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The Minimal Model Program for the Hilbert Scheme of Points on P2 and Bridgeland Stability
THE MINIMAL MODEL PROGRAM FOR THE HILBERT SCHEME OF POINTS ON P2 AND BRIDGELAND STABILITY DANIELE ARCARA, AARON BERTRAM, IZZET COSKUN, AND JACK HUIZENGA 2[n] 2 Abstract. In this paper, we study the birational geometry of the Hilbert scheme P of n-points on P . We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n ≤ 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions. Contents 1. Introduction 2 2. Preliminaries on the Hilbert scheme of points 4 2[n] 3. Effective divisors on P 5 2[n] 4. The stable base locus decomposition of the effective cone of P 8 5. Preliminaries on Bridgeland stability conditions 15 6. Potential walls 18 7. The quiver region 23 8. Moduli of stable objects are GIT quotients 26 9. Walls for the Hilbert scheme 27 10. Explicit Examples 30 2[2] 10.1. The walls for P 30 2[3] 10.2. The walls for P 31 2[4] 10.3. The walls for P 32 2[5] 10.4. The walls for P 33 2[6] 10.5. The walls for P 34 2[7] 10.6. -
Arxiv:2008.05229V2 [Math.AG]
A TORELLI THEOREM FOR MODULI SPACES OF PARABOLIC VECTOR BUNDLES OVER AN ELLIPTIC CURVE THIAGO FASSARELLA AND LUANA JUSTO Abstract. Let C be an elliptic curve, w ∈ C, and let S ⊂ C be a finite subset of cardinality at least 3. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle OC (w) over (C,S) which are semistable with respect to a weight vector 1 1 2 ,..., 2 . 1. Introduction Let C be a smooth complex curve of genus g ≥ 2 and fix w ∈ C. Let M be the corresponding moduli space of semistable rank two vector bundles having OC (w) as determinant line bundle. A classical Torelli type theorem of D. Mumford and P. Newstead [MN68] says that the isomorphism class of M determines the isomorphism class of C. This result has been extended first in [Tyu70, NR75, KP95] to higher rank and later to the parabolic context, which we now describe. We now assume g ≥ 0. Let S ⊂ C be a finite subset of cardinality n ≥ 1. Let Ma be the moduli space of rank two parabolic vector bundles on (C,S) with fixed determinant line bundle OC (w), and which are µa-semistable, see Section 2. The subscript a refers to a particular choice of a weight vector a = (a1, . , an) of real numbers, 0 ≤ ai ≤ 1, which gives the slope-stability condition. The moduli 1 1 space associated to the central weight aF = 2 ,..., 2 is particularly interesting, for instance when g = 0 and n ≥ 5 it is a Fano variety that is smooth if n is odd and has isolated singularities if n is even, see [Muk05, Cas15, AM16, AFKM19]. -
Easy Decision Diffie-Hellman Groups
EASY DECISION-DIFFIE-HELLMAN GROUPS STEVEN D. GALBRAITH AND VICTOR ROTGER Abstract. The decision-Di±e-Hellman problem (DDH) is a central compu- tational problem in cryptography. It is already known that the Weil and Tate pairings can be used to solve many DDH problems on elliptic curves. A natural question is whether all DDH problems are easy on supersingular curves. To answer this question it is necessary to have suitable distortion maps. Verheul states that such maps exist, and this paper gives an algorithm to construct them. The paper therefore shows that all DDH problems on the supersingular elliptic curves used in practice are easy. We also discuss the issue of which DDH problems on ordinary curves are easy. 1. Introduction It is well-known that the Weil and Tate pairings make many decision-Di±e- Hellman (DDH) problems on elliptic curves easy. This observation is behind ex- citing new developments in pairing-based cryptography. This paper studies the question of which DDH problems are easy and which are not necessarily easy. First we recall some de¯nitions. Decision Di±e-Hellman problem (DDH): Let G be a cyclic group of prime order r written additively. The DDH problem is to distinguish the two distributions in G4 D1 = f(P; aP; bP; abP ): P 2 G; 0 · a; b < rg and D2 = f(P; aP; bP; cP ): P 2 G; 0 · a; b; c < rg: 4 Here D1 is the set of valid Di±e-Hellman-tuples and D2 = G . By `distinguish' we mean there is an algorithm which takes as input an element of G4 and outputs \valid" or \invalid", such that if the input is chosen with probability 1/2 from each of D1 and D2 ¡ D1 then the output is correct with probability signi¯cantly more than 1/2. -
A Torelli Theorem for Moduli Spaces of Principal Bundles Over a Curve Tome 62, No 1 (2012), P
R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Indranil BISWAS & Norbert HOFFMANN A Torelli theorem for moduli spaces of principal bundles over a curve Tome 62, no 1 (2012), p. 87-106. <http://aif.cedram.org/item?id=AIF_2012__62_1_87_0> © Association des Annales de l’institut Fourier, 2012, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 62, 1 (2012) 87-106 A TORELLI THEOREM FOR MODULI SPACES OF PRINCIPAL BUNDLES OVER A CURVE by Indranil BISWAS & Norbert HOFFMANN (*) 0 Abstract. — Let X and X be compact Riemann surfaces of genus > 3, and 0 d let G and G be nonabelian reductive complex groups. If one component MG(X) of the coarse moduli space for semistable principal G–bundles over X is isomorphic d0 0 0 to another component MG0 (X ), then X is isomorphic to X . Résumé. — Soient X et X0 des surfaces de Riemann compactes de genre au moins 3, et G et G0 des groupes complexes réductifs non abéliens. -
The Lci Locus of the Hilbert Scheme of Points & the Cotangent Complex
The lci locus of the Hilbert scheme of points & the cotangent complex Peter J. Haine September 21, 2019 Abstract Weintroduce the basic algebraic geometry background necessary to understand the main results of the work of Elmanto, Hoyois, Khan, Sosnilo, and Yakerson on motivic infinite loop spaces [5]. After recalling some facts about the Hilbert func- tor of points, we introduce local complete intersection (lci) and syntomic morphisms. We then give an overview of the cotangent complex and discuss the relationship be- tween lci morphisms and the cotangent complex (in particular, Avramov’s charac- terization of lci morphisms). We then introduce the lci locus of the Hilbert scheme of points and the Hilbert scheme of framed points, and prove that the lci locus of the Hilbert scheme of points is formally smooth. Contents 1 Hilbert schemes 2 2 Local complete intersections via equations 3 Relative global complete intersections ...................... 3 Local complete intersections ........................... 4 3 Background on the cotangent complex 6 Motivation .................................... 6 Properties of the cotangent complex ....................... 7 4 Local complete intersections and the cotangent complex 9 Avramov’s characterization of local complete intersections ........... 11 5 The lci locus of the Hilbert scheme of points 13 6 The Hilbert scheme of framed points 14 1 1 Hilbert schemes In this section we recall the Hilbert functor of points from last time as well as state the basic representability properties of the Hilbert functor of points. 1.1 Recollection ([STK, Tag 02K9]). A morphism of schemes 푓∶ 푌 → 푋 is finite lo- cally free if 푓 is affine and 푓⋆풪푌 is a finite locally free 풪푋-module. -
Arxiv:1610.06871V4 [Math.AT] 13 May 2019 Se[A H.7506 N SG E.B.6.2.7]): Rem
THE MOREL–VOEVODSKY LOCALIZATION THEOREM IN SPECTRAL ALGEBRAIC GEOMETRY ADEEL A. KHAN Abstract. We prove an analogue of the Morel–Voevodsky localization theorem over spectral algebraic spaces. As a corollary we deduce a “derived nilpotent invariance” result which, informally speaking, says that A1-homotopy invariance kills all higher homotopy groups of a connective commutative ring spectrum. 1. Introduction ´et Let R be a connective E∞-ring spectrum, and denote by CAlgR the ∞-category of ´etale E∞-algebras over R. The starting point for this paper is the following fundamental result of J. Lurie, which says that the small ´etale topos of R is equivalent to the small ´etale topos of π0(R) (see [HA, Thm. 7.5.0.6] and [SAG, Rem. B.6.2.7]): Theorem 1.0.1 (Lurie). For any connective E∞-ring R, let π0(R) denote its 0-truncation E ´et ´et (viewed as a discrete ∞-ring). Then restriction along the canonical functor CAlgR → CAlgπ0(R) ´et induces an equivalence from the ∞-category of ´etale sheaves of spaces CAlgπ0(R) → Spc to the ´et ∞-category of ´etale sheaves of spaces CAlgR → Spc. Theorem 1.0.1 can be viewed as a special case of the following result (see [SAG, Prop. 3.1.4.1]): ′ Theorem 1.0.2 (Lurie). Let R → R be a homomorphism of connective E∞-rings that is ´et ´et surjective on π0. Then restriction along the canonical functor CAlgR → CAlgR′ defines a fully ´et faithful embedding of the ∞-category of ´etale sheaves CAlgR′ → Spc into the ∞-category of ´etale ´et sheaves CAlgR → Spc. -
Fundamental Algebraic Geometry
http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. -
Weighted Fano Varieties and Infinitesimal Torelli Problem 3
WEIGHTED FANO VARIETIES AND INFINITESIMAL TORELLI PROBLEM ENRICO FATIGHENTI, LUCA RIZZI AND FRANCESCO ZUCCONI Abstract. We solve the infinitesimal Torelli problem for 3-dimensional quasi-smooth Q-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively distribute themselves in examples and counterexamples for the infinitesimal Torelli claim and which share the analogue, and in some cases the same, Hodge-diagram properties as the length 3 Gushel- Mukai chain of prime smooth Fanos of coindex 3 and degree 10. 1. Introduction Minimal model program suggests us to formulate inside the category of varieties with terminal singularities many questions which were initially asked for smooth varieties. The construction of the period map, and the related Torelli type problems are definitely among these problems. 1.1. Infinitesimal Torelli. A Q-Fano variety is a projective variety X such that X has at worst Q-factorial terminal singularities, −KX is ample and Pic(X) has rank 1; cf. [CPR00]. In this paper we give a full answer to the infinitesimal Torelli problem in the case of quasi- smooth Q-Fano hypersurfaces of dimension 3 with terminal singularities and with Picard number 1. In subsections 2.1 of this paper the reader can find a basic dictionary and up to date references needed to understand the statement of the infinitesimal Torelli problem and where to find the meaning of infinitesimal variation of Hodge structures (IVHS; to short). Here we stress only that the method à la Griffiths based on the extension of the Macaulay’s theorem in algebra to the weighted case, cf. -
Algebraic Stacks (Winter 2020/2021) Lecturer: Georg Oberdieck
Algebraic Stacks (Winter 2020/2021) Lecturer: Georg Oberdieck Date: Wednesday 4-6, Friday 12-2. Oral Exam: February 19 and March 26, 2021. Exam requirement: You have to hand in solutions to at least half of all exercise problems to be admitted to the exam. Summary: The course is an introduction to the theory of algebraic stacks. We will discuss applications to the construction of moduli spaces. Prerequisites: Algebraic Geometry I and II (e.g. on the level of Hartshorne's book Chapter I and II plus some background on flat/etale morphisms). Some category theory (such as Vakil's Notes on Algebraic Geometry, Chapter 1). References: (1) Olsson, Algebraic Spaces and Stacks (2) Stacks Project (3) Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (available online) (4) Halpern-Leistner, The Moduli space, https://themoduli.space (5) Guide to the literature by Jarod Alper (6) Behrend, Introduction to algebraic stacks, lecture notes. The main reference of the coarse is Olsson and will be used throughout, in particular for many of the exercises. However, we will diverge in some details from Olsson's presentation, in particular on quasi-coherent sheaves on algebraic stacks where we follow the stacks project. The notes of Vistoli are also highly recommended for the first part which covers aspects of Grothendieck topologies, fibered categories and descent theory. In the second part we will use the notes of Halpern-Leistner. Exercises and student presentations: Friday's session will be used sometimes for informal discussion of the material covered in the problem sets. From time to time there will also be student presentations, see: http://www.math.uni-bonn.de/~georgo/stacks/presentations.pdf Schedule1 Oct 28: Motivation mostly. -
Algebraic Curves and Surfaces
Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces . -
Further Pathologies in Algebraic Geometry Author(S): David Mumford Source: American Journal of Mathematics , Oct., 1962, Vol
Further Pathologies in Algebraic Geometry Author(s): David Mumford Source: American Journal of Mathematics , Oct., 1962, Vol. 84, No. 4 (Oct., 1962), pp. 642-648 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.com/stable/2372870 JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics This content downloaded from 132.174.252.179 on Sat, 01 Aug 2020 11:36:26 UTC All use subject to https://about.jstor.org/terms FURTHER PATHOLOGIES IN ALGEBRAIC GEOMETRY.*1 By DAVID MUMFORD. The following note is not strictly a continuation of our previous note [1]. However, we wish to present two more examples of algebro-geometric phe- nomnena which seem to us rather startling. The first relates to characteristic ]) behaviour, and the second relates to the hypothesis of the completeness of the characteristic linear system of a maximal algebraic family. We will use the same notations as in [1]. The first example is an illustration of a general principle that might be said to be indicated by many of the pathologies of characteristic p: A non-singular characteristic p variety is analogous to a general non- K:ahler complex manifold; in particular, a projective embedding of such a variety is not as "strong" as a Kiihler metric on a complex manifold; and the Hodge-Lefschetz-Dolbeault theorems on sheaf cohomology break down in every possible way. -
The Tate Pairing for Abelian Varieties Over Finite Fields
Journal de Th´eoriedes Nombres de Bordeaux 00 (XXXX), 000{000 The Tate pairing for Abelian varieties over finite fields par Peter BRUIN Resum´ e.´ Nous d´ecrivons un accouplement arithm´etiqueassoci´e `aune isogenie entre vari´et´esab´eliennessur un corps fini. Nous montrons qu'il g´en´eralisel'accouplement de Frey et R¨uck, ainsi donnant une d´emonstrationbr`eve de la perfection de ce dernier. Abstract. In this expository note, we describe an arithmetic pairing associated to an isogeny between Abelian varieties over a finite field. We show that it generalises the Frey{R¨uck pairing, thereby giving a short proof of the perfectness of the latter. 1. Introduction Throughout this note, k denotes a finite field of q elements, and k¯ denotes an algebraic closure of k. If n is a positive integer, then µn denotes the group of n-th roots of unity in k¯×. Let C be a complete, smooth, geometrically connected curve over k, let J be the Jacobian variety of C, and let n be a divisor of q − 1. In [1], Frey and R¨uck defined a perfect pairing f ; gn : J[n](k) × J(k)=nJ(k) −! µn(k) as follows: if D and E are divisors on C with disjoint supports and f is a non-zero rational function with divisor nD, then (q−1)=n f[D]; [E] mod nJ(k)gn = f(E) ; where Y nx X f(E) = f(x) if E = nxx: x2C(k¯) x2C(k¯) Remark. We have composed the map as defined in [1] with the isomor- × × n phism k =(k ) ! µn(k) that raises elements to the power (q − 1)=n.