, t. 35, 2002, p. 773 à 875. THE SPECTRAL SEQUENCE RELATING ALGEBRAIC K-THEORY TO MOTIVIC COHOMOLOGY BY ERIC M. FRIEDLANDER 1 AND ANDREI SUSLIN 2 ABSTRACT. – Beginning with the Bloch–Lichtenbaum exact couple relating the motivic cohomology of a field F to the algebraic K-theory of F , the authors construct a spectral sequence for any smooth scheme X over F whose E2 term is the motivic cohomology of X and whose abutment is the Quillen K-theory of X. A multiplicative structure is exhibited on this spectral sequence. The spectral sequence is that associated to a tower of spectra determined by consideration of the filtration of coherent sheaves on X by codimension of support. 2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – Partant du couple exact de Bloch–Lichtenbaum, reliant la cohomologie motivique d’un corps F àsaK-théorie algébrique, on construit, pour tout schéma lisse X sur F , une suite spectrale dont le terme E2 est la cohomologie motivique de X et dont l’aboutissement est la K-théorie de Quillen de X. Cette suite spectrale, qui possède une structure multiplicative, est associée à une tour de spectres déterminée par la considération de la filtration des faisceaux cohérents sur X par la codimension du support. 2002 Éditions scientifiques et médicales Elsevier SAS The purpose of this paper is to establish in Theorem 13.6 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic K-theory of X: p,q p−q Z − −q − − ⇒ (13.6.1) E2 = H X, ( q) = CH (X, p q) K−p−q(X). -
Cyclic Homology, Cdh-Cohomology and Negative K-Theory
Annals of Mathematics, 167 (2008), 549–573 Cyclic homology, cdh-cohomology and negative K-theory By G. Cortinas,˜ C. Haesemeyer, M. Schlichting, and C. Weibel* Abstract We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero. Introduction The negative algebraic K-theory of a singular variety is related to its ge- ometry. This observation goes back to the classic study by Bass and Murthy [1], which implicitly calculated the negative K-theory of a curve X. By def- inition, the group K−n(X) describes a subgroup of the Grothendieck group 1 n K0(Y ) of vector bundles on Y = X × (A −{0}) . The following conjecture was made in 1980, based upon the Bass-Murthy calculations, and appeared in [38, 2.9]. Recall that if F is any contravariant functor on schemes, a scheme X is called F -regular if F (X) → F (X × Ar)is an isomorphism for all r ≥ 0. K-dimension Conjecture 0.1. Let X be a Noetherian scheme of di- mension d. Then Km(X)=0for m<−d and X is K−d-regular. In this paper we give a proof of this conjecture for X essentially of finite type over a field F of characteristic 0; see Theorem 6.2. -
V4A2 - Algebraic Geometry II (Sommersemester 2017)
V4A2 - Algebraic Geometry II (Sommersemester 2017) Taught by Prof. Dr. Peter Scholze Typed by Jack Davies Contents 1 Overview of the Course 20/04/2017 1 2 Flatness 24/04/2017 4 3 Faithfully Flat Descent 27/04/2017 9 4 Smoothness 04/05/2017 13 5 Kähler Differentials 08/05/2017 18 6 Smoothness and the Jacobi Criterion 11/05/2017 24 7 Smoothness is Local on Source 15/05/2017 30 8 The First Sheaf Cohomology Group 18/05/2017 34 9 Homological Algebra I (Derived Functors) 22/05/2017 38 10 Homological Algebra II (Homotopy Categories) 29/05/2017 43 11 Homological Algebra III (Derived Categories) 01/06/2017 48 12 Cechˇ Cohomology 12/06/2017 54 13 Finiteness in Cohomology I 19/06/2017 58 14 Finiteness in Cohomology II 22/06/2017 62 15 Affine Criterion and Base Change 26/06/2017 66 16 Generalised Base Change and ⊗L 03/07/2017 71 17 Finiteness of RΓ(X; M) and Riemann-Roch 06/07/2017 75 18 Ext Functor and Serre Duality 10/07/2017 79 19 Ext Functor and the Proof of Serre Duality 13/07/2017 84 20 Formal Functions 17/07/2017 89 21 Zariski’s Main Theorem and Consequences 20/07/2017 94 22 Relatively Ample Line Bundles 24/07/2017 98 23 Left Derived Functors and T or 27/07/2017 103 Introduction This course was taught in Bonn, Germany over the Sommersemester 2017, by Prof. Dr. Peter Scholze. We started by looking at properties of flat maps between schemes, as a way of parametrising nice families of schemes over the base scheme. -
On the K-Theory and Topological Cyclic Homology of Smooth Schemes Over a Discrete Valuation Ring
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 1, Pages 131–145 S 0002-9947(04)03599-8 Article electronically published on December 28, 2004 ON THE K-THEORY AND TOPOLOGICAL CYCLIC HOMOLOGY OF SMOOTH SCHEMES OVER A DISCRETE VALUATION RING THOMAS GEISSER AND LARS HESSELHOLT Abstract. We show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic (0,p), the p-adic ´etale K-theory and p-adic topological cyclic homology agree. Introduction In this paper, we consider a smooth scheme X over a henselian discrete valuation ring V of mixed characteristic (0,p). Let i: Y→ X denote the inclusion of the special fiber. We have previously showed that the cyclotomic trace et Z v −∼→ Z v Kq (Y, /p ) TCq(Y ; p, /p ) defines an isomorphism of the Dwyer-Friedlander ´etale K-theory of Y with Z/pv- coefficients and the topological cyclic homology of Y with Z/pv-coefficients [2]. Starting from this result, we show: Theorem A. Suppose that X is smooth and proper over V . Then for all integers q and all v ≥ 1, the cyclotomic trace et Z v −∼→ Z v Kq (X, /p ) TCq(X; p, /p ) is an isomorphism. We briefly outline the proof. The left-hand term in the statement of Theorem A is the abutment of a strongly convergent spectral sequence s,t s − Z v ˜ ⇒ et Z v E2 = H (Xet,K−t( , /p ) ) K−s−t(X, /p ) from the cohomology of the sheaf on the small ´etale site Xet associated with the v presheaf which to U → X assigns K−t(U, Z/p ) [12]. -
Notes for Topics in AG: Deformation Theory Instructor: Daniel Litt
Notes for Topics in AG: Deformation Theory Instructor: Daniel Litt Henry Liu December 11, 2017 Abstract These are my live-texed notes for the Fall 2017 offering of MATH GR6263 Topics in Algebraic Ge- ometry. Let me know when you find errors or typos. I'm sure there are plenty. 1 Introduction and motivation 1 2 Deformation functors 1 2.1 Tangent-obstruction theories . .2 2.2 Application: Lefschetz theorem for Pic . .3 2.3 Pro-representability and Schlessinger's criterion . .5 2.4 Application: smoothness of Hilbert scheme . .8 3 Tian{Todorov theorem 11 3.1 Deformations of a smooth variety . 12 3.2 T1-lifting theorem . 14 3.3 Proof of Tian{Todorov . 15 4 Generic vanishing 16 i 4.1 Analytic structure on S (X) ....................................... 17 4.2 The derivative complex via Hodge theory . 20 4.3 Proof of generic vanishing . 22 5 Unobstructedness of abelian varieties 23 5.1 Deformations of polarized AVs . 26 6 Formal GAGA 28 7 Bend and break 30 7.1 Moving curves in varieties . 31 7.2 Bend and break . 32 8 Isomonodromic deformations 33 9 Cotangent complex 36 9.1 Naive cotangent complex . 36 9.2 Using the cotangent complex . 38 9.3 Construction and properties . 38 9.4 More applications . 40 10 Gauss{Manin connection 40 a 1 Introduction and motivation Deformation theory is the study of the local structure of a space, e.g. a representable functor, near a point. Later on, a functor will not be sufficiently general, so we will work with categories co-fibered in groupoids. -
Modules Over Algebraic Cobordism
MODULES OVER ALGEBRAIC COBORDISM ELDEN ELMANTO, MARC HOYOIS, ADEEL A. KHAN, VLADIMIR SOSNILO, AND MARIA YAKERSON Abstract. We prove that the 1-category of MGL-modules over any scheme is equivalent to the 1-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite P1-loop spaces, we deduce that very effective MGL-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that Ω1 MGL is the A1-homotopy type of the moduli stack of virtual finite flat P1 local complete intersections, and that for n > 0, Ω1 Σn MGL is the A1-homotopy type of P1 P1 the moduli stack of finite quasi-smooth derived schemes of virtual dimension −n. Contents 1. Introduction 2 1.1. Modules over algebraic cobordism 2 1.2. Framed correspondences and the motivic recognition principle 2 1.3. Geometric models of motivic Thom spectra 3 1.4. Analogies with topology 4 1.5. Related work 5 1.6. Conventions and notation 5 1.7. Acknowledgments 6 2. Twisted framed correspondences 6 2.1. Stable vector bundles and K-theory 6 2.2. Presheaves of twisted framed correspondences 7 2.3. Descent and additivity 9 2.4. Comparison theorems 10 2.5. Base change 15 3. Geometric models of motivic Thom spectra 16 3.1.