TROPICAL BRILL-NOETHER THEORY
|
|
- Claud Miller
- 5 years ago
- Views:
Transcription
1 TROPICAL BRILL-NOETHER THEORY 10. Clifford s Theorem In this section we consider natural relations between the degree and rank of a divisor on a metric graph. Our primary reference is Yoav Len s Hyperelliptic graphs on metrized complexes (2016). See also Melody Chan s Tropical hyperelliptic curves (2013), Marc Coppens s Clifford s theorem for graphs (2016), and Kawaguchi and Yamaki s Rank of divisors on hyperelliptic curves and graphs under specialization (2015). By Riemann-Roch, the rank of a divisor D is bounded below by min{ 1, deg(d) g}, with equality if deg(d) < 0 or deg(d) > 2g 2. On the other hand, the rank of a divisor is bounded above by its degree. The first part of Clifford s Theorem provides a stronger upper bound on the rank of a divisor in terms of its degree. Proposition Let Γ be a metric graph of genus g, and let D be a divisor on Γ satisfying 0 < deg(d) < 2g 2. Then deg(d) 2r(D). Proof. When D has negative rank, the result is obvious, so we assume throughout that r(d) 0. First, consider the case where r(k D) = 1. Since deg(d) < 2g, by Riemann-Roch we have r(d) = deg(d) g < 1 2 deg(d). Now, consider the case where r(k D) 0. Let E be an effective divisor of degree r(d)+r(k D) and let E and E be effective divisors of degree r(d) and r(k D), respectively, such that E = E +E. Note that D E and K D E are equivalent to effective divisors, so K E is equivalent to an effective divisor. It follows that By Riemann-Roch, we have Adding these, we obtain r(d) + r(k D) r(k) = g 1. r(d) r(k D) = deg(d) g r(D) deg(d). Classically, Clifford s Theorem for algebraic curves consists of two parts. The first part is the inequality above, while the second part states that equality is obtained if and only if the curve is hyperelliptic. This second statement is less straightforward in the case of metric graphs, and will be our primary focus for the remainder of this section. Definition A metric graph Γ is hyperelliptic if it has a divisor of degree 2 and rank 1. Date: March 6, 2016, Speaker: Nicholas Wawrykow, Scribe: Netanel Friedenberg. 1
2 2 TROPICAL BRILL-NOETHER THEORY Figure 1. A divisor of degree 2 and rank 1 on a metric graph Example Consider the graph pictured in Figure 1 with three edges of arbitrary length. Let D be the divisor of degree 2 with one chip on each of the trivalent vertices. It is easy to see that after a chip is subtracted at any point on the graph, the resulting divisor is still equivalent to an effective divisor. Thus, the divisor D has rank 1, and the metric graph is hyperelliptic. Before we proceed to Clifford s theorem for metric graphs, we note the following simple fact about hyperelliptic graphs. Proposition Let Γ be a hyperelliptic metric graph of genus g > 1. If D and D are divisors of degree 2 and rank 1 on Γ, then D and D are equivalent. Proof. Since rank is a superadditive function for divisors of nonnegative rank, r((g 1)D) g 1 and r((g 2)D + D ) g 1. By Riemann-Roch, it follows that (g 1)D (g 2)D + D K, hence D D. Our goal is to prove the following theorem. Theorem Let Γ be a metric graph of genus g. If there exists a divisor D of rank r on Γ such that 0 < r < g 1 and deg(d) = 2r, then Γ is hyperelliptic. By the classification of special divisors on a hyperelliptic graph, the class of any such D must be equal to r times the class of the unique g 1 2. See the paper of Kawaguchi and Yamaki (2015). The proof will use geometric properties of the Jacobian Jac(Γ) and the theta divisor W g 1 (Γ). Interestingly, however, in the case where g 1 is prime, we can provide a purely combinatorial proof. Proof of Theorem 10.5 in the case that g 1 is prime. We prove this by induction on r, the case r = 1 being obvious. If r 1, then since g 1 is prime, there is an integer n such that nr < g 1 < (n + 1)r. Note that r(nd) nr, and since deg(nd) = 2nr, by Proposition 10.1 we see that r(nd) = nr. Let r = g 1 nr. By Riemann-Roch, we then have r(k nd) = r and deg(k nd) = 2r. Since r < r, the result follows by induction.
3 TROPICAL BRILL-NOETHER THEORY 3 Definition Let Γ be a metric graph, and let D 1 and D 2 be divisors on Γ. We say that D 1 contains D 2 if D 1 D 2 is an effective divisor. Definition For a metric graph Γ we define Wd r (Γ) to be the polyhedral subset of Pic d (Γ) of equivalence classes of divisors of degree d and rank at least r. For the sake of notational convenience we write W d (Γ) for Wd 0(Γ). Definition A divisor D on a metric graph Γ is said to be rigid if it is the unique effective divisor in its class. We say that a divisor class [D] is rigid if it contains a unique effective representative. Remark Every rigid divisor has rank 0. On an algebraic curve, a divisor is rigid if and only if it has rank 0, but on a metric graph this is not the case. The following lemma tells us that for a metric graph Γ of genus g, the set of rigid divisors of degree g 1 is open and dense in W g 1 (Γ). Lemma Let Γ be a metric graph of genus g > 0. Then there exists a divisor P of degree g 1 such that both [P ] and [K P ] are rigid. Moreover, the set of such divisors is open and dense in W g 1 (Γ). Proof. Note that W g 1 (Γ) is a connected polyhedral subset of pure dimension g 1. Let B Γ be the set of points of valence different from 2. We show that, if D W g 1 (Γ) is not rigid, then Supp D B. To see this, recall that (Supp D ) c is a disjoint union of YL sets. Let U be a YL set and X a connected component of Γ U. If X B =, then by definition there exists v X such that outdeg X (v) = val(v) = 2, hence X = {v} is an isolated point. It follows that, if Supp D B =, then Supp D is a finite union of isolated points, so D is rigid. In other words, the set of non-rigid divisors is contained in the image of B Sym g 2 Γ under the Abel-Jacobi map. Since B is finite, it follows that this set has dimension at most g 2, hence its complement is dense in W g 1 (Γ). The same argument shows that the set of divisors P W g 1 (Γ) such that [K P ] is rigid is open and dense. Since any two open dense sets must intersect, the claim follows. We will use σ to denote the open subset of W g 1 (Γ) consisting of divisor classes [P ] such that both [P ] and [K P ] are rigid. Let µ : σ σ denote the map that sends [P ] in σ to [K P ] in σ. From now on [P ] will denote an element of σ, [Q] will denote [K P ], P will denote the rigid divisor in [P ], and Q will denote the rigid divisor in [Q]. We now define the intersection and union of two divisors. Definition For divisors D and D let (D D )(v) = min(d(v), D (v)) and (D D )(v) = max(d(v), D (v)). For a divisor D, define its P -part as D P = D P and its Q-part as D Q = D Q. For the rest of the paper let Γ be a metric graph of genus g. We assume that there exists a divisor class δ of degree 2r and rank r where 0 < r < g 1. We aim to show that Γ is hyperelliptic.
4 4 TROPICAL BRILL-NOETHER THEORY Lemma Let D δ such that deg(d P ) = r. Then D is supported on P + Q. In particular, D = D P + D Q. Proof. By Riemann-Roch we have r(k D) = g 1 r. Hence there exists a divisor E K D such that E (P D P ) is effective. We wish to show that D + E P = Q. We may write D + E P = (D D P ) + (E P + D P ). By the definition of the P -part, D contains D P, and by the definition of E we know that E contains P D P. Thus, D + E P is effective. Because D + E P Q is rigid, we see that in fact D + E P = Q. Note that, in Lemma 10.12, the Q-part D Q is determined uniquely by the P -part D P. Indeed, if D P + D Q D P + D Q, then D Q D Q implies D Q = D Q by the rigidity of Q. Lemma therefore provides us with a correspondence between divisors of degree r supported on P and divisors of degree r supported on Q. We want to know that this correspondence is respected by the intersection of two or more divisors. To see this, we will consider how the correspondence changes as we deform the divisor P. Proposition Let D 1,..., D n be effective representatives of δ such that deg(di P ) = r for all i. Then n deg( D Q i ) = deg( Di P ). Proof. Let σ i be the subset of σ that consists of rigid divisors classes whose rigid representatives contain P Di P. Similarly, let τ i be the subset of σ that consists of rigid divisor classes whose rigid representatives contain Q D Q i. Note that dim( σ i ) = deg( Di P ), and similarly for the τ i. It therefore suffices to show that dim( τ i ) = dim( σ i ). In order to see this, we first prove that µ(σ i ) τ i for all i. Let P be a rigid divisor of degree g 1 that contains P Di P. That is, P is an element of σ i. Set E i = P P + Di P. Since P contains P Di P, E i is effective, and since deg(e i ) = deg(p ) deg(p ) + deg(di P ), deg(e i) = r. Since δ has degree 2r and rank r, there exists a divisor D i δ such that D i contains E i. By Lemma 10.12, D i is contained in P + µ(p ), so P + µ(p ) + D i D i is effective and equivalent to K. By looking at the definition of µ, we note that P + µ(p ) [K]. Moreover, since P + µ(p ) + D i D i contains P and both P and µ(p ) are rigid, this divisor must be equal to P + µ(p ). Thus, µ(p ) contains Q D Q i, and is therefore contained in τ i. It follows that µ( σ i ) τ i.
5 TROPICAL BRILL-NOETHER THEORY 5 Since µ is a bijection, we may argue similarly that µ( τ i ) σ i. Hence dim( τ i ) = dim( σ i ). Now we are in a position to prove Clifford s theorem for metric graphs. Proof of Theorem Choose [P ] σ, and let P be its rigid representative. For each point p i SuppP, let S i = {D δ deg(d P ) = r, p i D}. By Lemma 10.12, we can decompose D S i as D = D P + D Q, where Q is the rigid representative of [Q] = [K P ]. Let φ be the map that assigns a rigid subdivisor A of P of degree r to the unique rigid subdivisor B of Q of degree r such that A + B δ. By Proposition 10.13, D Si D Q consists of a single point q i. Thus, a divisor D δ with P -part of degree r contains p i if and only if it contains q i. We now show that the divisors p i + q i are equivalent for all i. Let p i p j. Choose D i, D j δ such that D i S i S j, D j S j S i, and deg(di P Dj P ) = r 1. By Proposition 10.13, deg(d Q i D Q j ) = r 1, and by Lemma 10.12, D i and D j are supported on P + Q. Thus, deg(d i D j ) = deg(di P DP j ) + deg(dq i DQ j ) = 2r 2. Since D i and D j are equivalent, D i does not contain p j +q j, and D j does not contain p i + q i, we have the following equivalence: 0 D i D j = q i + p i q j p j. Therefore, p i + q i p j + q j. To show that p 1 + q 1 has rank 1, we need to extend SuppP = {p 1,..., p g 1 } to a rank determining set. Choose a point p g such that Γ {p 1,..., p g } is a tree, and such that P = P p 1 + p g σ. Note that the maps sending D P to D Q and D P to D Q coincide for any divisor D whose P -part D P is contained in P P. It follows that, for any p i P P, the associated point q i does not depend on which of the two sets P, P we use to define the bijection. Thus p 1 + q 1 p i + q i p g + q g. Since p 1 + q 1 moves, Supp(p 1 + q 1 ) contains a point p g+1 / {p 1,..., p g }. Since {p 1,..., p g+1 } is a rank determining set contained in Supp p 1 + q 1, we see that p 1 + q 1 must have rank at least 1. References [Cha13] M. Chan. Tropical hyperelliptic curves. J. Algebraic Combin., 37(2): , [Cop16] M. Coppens. Clifford s theorem for graphs. Adv. Geom., 16(3): , [KY15] S. Kawaguchi and K. Yamaki. Rank of divisors on hyperelliptic curves and graphs under specialization. Int. Math. Res. Not. IMRN, (12): , [Len16] Y. Len. Hyperelliptic graphs and metrized complexes. preprint arxiv: , 2016.
Free divisors on metric graphs
Free divisors on metric graphs Marc Coppens Abstract On a metric graph we introduce the notion of a free divisor as a replacement for the notion of a base point free complete linear system on a curve.
More informationMATH 665: TROPICAL BRILL-NOETHER THEORY
MATH 665: TROPICAL BRILL-NOETHER THEORY 2. Reduced divisors The main topic for today is the theory of v-reduced divisors, which are canonical representatives of divisor classes on graphs, depending only
More informationTROPICAL BRILL-NOETHER THEORY
TROPICAL BRILL-NOETHER THEORY 5. Special divisors on a chain of loops For this lecture, we will study special divisors a generic chain of loops. specifically, when g, r, d are nonnegative numbers satisfying
More informationA RIEMANN-ROCH THEOREM IN TROPICAL GEOMETRY
A RIEMANN-ROCH THEOREM IN TROPICAL GEOMETRY ANDREAS GATHMANN AND MICHAEL KERBER ABSTRACT. Recently, Baker and Norine have proven a Riemann-Roch theorem for finite graphs. We extend their results to metric
More informationA RIEMANN-ROCH THEOREM FOR EDGE-WEIGHTED GRAPHS
A RIEMANN-ROCH THEOREM FOR EDGE-WEIGHTED GRAPHS RODNEY JAMES AND RICK MIRANDA Contents 1. Introduction 1 2. Change of Rings 3 3. Reduction to Q-graphs 5 4. Scaling 6 5. Reduction to Z-graphs 8 References
More informationarxiv: v1 [math.ag] 15 Apr 2014
LIFTING DIVISORS ON A GENERIC CHAIN OF LOOPS DUSTIN CARTWRIGHT, DAVID JENSEN, AND SAM PAYNE arxiv:1404.4001v1 [math.ag] 15 Apr 2014 Abstract. Let C be a curve over a complete valued field with infinite
More informationCombinatorial and inductive methods for the tropical maximal rank conjecture
Journal of Combinatorial Theory, Series A 152 (2017) 138 158 Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series A www.elsevier.com/locate/jcta Combinatorial and inductive
More informationDIVISOR THEORY ON TROPICAL AND LOG SMOOTH CURVES
DIVISOR THEORY ON TROPICAL AND LOG SMOOTH CURVES MATTIA TALPO Abstract. Tropical geometry is a relatively new branch of algebraic geometry, that aims to prove facts about algebraic varieties by studying
More informationarxiv: v1 [math.co] 19 Oct 2018
GRAPHS OF GONALITY THREE IVAN AIDUN, FRANCES DEAN, RALPH MORRISON, TERESA YU, JULIE YUAN arxiv:1810.08665v1 [math.co] 19 Oct 2018 Abstract. In 2013, Chan classified all metric hyperelliptic graphs, proving
More informationWhat is a Weierstrass Point?
What is a Weierstrass Point? The Boys July 12, 2017 Abstract On a tropical curve (a metric graph with unbounded edges), one may introduce the so-called chip-firing game. Given a configuration D of chips
More informationThe Riemann Roch theorem for metric graphs
The Riemann Roch theorem for metric graphs R. van Dobben de Bruyn 1 Preface These are the notes of a talk I gave at the graduate student algebraic geometry seminar at Columbia University. I present a short
More informationGeometry of the theta divisor of a compactified jacobian
J. Eur. Math. Soc. 11, 1385 1427 c European Mathematical Society 2009 Lucia Caporaso Geometry of the theta divisor of a compactified jacobian Received October 16, 2007 and in revised form February 21,
More informationOn Weierstrass semigroups arising from finite graphs
On Weierstrass semigroups arising from finite graphs Justin D. Peachey Department of Mathematics Davidson College October 3, 2013 Finite graphs Definition Let {P 1, P 2,..., P n } be the set of vertices
More informationarxiv:math/ v4 [math.nt] 5 Jul 2007
SPECIALIZATION OF LINEAR SYSTEMS FROM CURVES TO GRAPHS MATTHEW BAKER arxiv:math/0701075v4 [math.nt] 5 Jul 2007 Abstract. We investigate the interplay between linear systems on curves and graphs in the
More informationAN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES
AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,
More informationarxiv: v3 [math.co] 6 Aug 2016
Computing Linear Systems on Metric Graphs arxiv:1603.00547v3 [math.co] 6 Aug 2016 Bo Lin Abstract The linear system D of a divisor D on a metric graph has the structure of a cell complex. We introduce
More informationRIEMANN SURFACES. max(0, deg x f)x.
RIEMANN SURFACES 10. Weeks 11 12: Riemann-Roch theorem and applications 10.1. Divisors. The notion of a divisor looks very simple. Let X be a compact Riemann surface. A divisor is an expression a x x x
More informationGeometry of the Theta Divisor of a compactified Jacobian
Geometry of the Theta Divisor of a compactified Jacobian Lucia Caporaso 1 Abstract. The object of this paper is the theta divisor of the compactified Jacobian of a nodal curve. We determine its irreducible
More informationarxiv: v4 [math.co] 14 Apr 2017
RIEMANN-ROCH THEORY FOR GRAPH ORIENTATIONS SPENCER BACKMAN arxiv:1401.3309v4 [math.co] 14 Apr 2017 Abstract. We develop a new framework for investigating linear equivalence of divisors on graphs using
More informationTROPICAL BRILL-NOETHER THEORY
TROPICAL BRILL-NOETHER THEORY 11. Berkovich Analytification and Skeletons of Curves We discuss the Berkovich analytification of an algebraic curve and its skeletons, which have the structure of metric
More informationarxiv: v1 [math.co] 25 Jun 2014
THE NON-PURE VERSION OF THE SIMPLEX AND THE BOUNDARY OF THE SIMPLEX NICOLÁS A. CAPITELLI arxiv:1406.6434v1 [math.co] 25 Jun 2014 Abstract. We introduce the non-pure versions of simplicial balls and spheres
More informationTROPICALIZATION OF THETA CHARACTERISTICS, DOUBLE COVERS, AND PRYM VARIETIES
TROPICALIZATION OF THETA CHARACTERISTICS, DOUBLE COVERS, AND PRYM VARIETIES DAVID JENSEN AND YOAV LEN Abstract. We study the behavior of theta characteristics on an algebraic curve under the specialization
More informationTORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS
TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS YUSUKE SUYAMA Abstract. We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be
More informationINDRANIL BISWAS AND GEORG HEIN
GENERALIZATION OF A CRITERION FOR SEMISTABLE VECTOR BUNDLES INDRANIL BISWAS AND GEORG HEIN Abstract. It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed
More informationTROPICAL SCHEME THEORY
TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationRational points on curves and tropical geometry.
Rational points on curves and tropical geometry. David Zureick-Brown (Emory University) Eric Katz (Waterloo University) Slides available at http://www.mathcs.emory.edu/~dzb/slides/ Specialization of Linear
More informationMULTIPLICITIES OF MONOMIAL IDEALS
MULTIPLICITIES OF MONOMIAL IDEALS JÜRGEN HERZOG AND HEMA SRINIVASAN Introduction Let S = K[x 1 x n ] be a polynomial ring over a field K with standard grading, I S a graded ideal. The multiplicity of S/I
More informationConstructions of digital nets using global function fields
ACTA ARITHMETICA 105.3 (2002) Constructions of digital nets using global function fields by Harald Niederreiter (Singapore) and Ferruh Özbudak (Ankara) 1. Introduction. The theory of (t, m, s)-nets and
More informationMath 213br HW 12 solutions
Math 213br HW 12 solutions May 5 2014 Throughout X is a compact Riemann surface. Problem 1 Consider the Fermat quartic defined by X 4 + Y 4 + Z 4 = 0. It can be built from 12 regular Euclidean octagons
More informationSome consequences of the Riemann-Roch theorem
Some consequences of the Riemann-Roch theorem Proposition Let g 0 Z and W 0 D F be such that for all A D F, dim A = deg A + 1 g 0 + dim(w 0 A). Then g 0 = g and W 0 is a canonical divisor. Proof We have
More informationBRILL-NOETHER THEORY FOR CURVES OF FIXED GONALITY (AFTER JENSEN AND RANGANATHAN)
BRILL-NOETHER THEORY FOR CURVES OF FIXED GONALITY (AFTER JENSEN AND RANGANATHAN) SAM PAYNE Abstract. This talk, presented at the 2017 Simon Symposium on Nonarchimedean and Tropical Geometry, reports on
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 As usual, a curve is a smooth projective (geometrically irreducible) variety of dimension one and k is a perfect field. 23.1
More information2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d
ON THE BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Abstract. On an arbitrary compact Riemann surface, necessary and sucient conditions are found for the
More informationMODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS
MODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS MIHNEA POPA 1. Lecture II: Moduli spaces and generalized theta divisors 1.1. The moduli space. Back to the boundedness problem: we want
More informationRIEMANN-ROCH AND ABEL-JACOBI THEORY ON A FINITE GRAPH
RIEMANN-ROCH AND ABEL-JACOBI THEORY ON A FINITE GRAPH MATTHEW BAKER AND SERGUEI NORINE Abstract. It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann
More informationarxiv: v1 [math.co] 17 Feb 2016
ON METRIC GRAPHS WITH PRESCRIBED GONALITY FILIP COOLS AND JAN DRAISMA arxiv:160.0554v1 [math.co] 17 Feb 016 Abstract. We prove that in the moduli space of genus-g metric graphs the locus of graphs with
More informationExistence of coherent systems of rank two and dimension four
Collect. Math. 58, 2 (2007), 193 198 c 2007 Universitat de Barcelona Existence of coherent systems of rank two and dimension four Montserrat Teixidor i Bigas Mathematics Department, Tufts University, Medford
More informationRiemann s goal was to classify all complex holomorphic functions of one variable.
Math 8320 Spring 2004, Riemann s view of plane curves Riemann s goal was to classify all complex holomorphic functions of one variable. 1) The fundamental equivalence relation on power series: Consider
More informationThe Maximal Rank Conjecture
The Maximal Ran Conjecture Eric Larson Abstract Let C be a general curve of genus g, embedded in P r via a general linear series of degree d. In this paper, we prove the Maximal Ran Conjecture, which determines
More informationBRILL-NOETHER THEORY. This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve.
BRILL-NOETHER THEORY TONY FENG This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve." 1. INTRODUCTION Brill-Noether theory is concerned
More informationRIEMANN-ROCH THEORY FOR WEIGHTED GRAPHS AND TROPICAL CURVES
RIEMANN-ROCH THEORY FOR WEIGHTED GRAPHS AND TROPICAL CURVES OMID AMINI AND LUCIA CAPORASO Abstract. We define a divisor theory for graphs and tropical curves endowed with a weight function on the vertices;
More informationThe cocycle lattice of binary matroids
Published in: Europ. J. Comb. 14 (1993), 241 250. The cocycle lattice of binary matroids László Lovász Eötvös University, Budapest, Hungary, H-1088 Princeton University, Princeton, NJ 08544 Ákos Seress*
More informationA Note on Jacobians, Tutte Polynomials, and Two-Variable Zeta Functions of Graphs
Experimental Mathematics, 24:1 7,2015 Copyright C Taylor & Francis Group, LLC ISSN: 1058-6458 print / 1944-950X online DOI: 10.1080/10586458.2014.917443 A Note on Jacobians, Tutte Polynomials, and Two-Variable
More informationMaximal Independent Sets In Graphs With At Most r Cycles
Maximal Independent Sets In Graphs With At Most r Cycles Goh Chee Ying Department of Mathematics National University of Singapore Singapore goh chee ying@moe.edu.sg Koh Khee Meng Department of Mathematics
More informationMath 730 Homework 6. Austin Mohr. October 14, 2009
Math 730 Homework 6 Austin Mohr October 14, 2009 1 Problem 3A2 Proposition 1.1. If A X, then the family τ of all subsets of X which contain A, together with the empty set φ, is a topology on X. Proof.
More informationA RIEMANN-ROCH THEOREM FOR EDGE-WEIGHTED GRAPHS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 11, November 2013, Pages 3793 3802 S 0002-9939(2013)11671-0 Article electronically published on July 26, 2013 A RIEMANN-ROCH THEOREM
More informationChapter 4. Measure Theory. 1. Measure Spaces
Chapter 4. Measure Theory 1. Measure Spaces Let X be a nonempty set. A collection S of subsets of X is said to be an algebra on X if S has the following properties: 1. X S; 2. if A S, then A c S; 3. if
More informationNOTES ON DIVISORS AND RIEMANN-ROCH
NOTES ON DIVISORS AND RIEMANN-ROCH NILAY KUMAR Recall that due to the maximum principle, there are no nonconstant holomorphic functions on a compact complex manifold. The next best objects to study, as
More informationD-bounded Distance-Regular Graphs
D-bounded Distance-Regular Graphs CHIH-WEN WENG 53706 Abstract Let Γ = (X, R) denote a distance-regular graph with diameter D 3 and distance function δ. A (vertex) subgraph X is said to be weak-geodetically
More information12. Hilbert Polynomials and Bézout s Theorem
12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of
More informationOn a theorem of Ziv Ran
INSTITUTUL DE MATEMATICA SIMION STOILOW AL ACADEMIEI ROMANE PREPRINT SERIES OF THE INSTITUTE OF MATHEMATICS OF THE ROMANIAN ACADEMY ISSN 0250 3638 On a theorem of Ziv Ran by Cristian Anghel and Nicolae
More informationarxiv: v2 [math.ag] 25 Nov 2014
ALGEBRAIC AND COMBINATORIAL RANK OF DIVISORS ON FINITE GRAPHS LUCIA CAPORASO, YOAV LEN, AND MARGARIDA MELO arxiv:1401.5730v2 [math.ag] 25 Nov 2014 Abstract. We study the algebraic rank of a divisor on
More informationWe want to show P (n) is true for all integers
Generalized Induction Proof: Let P (n) be the proposition 1 + 2 + 2 2 + + 2 n = 2 n+1 1. We want to show P (n) is true for all integers n 0. Generalized Induction Example: Use generalized induction to
More informationPERIODIC POINTS OF THE FAMILY OF TENT MAPS
PERIODIC POINTS OF THE FAMILY OF TENT MAPS ROBERTO HASFURA-B. AND PHILLIP LYNCH 1. INTRODUCTION. Of interest in this article is the dynamical behavior of the one-parameter family of maps T (x) = (1/2 x
More informationMeasurable Choice Functions
(January 19, 2013) Measurable Choice Functions Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/choice functions.pdf] This note
More informationSome remarks on symmetric correspondences
Some remarks on symmetric correspondences H. Lange Mathematisches Institut Universitat Erlangen-Nurnberg Bismarckstr. 1 1 2 D-91054 Erlangen (Germany) E. Sernesi Dipartimento di Matematica Università Roma
More informationFast arithmetic and pairing evaluation on genus 2 curves
Fast arithmetic and pairing evaluation on genus 2 curves David Freeman University of California, Berkeley dfreeman@math.berkeley.edu November 6, 2005 Abstract We present two algorithms for fast arithmetic
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationChip Firing Games and Riemann-Roch Properties for Directed Graphs
Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2013 Chip Firing Games and Riemann-Roch Properties for Directed Graphs Joshua Z. Gaslowitz Harvey Mudd College Recommended
More informationMaximal non-commuting subsets of groups
Maximal non-commuting subsets of groups Umut Işık March 29, 2005 Abstract Given a finite group G, we consider the problem of finding the maximal size nc(g) of subsets of G that have the property that no
More informationPartial cubes: structures, characterizations, and constructions
Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov San Francisco State University, Mathematics Department, 1600 Holloway Ave., San Francisco, CA 94132 Abstract Partial cubes
More informationTheta divisors and the Frobenius morphism
Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following
More informationINVERSE LIMITS AND PROFINITE GROUPS
INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological
More informationComputing discrete logarithms with pencils
Computing discrete logarithms with pencils Claus Diem and Sebastian Kochinke Augst 25, 2017 1 Introduction Let us consider the discrete logarithm problem for curves of a fixed genus g at least 3. In [7]
More informationInterpolation with Bounded Error
Interpolation with Bounded Error Eric Larson arxiv:1711.01729v1 [math.ag] 6 Nov 2017 Abstract Given n general points p 1,p 2,...,p n P r it is natural to ask whether there is a curve of given degree d
More informationCURVES WITH MAXIMALLY COMPUTED CLIFFORD INDEX
CURVES WITH MAXIMALLY COMPUTED CLIFFORD INDEX TAKAO KATO AND GERRIET MARTENS Abstract. We say that a curve X of genus g has maximally computed Clifford index if the Clifford index c of X is, for c > 2,
More informationMath 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement.
Math 421, Homework #6 Solutions (1) Let E R n Show that (Ē) c = (E c ) o, i.e. the complement of the closure is the interior of the complement. 1 Proof. Before giving the proof we recall characterizations
More informationThe Rationality of Certain Moduli Spaces of Curves of Genus 3
The Rationality of Certain Moduli Spaces of Curves of Genus 3 Ingrid Bauer and Fabrizio Catanese Mathematisches Institut Universität Bayreuth, NW II D-95440 Bayreuth, Germany Ingrid.Bauer@uni-bayreuth.de,
More informationThus, X is connected by Problem 4. Case 3: X = (a, b]. This case is analogous to Case 2. Case 4: X = (a, b). Choose ε < b a
Solutions to Homework #6 1. Complete the proof of the backwards direction of Theorem 12.2 from class (which asserts the any interval in R is connected). Solution: Let X R be a closed interval. Case 1:
More informationIndeed, if we want m to be compatible with taking limits, it should be countably additive, meaning that ( )
Lebesgue Measure The idea of the Lebesgue integral is to first define a measure on subsets of R. That is, we wish to assign a number m(s to each subset S of R, representing the total length that S takes
More informationTHE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS
THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS AILANA FRASER AND JON WOLFSON Abstract. In this paper we study the topology of compact manifolds of positive isotropic
More informationThe Structure of the Jacobian Group of a Graph. A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College
The Structure of the Jacobian Group of a Graph A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College In Partial Fulfillment of the Requirements for the Degree Bachelor of
More informationCONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP
CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat
More informationWelsh s problem on the number of bases of matroids
Welsh s problem on the number of bases of matroids Edward S. T. Fan 1 and Tony W. H. Wong 2 1 Department of Mathematics, California Institute of Technology 2 Department of Mathematics, Kutztown University
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationThe Skorokhod reflection problem for functions with discontinuities (contractive case)
The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection
More informationON A THEOREM OF CAMPANA AND PĂUN
ON A THEOREM OF CAMPANA AND PĂUN CHRISTIAN SCHNELL Abstract. Let X be a smooth projective variety over the complex numbers, and X a reduced divisor with normal crossings. We present a slightly simplified
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationMath 6510 Homework 10
2.2 Problems 9 Problem. Compute the homology group of the following 2-complexes X: a) The quotient of S 2 obtained by identifying north and south poles to a point b) S 1 (S 1 S 1 ) c) The space obtained
More informationSegre classes of tautological bundles on Hilbert schemes of surfaces
Segre classes of tautological bundles on Hilbert schemes of surfaces Claire Voisin Abstract We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande
More informationA finite universal SAGBI basis for the kernel of a derivation. Osaka Journal of Mathematics. 41(4) P.759-P.792
Title Author(s) A finite universal SAGBI basis for the kernel of a derivation Kuroda, Shigeru Citation Osaka Journal of Mathematics. 4(4) P.759-P.792 Issue Date 2004-2 Text Version publisher URL https://doi.org/0.890/838
More informationPencils on real curves
Math. Nachr. 286, No. 8 9, 799 816 (2013) / DOI 10.1002/mana.201100196 Pencils on real curves Marc Coppens 1 and Johannes Huisman 2 1 Katholieke Hogeschool Kempen, Departement Industrieel Ingenieur en
More informationDISTINGUISHING PARTITIONS AND ASYMMETRIC UNIFORM HYPERGRAPHS
DISTINGUISHING PARTITIONS AND ASYMMETRIC UNIFORM HYPERGRAPHS M. N. ELLINGHAM AND JUSTIN Z. SCHROEDER In memory of Mike Albertson. Abstract. A distinguishing partition for an action of a group Γ on a set
More informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationHow to compute regulators using Arakelov intersection theory
How to compute regulators using Arakelov intersection theory Raymond van Bommel 25 October 2018 These are notes for a talk given in the SFB/TRR45 Kolloquium held in Mainz, Germany, in the autumn of 2018.
More informationA generalization of the Weierstrass semigroup
Journal of Pure and Applied Algebra 207 (2006) 243 260 www.elsevier.com/locate/jpaa A generalization of the Weierstrass semigroup Peter Beelen a,, Nesrin Tutaş b a Department of Mathematics, Danish Technical
More informationNAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key
NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)
More informationLINKED HOM SPACES BRIAN OSSERMAN
LINKED HOM SPACES BRIAN OSSERMAN Abstract. In this note, we describe a theory of linked Hom spaces which complements that of linked Grassmannians. Given two chains of vector bundles linked by maps in both
More informationReachability of recurrent positions in the chip-firing game
Egerváry Research Group on Combinatorial Optimization Technical reports TR-2015-10. Published by the Egerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More information1 Basic Combinatorics
1 Basic Combinatorics 1.1 Sets and sequences Sets. A set is an unordered collection of distinct objects. The objects are called elements of the set. We use braces to denote a set, for example, the set
More informationOn zero-sum partitions and anti-magic trees
Discrete Mathematics 09 (009) 010 014 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: wwwelseviercom/locate/disc On zero-sum partitions and anti-magic trees Gil Kaplan,
More informationCHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998
CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic
More informationKLEIN-FOUR COVERS OF THE PROJECTIVE LINE IN CHARACTERISTIC TWO
ALBANIAN JOURNAL OF MATHEMATICS Volume 1, Number 1, Pages 3 11 ISSN 1930-135(electronic version) KLEIN-FOUR COVERS OF THE PROJECTIVE LINE IN CHARACTERISTIC TWO DARREN GLASS (Communicated by T. Shaska)
More informationParameterizing orbits in flag varieties
Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about
More informationSets and Functions. (As we will see, in describing a set the order in which elements are listed is irrelevant).
Sets and Functions 1. The language of sets Informally, a set is any collection of objects. The objects may be mathematical objects such as numbers, functions and even sets, or letters or symbols of any
More informationPERVERSE SHEAVES ON A TRIANGULATED SPACE
PERVERSE SHEAVES ON A TRIANGULATED SPACE A. POLISHCHUK The goal of this note is to prove that the category of perverse sheaves constructible with respect to a triangulation is Koszul (i.e. equivalent to
More information