Rational Points on Curves
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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 . -
Arxiv:2002.02220V2 [Math.AG] 8 Jun 2020 in the Early 80’S, V
Contemporary Mathematics Toward good families of codes from towers of surfaces Alain Couvreur, Philippe Lebacque, and Marc Perret To our late teacher, colleague and friend Gilles Lachaud Abstract. We introduce in this article a new method to estimate the mini- mum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be \liftable" under finite morphisms, paving the way toward the construction of good codes from towers of surfaces. In the same direction, we establish a criterion for a surface with a fixed finite set of closed points P to have an infinite tower of `{´etalecovers in which P splits totally. We conclude by stating several open problems. In par- ticular, we relate the existence of asymptotically good codes from general type surfaces with a very ample canonical class to the behaviour of their number of rational points with respect to their K2 and coherent Euler characteristic. Contents Introduction 1 1. Codes from surfaces 3 2. Infinite ´etaletowers of surfaces 15 3. Open problems 26 Acknowledgements 29 References 29 Introduction arXiv:2002.02220v2 [math.AG] 8 Jun 2020 In the early 80's, V. D. Goppa proposed a construction of codes from algebraic curves [21]. A pleasant feature of these codes from curves is that they benefit from an elementary but rather sharp lower bound for the minimum distance, the so{called Goppa designed distance. In addition, an easy lower bound for the dimension can be derived from Riemann-Roch theorem. The very simple nature of these lower bounds for the parameters permits to formulate a concise criterion for a sequence of curves to provide asymptotically good codes : roughly speaking, the curves should have a large number of rational points compared to their genus. -
Moduli of Curves 1
THE MODULI SPACE OF CURVES In this section, we will give a sketch of the construction of the moduli space Mg of curves of genus g and the closely related moduli space Mg;n of n-pointed curves of genus g using two different approaches. Throughout this section we always assume that 2g − 2 + n > 0. In the first approach, we will embed curves in a projective space PN using a sufficiently high power n of their dualizing sheaf. Then a locus in the Hilbert scheme parameterizes n-canonically embedded curves of genus g. The automorphism group PGL(N + 1) acts on the Hilbert scheme. Using Mumford's Geometric Invariant Theory, one can take the quotient of the appropriate locus in the Hilbert scheme by the action of PGL(N + 1) to construct Mg. In the second approach, one first constructs the moduli space of curves as a Deligne-Mumford stack. One then exhibits an ample line bundle on the coarse moduli scheme of this stack. 1. Basics about curves We begin by collecting basic facts and definitions about stable curves. A curve singularity (C; p) is called a node if locally analytically the singularity is isomophic to the plane curve singularity xy = 0. A curve C is called at-worst-nodal or more simply nodal if the only singularities of C are nodes. The dualizing sheaf !C of an at-worst-nodal curve C is an invertible sheaf that has a simple description. Let ν : Cν ! C be the nor- malization of the curve C. Let p1; : : : ; pδ be the nodes of C and let −1 fri; sig = ν (pi). -
Log Minimal Model Program for the Moduli Space of Stable Curves: the Second Flip
LOG MINIMAL MODEL PROGRAM FOR THE MODULI SPACE OF STABLE CURVES: THE SECOND FLIP JAROD ALPER, MAKSYM FEDORCHUK, DAVID ISHII SMYTH, AND FREDERICK VAN DER WYCK Abstract. We prove an existence theorem for good moduli spaces, and use it to construct the second flip in the log minimal model program for M g. In fact, our methods give a uniform self-contained construction of the first three steps of the log minimal model program for M g and M g;n. Contents 1. Introduction 2 Outline of the paper 4 Acknowledgments 5 2. α-stability 6 2.1. Definition of α-stability6 2.2. Deformation openness 10 2.3. Properties of α-stability 16 2.4. αc-closed curves 18 2.5. Combinatorial type of an αc-closed curve 22 3. Local description of the flips 27 3.1. Local quotient presentations 27 3.2. Preliminary facts about local VGIT 30 3.3. Deformation theory of αc-closed curves 32 3.4. Local VGIT chambers for an αc-closed curve 40 4. Existence of good moduli spaces 45 4.1. General existence results 45 4.2. Application to Mg;n(α) 54 5. Projectivity of the good moduli spaces 64 5.1. Main positivity result 67 5.2. Degenerations and simultaneous normalization 69 5.3. Preliminary positivity results 74 5.4. Proof of Theorem 5.5(a) 79 5.5. Proof of Theorem 5.5(b) 80 Appendix A. 89 References 92 1 2 ALPER, FEDORCHUK, SMYTH, AND VAN DER WYCK 1. Introduction In an effort to understand the canonical model of M g, Hassett and Keel introduced the log minimal model program (LMMP) for M . -
An Introduction to Moduli Spaces of Curves and Its Intersection Theory
An introduction to moduli spaces of curves and its intersection theory Dimitri Zvonkine∗ Institut math´ematiquede Jussieu, Universit´eParis VI, 175, rue du Chevaleret, 75013 Paris, France. Stanford University, Department of Mathematics, building 380, Stanford, California 94305, USA. e-mail: [email protected] Abstract. The material of this chapter is based on a series of three lectures for graduate students that the author gave at the Journ´eesmath´ematiquesde Glanon in July 2006. We introduce moduli spaces of smooth and stable curves, the tauto- logical cohomology classes on these spaces, and explain how to compute all possible intersection numbers between these classes. 0 Introduction This chapter is an introduction to the intersection theory on moduli spaces of curves. It is meant to be as elementary as possible, but still reasonably short. The intersection theory of an algebraic variety M looks for answers to the following questions: What are the interesting cycles (algebraic subvarieties) of M and what cohomology classes do they represent? What are the interesting vector bundles over M and what are their characteristic classes? Can we describe the full cohomology ring of M and identify the above classes in this ring? Can we compute their intersection numbers? In the case of moduli space, the full cohomology ring is still unknown. We are going to study its subring called the \tautological ring" that contains the classes of most interesting cycles and the characteristic classes of most interesting vector bundles. To give a sense of purpose to the reader, we assume the following goal: after reading this paper, one should be able to write a computer program evaluating ∗Partially supported by the NSF grant 0905809 \String topology, field theories, and the topology of moduli spaces". -
Tropicalization and Semistable Reduction of Curves
TROPICALIZATION AND STABLE REDUCTION OF CURVES CHRISTOPHER EUR Abstract. The stable reduction theorem implies that Mg, moduli space of curves of genus g, has a compactification to Mg, the Deligne-Mumford stable compactification. The space Mg has a stratification where each stratum corresponds to a combinatorial type of the stable reduction. The combinatorial data of this stratification can be encoded as the moduli of tropical curves. Here we survey a relationship between these tropical curves associated to stable reductions and tropicalizations of curves. Contents 1. The moduli space Mg 1 1.1. What is a moduli space?1 1.2. The spaces Mg; Mg and (semi)stable curves2 1.3. Stratification of Mg and dual graphs3 trop 2. Mg and the moduli space Mg 5 2.1. Moduli of tropical curves5 2.2. Berkovich analyfication and tropicalization6 trop 2.3. Connecting Mg to Mg 7 2.4. Computing the (semi)stable model8 References 10 Notation. Throughout this paper, let k denote an algebraically closed field. For a scheme X, we denote by X(A) := MorSch(Spec A; X) the A-valued points of X.A curve over k is a pure 1-dimensional reduced, separated, and finite type k-scheme. 1. The moduli space Mg 1.1. What is a moduli space? Informally, a moduli space is a geometric space whose points correspond to algebro- geometric objects of specified kind. For example, the space n is a \moduli space of lines PC through the origin in (n + 1)-space over ," since the classical points of n corresponds to C PC lines in Cn+1 through the origin. -
On the Density of Rational Points on Elliptic Fibrations
On the density of rational points on elliptic fibrations F. A. Bogomolov Courant Institute of Mathematical Sciences, N.Y.U. 251 Mercer str. New York, (NY) 10012, U.S.A. e-mail: [email protected] and Yu. Tschinkel Dept. of Mathematics, U.I.C. 851 South Morgan str. Chicago, (IL) 60607-7045, U.S.A. e-mail: [email protected] arXiv:math/9811043v1 [math.AG] 7 Nov 1998 1 Introduction Let X be an algebraic variety defined over a number field F . We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K-rational points X(K) is Zariski dense in X. The main problem is to relate this property to geometric invariants of X. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]). Varieties which are not of general type can be thought of as triple fibra- tions X → Y → Z, where the generic fiber of X → Y is rationally connected, Y → Z is a Kodaira fibration with generic fiber of Kodaira dimension 0 and the base has Kodaira dimension ≤ 0 (cf. [3]). In this paper we study mostly varieties of dimension 2 and 3. In dimension 2 the picture is as follows: Rationally connected surfaces are rational over some finite extension of F and therefore the problem has an easy solution in this case. -
Rational Points and Obstructions to Their Existence 2015 Arizona Winter School Problem Set Extended Version
RATIONAL POINTS AND OBSTRUCTIONS TO THEIR EXISTENCE 2015 ARIZONA WINTER SCHOOL PROBLEM SET EXTENDED VERSION KĘSTUTIS ČESNAVIČIUS The primary goal of the problems below is to build up familiarity with some useful lemmas and examples that are related to the theme of the Winter School. Notation. For a field k, we denote by k a choice of its algebraic closure, and by ks Ă k the resulting separable closure. If k is a number field and v is its place, we write kv for the corresponding completion. If k “ Q, we write p ¤ 8 to emphasize that p is allowed be the infinite place; for this particular p, we write Qp to mean R. For a base scheme S and S-schemes X and Y , we write XpY q for the set of S-morphisms Y Ñ X. When dealing with affine schemes we sometimes omit Spec for brevity: for instance, we write Br R in place of BrpSpec Rq. A ‘torsor’ always means a ‘right torsor.’ Acknowledgements. I thank the organizers of the Arizona Winter School 2015 for the opportunity to design this problem set. I thank Alena Pirutka for helpful comments. 1. Rational points In this section, k is a field and X is a k-scheme. A rational point of X is an element x P Xpkq, i.e., a section x: Spec k Ñ X of the structure map X Ñ Spec k. 1.1. Suppose that X “ Spec krT1;:::;Tns . Find a natural bijection pf1;:::;fmq n Xpkq ÐÑ tpx1; : : : ; xnq P k such that fipx1; : : : ; xnq “ 0 for every i “ 1; : : : ; mu: krT1;:::;Tns Hint. -
IV.5 Arithmetic Geometry Jordan S
i 372 IV. Branches of Mathematics where the aj,i1,...,in are indeterminates. If we write with many nice pictures and reproductions. A Scrap- g1f1 + ··· + gmfm as a polynomial in the variables book of Complex Curve Theory (American Mathemat- x1,...,xn, then all the coefficients must vanish, save ical Society, Providence, RI, 2003), by C. H. Clemens, the constant term which must equal 1. Thus we get and Complex Algebraic Curves (Cambridge University a system of linear equations in the indeterminates Press, Cambridge, 1992), by F. Kirwan, also start at an easily accessible level, but then delve more quickly into aj,i1,...,in . The solvability of systems of linear equations is well-known (with good computer implementations). advanced subjects. Thus we can decide if there is a solution with deg gj The best introduction to the techniques of algebraic 100. Of course it is possible that 100 was too small geometry is Undergraduate Algebraic Geometry (Cam- a guess, and we may have to repeat the process with bridge University Press, Cambridge, 1988), by M. Reid. larger and larger degree bounds. Will this ever end? For those wishing for a general overview, An Invitation The answer is given by the following result, which was to Algebraic Geometry (Springer, New York, 2000), by proved only recently. K. E. Smith, L. Kahanpää, P. Kekäläinen, and W. Traves, is a good choice, while Algebraic Geometry (Springer, New Effective Nullstellensatz. Let f1,...,fm be polyno- York, 1995), by J. Harris, and Basic Algebraic Geometry, mials of degree less than or equal to d in n variables, volumes I and II (Springer, New York, 1994), by I. -
Moduli of Elliptic Curves Via Twisted Stable Maps by Andrew Norton Niles a Dissertation Submitted in Partial Satisfaction Of
Moduli of Elliptic Curves via Twisted Stable Maps by Andrew Norton Niles A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Martin Olsson, Chair Professor Arthur Ogus Professor Mary Gaillard Spring 2014 Moduli of Elliptic Curves via Twisted Stable Maps Copyright 2014 by Andrew Norton Niles Includes substantial material appearing in the author's article Moduli of elliptic curves via twisted stable maps, first published in 2013 in Algebra & Number Theory Vol. 7 No. 9, published by Mathematical Sciences Publishers. The copyright for this article is owned solely by MSP, and the contents of this article are reproduced here with the permission of MSP. 1 Abstract Moduli of Elliptic Curves via Twisted Stable Maps by Andrew Norton Niles Doctor of Philosophy in Mathematics University of California, Berkeley Professor Martin Olsson, Chair Abramovich, Corti and Vistoli have studied modular compactifications of stacks of curves equipped with abelian level structures arising as substacks of the stack of twisted stable maps into the classifying stack of a finite group, provided the order of the group is invertible on the base scheme. Recently Abramovich, Olsson and Vistoli extended the notion of twisted stable maps to allow arbitrary base schemes, where the target is a tame stack, not necessarily Deligne-Mumford. In this dissertation, we use this extended notion of twisted stable maps to extend the results of Abramovich, Corti and Vistoli to the case of elliptic curves with level structures over arbitrary base schemes. -
Self-Points on Elliptic Curves
Self-points on elliptic curves Christian Wuthrich March 17, 2008 Abstract Let E/Q be an elliptic curve of conductor N. We consider trace-compatible towers of modular points in the non-commutative division tower Q(E[p∞]). Under weak assumption we can prove that all these points are of infinite order. Furthermore, we use Kolyvagin’s construction of derivate classes to find explicit elements in certain Tate-Shafarevich groups. Mathematics Subject Classification 2000: 11G05, 11G18, 11G40 1 Introduction 1.1 Definition of self-points Let E/Q be an elliptic curve. Write N for its conductor. As proved in [BCDT01], there exists a modular parametrisation ϕE : X (N) E 0 ≻ which is a surjective morphism defined over Q mapping the cusp on the modular curve X (N) ∞ 0 to O. The open subvariety Y0(N) in X0(N) is a moduli space for the set of couples (A, C) where A is an elliptic curve and C is a cyclic subgroup in A of order N. More precisely, if k/Q is a field, then Y0(N)(k) is in bijection with the set of such couples (A, C) with A and C defined over k, up to isomorphism over the algebraic closure k¯. In particular, we may consider the couple xC = (E, C) for any given cyclic subgroup C of order N in E as a point in Y0(N)(C). Its image PC = ϕE (xC ) under the modular parametrisation is called a self-point of E. The field of definition of the point PC on E is the same as the field of definition Q(C) of C. -
Arxiv:1803.02069V1 [Math.NT] 6 Mar 2018 Ayelpi Uvsdfie Yeutoso H Form the of Equations by Defined Curves Elliptic Many Be Have Respectively
SEQUENCES OF CONSECUTIVE SQUARES ON QUARTIC ELLIPTIC CURVES MOHAMED KAMEL AND MOHAMMAD SADEK Abstract. Let C : y2 = ax4 + bx2 + c, be an elliptic curve defined over Q. A set of rational points (xi,yi) ∈ C(Q), i = 1, 2, ··· , is said to be a sequence 2 of consecutive squares if xi = (u + i) , i = 1, 2, ··· , for some u ∈ Q. Using ideas of Mestre, we construct infinitely many elliptic curves C with sequences of consecutive squares of length at least 6. It turns out that these 6 rational points are independent. We then strengthen this result by proving that for a fixed 6- term sequence of consecutive squares, there are infinitely many elliptic curves C with the latter sequence forming the x-coordinates of six rational points in C(Q). 1. Introduction In [3], Bremner started discussing the existence of long sequences on elliptic curves. He produced an infinite family of elliptic curves with arithmetic progres- sion sequences of length 8. Several authors displayed infinite families of elliptic curves with long arithmetic progression sequences, see [5, 7, 10]. A geometric progression sequence is another type of sequence that has been studied on elliptic and hyperelliptic curves. Infinitely many (hyper)elliptic curves with 5-term and 8-term geometric progression sequences have been introduced in [4] and [1] respectively. arXiv:1803.02069v1 [math.NT] 6 Mar 2018 In [6], sequences of consecutive squares on elliptic curves were studied. Infinitely many elliptic curves defined by equations of the form E : y2 = ax3 + bx + c, a, b, c ∈ Q, with 5-term sequences of consecutive squares were presented.