DOCSLIB.ORG

Adjunction Formula, Poincaré Residue and Holomorphic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Adjunction Formula, Poincaré Residue and Holomorphic Methods of Functional Analysis and Topology Vol. 24 (2018), no. 1, pp. 41{52 ADJUNCTION FORMULA, POINCARE´ RESIDUE AND HOLOMORPHIC DIFFERENTIALS ON RIEMANN SURFACES A. LESFARI Abstract. There is still a big gap between knowing that a Riemann surface of genus g has g holomorphic differential forms and being able to find them explicitly. The aim of this paper is to show how to construct holomorphic differential forms 1 on compact Riemann surfaces. As known, the dimension of the space H (D; C) of holomorphic differentials of a compact Riemann surface D is equal to its genus, 1 dim H (D; C) = g(D) = g. When the Riemann surface is concretely described, we show that one can usually present a basis of holomorphic differentials explicitly. We apply the method to the case of relatively complicated Riemann surfaces. 1. Adjunction formula and Poincare´ residue Let M be a compact complex manifold of dimension n and let D ⊂ M be a smooth hypersurface. Let ND be the normal bundle on D which can be defined as the quotient 0 TM jD 0 0 ND = 0 , where TM and TD are the tangent bundles on M and D respectively (The TD 0 sign is added to recall the adjective holomorphic). The dual of ND is called the conormal ∗ 0∗ bundle and we denote it by ND. This is a sub-bundle of the cotangent bundle TM jD; i.e., 0 0 the set of cotangent vectors on M vanishing on TD ⊂ TM jD. We suppose that D is locally defined by functions fα 2 O(Uα). Therefore, the line bundle [D] on M is defined by the fα transition functions gαβ = . Since fα ≡ 0 on D\Uα, dfα is a section of the conormal fβ ∗ bundle ND on D\Uα. Moreover, by assumption, D is smooth so dfα does not vanish on D\Uα. Moreover, we have on Uα \Uβ \D, that dfα = d(gαβfβ) = gαβdfβ, and since D is 1 ∗ smooth, dfα is non-zero on D. So this shows that the sections dfα 2 H (Uα \D; OD(ND)) ∗ determine a global section, which does not vanish on D, of the line bundle ND ⊗ [D]jD. Therefore, the latter is trivial (a bundle that has its holomorphic sections is trivial) and we have the adjunction formula ∗ (1) ND = [−D]jD: n Now let KM = Λ TM be the canonical bundle of M where n is the dimension of M. This is a line bundle contrary to tangent and cotangent bundles. Recall that the sections n of KM are the holomorphic n-forms, i.e., O(KM ) = ΩM . To determine the canonical bundle KD of D as a function of KM , one proceeds as follows: consider the exact sequence ∗ 0∗ 0∗ 0 −! ND −! TM jD −! TD −! 0: We deduce that 0∗ ∗ 0∗ n 0∗ n−1 0∗ ∗ ∗ TM jD = ND ⊕ TD ; Λ TM ' Λ TD ⊗ ND;KM jD = KD ⊗ ND; D 2010 Mathematics Subject Classification. 34M45, 34M05, 11G10, 14H40, 14K02, 14H70. Key words and phrases. Adjunction formula, Poincar´e residue, Riemann surfaces, holomorphic differentials. 41 42 A. LESFARI and taking account of formula (1), one obtains KD = KM jD ⊗ ND = KM jD ⊗ [D]jD = (KM ⊗ [D])jD: Therefore, we get the following. Proposition 1. Let D ⊂ M be a smooth hypersurface, N ∗D be the dual bundle of the normal bundle ND on D and KD (resp. KM ) the canonical bundle on D (respectively M). Then we have the following adjunction formulas: ∗ ND = [−D]jD;KD = (KM ⊗ [D])jD: n Example 1. Let Z0;:::;Zn be homogeneous coordinates in P (C) and Z1 Z2 Zn x1 = ; x2 = ; : : : ; xn = ; Z0 Z0 Z0 n the affine coordinates in P (C)nfZ0 = 0g. Let us define the divisor of the n-meromorphic form dx dx ! = 1 ^ · · · ^ n : x1 xn n To do this, consider on P (C)nfZn = 0g the following affine coordinates: Z0 Z1 Zn−1 u0 = ; u1 = ; : : : ; un−1 = : Zn Zn Zn We have 1 x1 xn−1 u0 = ; u1 = ; : : : ; un−1 = ; xn xn xn and 1 ! = n+1 du0 ^ du1 ^ · · · ^ dun−1: u0 n Hence the divisor of ! is (!) = −(n + 1)H1, where H1 is a hyperplane in P (C) and n KP (C) = [(!)] = [−(n + 1)H1]: n Let ! 2 ΩM (D) be a meromorphic n-form on D with at worst a simple pole along D. Let (z1; : : : ; zn) be complex local coordinates on an open subset of M and assume that D is defined locally by functions f(z1; : : : ; zn). Therefore, g(z1; : : : ; zn)dz1 ^ · · · ^ dzn ! = 2 KM ; f(z1; : : : ; zn) where g(z1; : : : ; zn) is a holomorphic function. Taking into account the second adjunction formula (Proposition 1), one can write ! in the form df ! = ^ ! 2 K ; f e M where !e is a holomorphic (n − 1)-form. Therefore, g(z1; : : : ; zn)dz1 ^ · · · ^ dzcj ^ · · · ^ dzn ! = (−1)j−1 ; e @f @zj @f where j is such that 6= 0. It is therefore possible to define an application (corre- @zj sponding to the second adjunction formula) by setting n n−1 PR :ΩM (D) −! ΩV ;! 7−! PR(!) = !ejf=0; and we call PR(!) the Poincar´eresidue of !. It is a useful formula as we shall see later that under certain conditions it allows explicitly to find holomorphic differentials on submanifolds. ADJUNCTION FORMULA, POINCARE´ RESIDUE AND HOLOMORPHIC DIFFERENTIALS ::: 43 From the exact sequence of sheaves n n PR n−1 0 −! ΩM −! ΩM (D) −! ΩD −! 0; we obtain the long exact cohomology sequence 0 n 0 n PR 0 n−1 0 −! H (M; ΩM ) −! H (M; ΩM (D)) −! H (D; ΩD ) 1 n 1 n −! H (M; ΩM ) −! H (M; ΩM (D)) −! · · · It should be noted that the kernel of the application of the Poincar´eresidue consists 1 n simply of an n-holomorphic form on M and this application is surjective if H (M; ΩM ) = 0. 2. How to construct holomorphic differential forms on some compact Riemann surfaces Let now M be an Abelian surface (a complex algebraic torus) and let D be a positive P divisor on M. Define L(D) = ff meromorphic onM :(f) ≥ −Dg, i.e., for D = kjDj a function f 2 L(D) has at worst a kj-fold pole along Dj: The divisor D is called ample N when a basis (f0; : : : ; fN ) of L(kD) embeds M smoothly into P (C) for some k via the map N M −! P (C); p 7−! [1 : f1(p): ::: : fN (p)]; then kD is called very ample. If we assume D to be very ample, then M is embedded into PN (C) using the functions of L(D) where N = dim L(D) − 1 and in particular D ⊆ PN (C), g = N + 2, (the geometric genus of D). n Letz _ = f(z), z 2 C , be a system of differential equations governing a flow X1 and let X2 be any other flow on M commuting with the first (X1 and X2 are linearly independent vector fields in a neighborhood of a point p in M). Let the vector field have Laurent solutions −1 (0) (1) (2) 2 (2) zj(t) = t zj + zj t + +zj t + ··· ; 1 ≤ j ≤ n whose coefficients are rational functions on an (n − 1)-dimensional affine variety. Con- sider the complex affine invariant manifold (thought of as lying in Cn) defined by the intersection of the constants of the motion of this system and let D be the pole solutions (2) restricted to this invariant manifold, i.e., the set of Laurent solution which remain confined to a fixed affine invariant manifold, related to specific values of the constants (generic constants). Consider a basis f0 = 1, f1,. ,fN of L(D); these functions will be polynomials in z1; : : : ; zn and if the Laurent series zj (see (2)) are substituted into these polynomials fj, they have the following property −1 (0) (1) (2) 2 (3) fj(z1(t); : : : ; zn(t)) = t fj + fj t + +fj t + ··· : (0) (0) Let W = ff1 ; : : : ; fN g be the span of the residues of the functions f0 = 1, f1,. ,fN in L(D), the residues being meromorphic functions on D. Let dt1, dt2 be the holomorphic 1-forms on M defined by dti(Xj) = δij. Giving the vector fields X1 and X2 above and associated times t1 and t2 defines on D the following differentials !i = dtijD, restrictions of dt1 and dt2 to D: (0) ! 1 f 1 ! = dt j = d j ;! = dt j = −f (0)X ! 1 1 D (0) (0) 2 2 D 2 1 fi ∆fi fi 44 A. LESFARI with 0 n 1 o n 1 o 1 X1 X2 fi fi ∆ = lim det @ n o n o A : t!0 fj fj X1 f X2 f i i Laurent solutions z(t;p;D) Taking into account the adjunction formula for the Poincar´eresidue applied to the 2-form ! = fjdt1 ^ dt2, with fj 2 L(D), we have dt1 ^ dt2 dt1 ! = −! R´es !jD = − ; 1 @ 1 fj @t f 2 j D dt2 = ; @ 1 @t f 1 j D dt2 = ; @ t1 2 (0) + ◦(t ) @t1 f 1 j D (0) = fj dt2 ; D (0) = fj !2; (0) @ where fj is the residue appearing in (3) and is the derivative according to the @tj (0) (0) vector field Xj. Similarly, using the second vector field X2, we find ! = fej !1, with fej the residue. Since M is an Abelian surface and D a positive divisor, we deduce from the Kodaira- 1 2 Nakano vanishing theorem [5, 11] that H (M; ΩM (D) = 0 and again from the above exact sequence of sheaves 2 2 PR 1 0 −! ΩM −! ΩM (D) −! ΩD −! 0; we get a long exact sequence 0 2 0 2 PR 0 1 1 2 0 −! H (M; ΩM ) −! H (M; ΩM (D)) −! H (D; ΩD) −! H (M; ΩM ) −! 0: Note that dt1 ^ dt2 is the only holomorphic 2-form on the Abelian surface M.
Load more

Recommended publications
  • 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 .
    [Show full text]
  • Counting Bitangents with Stable Maps David Ayalaa, Renzo Cavalierib,∗ Adepartment of Mathematics, Stanford University, 450 Serra Mall, Bldg
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Expo. Math. 24 (2006) 307–335 www.elsevier.de/exmath Counting bitangents with stable maps David Ayalaa, Renzo Cavalierib,∗ aDepartment of Mathematics, Stanford University, 450 Serra Mall, Bldg. 380, Stanford, CA 94305-2125, USA bDepartment of Mathematics, University of Utah, 155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, USA Received 9 May 2005; received in revised form 13 December 2005 Abstract This paper is an elementary introduction to the theory of moduli spaces of curves and maps. As an application to enumerative geometry, we show how to count the number of bitangent lines to a projective plane curve of degree d by doing intersection theory on moduli spaces. ᭧ 2006 Elsevier GmbH. All rights reserved. MSC 2000: 14N35; 14H10; 14C17 Keywords: Moduli spaces; Rational stable curves; Rational stable maps; Bitangent lines 1. Introduction 1.1. Philosophy and motivation The most apparent goal of this paper is to answer the following enumerative question: What is the number NB(d) of bitangent lines to a generic projective plane curve Z of degree d? This is a very classical question, that has been successfully solved with fairly elementary methods (see for example [1, p. 277]). Here, we propose to approach it from a very modern ∗ Corresponding author. E-mail addresses: [email protected] (D. Ayala), [email protected] (R. Cavalieri). 0723-0869/$ - see front matter ᭧ 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.exmath.2006.01.003 308 D.
    [Show full text]
  • Geometry of Algebraic Curves
    Geometry of Algebraic Curves Fall 2011 Course taught by Joe Harris Notes by Atanas Atanasov One Oxford Street, Cambridge, MA 02138 E-mail address: [email protected] Contents Lecture 1. September 2, 2011 6 Lecture 2. September 7, 2011 10 2.1. Riemann surfaces associated to a polynomial 10 2.2. The degree of KX and Riemann-Hurwitz 13 2.3. Maps into projective space 15 2.4. An amusing fact 16 Lecture 3. September 9, 2011 17 3.1. Embedding Riemann surfaces in projective space 17 3.2. Geometric Riemann-Roch 17 3.3. Adjunction 18 Lecture 4. September 12, 2011 21 4.1. A change of viewpoint 21 4.2. The Brill-Noether problem 21 Lecture 5. September 16, 2011 25 5.1. Remark on a homework problem 25 5.2. Abel's Theorem 25 5.3. Examples and applications 27 Lecture 6. September 21, 2011 30 6.1. The canonical divisor on a smooth plane curve 30 6.2. More general divisors on smooth plane curves 31 6.3. The canonical divisor on a nodal plane curve 32 6.4. More general divisors on nodal plane curves 33 Lecture 7. September 23, 2011 35 7.1. More on divisors 35 7.2. Riemann-Roch, finally 36 7.3. Fun applications 37 7.4. Sheaf cohomology 37 Lecture 8. September 28, 2011 40 8.1. Examples of low genus 40 8.2. Hyperelliptic curves 40 8.3. Low genus examples 42 Lecture 9. September 30, 2011 44 9.1. Automorphisms of genus 0 an 1 curves 44 9.2.
    [Show full text]
  • The Genus of a Riemann Surface in the Plane Let S ⊂ CP 2 Be a Plane
    The Genus of a Riemann Surface in the Plane 2 Let S ⊂ CP be a plane curve of degree δ. We prove here that: (δ − 1)(δ − 2) g = 2 We'll start with a version of the: n B´ezoutTheorem. If S ⊂ CP has degree δ and G is a homogeneous polynomial of degree d not vanishing identically on S (i.e. G 62 Id), then deg(div(G)) = dδ Proof. The intersection V (G) \ S is a finite set. We choose a linear form l that is non-zero on all the points of V (G) \ S, and then φ = G=ld is meromorphic when restricted to S, and div(G) − d · div(l) = div(φ) has degree zero. But div(l) has degree δ, by the definition of δ. 2 Now let S ⊂ CP be a planar embedding of a Riemann surface, and let F 2 C[x0; x1; x2]δ generate the ideal I of S and assume, changing coordinates if necessary, that (0 : 1 : 0) 62 S and that the line x2 = 0 is transverse to S, necessarily meeting S in δ points. Proof Sketch. Projecting from p = (0 : 1 : 0) gives a holomorphic map: 1 π : S ! CP 2 1 (as a map from CP to CP , the projection is defined at every point except 2 for p itself). In the open set U2 = f(x0 : x1 : x2) j x2 6= 0g = C with local coordinates x = x0=x2 and y = x1=x2, π is the projection onto the x-axis. 1 In these coordinates, the line at infinity is x2 = 0 and maps to 1 2 CP and does not contribute to the ramification of the map π.
    [Show full text]
  • The Adjunction Conjecture and Its Applications
    The Adjunction Conjecture and its applications Florin Ambro Abstract Adjunction formulas are fundamental tools in the classification theory of algebraic va- rieties. In this paper we discuss adjunction formulas for fiber spaces and embeddings, extending the known results along the lines of the Adjunction Conjecture, independently proposed by Y. Kawamata and V. V. Shokurov. As an application, we simplify Koll´ar’s proof for the Anghern and Siu’s quadratic bound in the Fujita’s Conjecture. We also connect adjunction and its precise inverse to the problem of building isolated log canonical singularities. Contents 1 The basics of log pairs 5 1.1 Prerequisites ................................... 5 1.2 Logvarietiesandpairs. .. .. .. .. .. .. .. .. 5 1.3 Singularities and log discrepancies . ....... 6 1.4 Logcanonicalcenters. .. .. .. .. .. .. .. .. 7 1.5 TheLCSlocus .................................. 8 2 Minimal log discrepancies 9 2.1 The lower semicontinuity of minimal log discrepancies . ........... 10 2.2 Preciseinverseofadjunction. ..... 11 3 Adjunction for fiber spaces 14 3.1 Thediscriminantofalogdivisor . 14 3.2 Base change for the divisorial push forward . ....... 16 3.3 Positivityofthemodulipart . 18 arXiv:math/9903060v3 [math.AG] 30 Mar 1999 4 Adjunction on log canonical centers 20 4.1 Thedifferent ................................... 20 4.2 Positivityofthemodulipart . 22 5 Building singularities 25 5.1 Afirstestimate.................................. 26 5.2 Aquadraticbound ................................ 27 5.3 Theconjecturedoptimalbound . 29 5.4 Global generation of adjoint line bundles . ........ 30 References 32 1 Introduction The rough classification of projective algebraic varieties attempts to divide them according to properties of their canonical class KX . Therefore, whenever two varieties are closely related, it is essential to find formulas comparing their canonical divisors. Such formulas are called adjunction formulas.
    [Show full text]
  • Blow-Ups Global Picture
    Synopsis for Thursday, October 18 Blow-ups Global picture 2 1 Consider the projection π : P → P given by (x0,x1,x2) → (x1,x2). This is well-defined except at the center of projection (1, 0, 0). If we ignore that point we can consider the graph Γ of the well-defined map P 2 \ {(1, 0, 0)} → P 1 2 1 2 1 inside P \ {(1, 0, 0)} × P . If(x0,x1,x2; y0,y1) is a point in P × P then clearly y0x2 − y1x1 = 0 for points in Γ, because that is just another way of saying that (x1,x2) = λ(y0,y1) for some λ 6= 0. But this polynomial equation makes sense for the whole product and its zeroes make up the closure Γ of the graph. Now there is a projection π : Γ → P 1 given by the projection onto the second factor, which is defined on the whole of Γ as it is defined on the whole of P 2 × P 1. But what is the relation of Γ to P 2? There is of course a projection of P 2 × P 1 to the first factor that can be restricted to Γ. What are its fibers? If (x0,x1,x2) 6= (1, 0, 0) then not both x1,x2 can be zero, thus there is a unique 1 point (y0,y1) ∈ P such that (x0,x1,x2; y0,y1) ∈ Γ. Thus the projection is 1 : 1. However over (1, 0, 0) there will be no restriction on y0,y1 so the fiber is the entire line P 1! Thus Γ is the same as P 2 except over the point P where it has been replaced by an entire line.
    [Show full text]
  • The Geometry of Smooth Quartics
    The Geometry of Smooth Quartics Jakub Witaszek Born 1st December 1990 in Pu lawy, Poland June 6, 2014 Master's Thesis Mathematics Advisor: Prof. Dr. Daniel Huybrechts Mathematical Institute Mathematisch-Naturwissenschaftliche Fakultat¨ der Rheinischen Friedrich-Wilhelms-Universitat¨ Bonn Contents 1 Preliminaries 8 1.1 Curves . .8 1.1.1 Singularities of plane curves . .8 1.1.2 Simultaneous resolutions of singularities . 11 1.1.3 Spaces of plane curves of fixed degree . 13 1.1.4 Deformations of germs of singularities . 14 1.2 Locally stable maps and their singularities . 15 1.2.1 Locally stable maps . 15 1.2.2 The normal-crossing condition and transversality . 16 1.3 Picard schemes . 17 1.4 The enumerative theory . 19 1.4.1 The double-point formula . 19 1.4.2 Polar loci . 20 1.4.3 Nodal rational curves on K3 surfaces . 21 1.5 Constant cycle curves . 22 3 1.6 Hypersurfaces in P .......................... 23 1.6.1 The Gauss map . 24 1.6.2 The second fundamental form . 25 3 2 The geometry of smooth quartics in P 27 2.1 Properties of Gauss maps . 27 2.1.1 A local description . 27 2.1.2 A classification of tangent curves . 29 2.2 The parabolic curve . 32 2.3 Bitangents, hyperflexes and the flecnodal curve . 34 2.3.1 Bitangents . 34 2.3.2 Global properties of the flecnodal curve . 35 2.3.3 The flecnodal curve in local coordinates . 36 2 The Geometry of Smooth Quartics 2.4 Gauss swallowtails . 39 2.5 The double-cover curve .
    [Show full text]
  • Introduction
    Introduction There will be no specific text in mind. Sources include Griffiths and Harris as well as Hartshorne. Others include Mumford's "Red Book" and "AG I- Complex Projective Varieties", Shafarevich's Basic AG, among other possibilities. We will be taught to the test: that is, we will be being prepared to take orals on AG. This means we will de-emphasize proofs and do lots of examples and applications. So the focus will be techniques for the first semester, and the second will be topics from modern research. 1. Techniques of Algebraic Geometry (a) Varieties and Sheaves on Varieties i. Cohomology, the basic tool for studying these (we will start this early) ii. Direct Images (and higher direct images) iii. Base Change iv. Derived Categories (b) Topology of Complex Algebraic Varieties i. DeRham Theorem ii. Hodge Theorem iii. K¨ahlerPackage iv. Spectral Sequences (Specifically Leray) v. Grothendieck-Riemann-Roch (c) Deformation Theory and Moduli Spaces (d) Toric Geometry (e) Curves, Jacobians, Abelian Varieties and Analytic Theory of Theta Functions (f) Elliptic Curves and Elliptic fibrations (g) Classification of Surfaces (h) Singularities, Blow-Up, Resolution of Singularities 2. Problems in AG (mostly second semester) (a) Torelli and Schottky Problems: The Torelli question is "Is the map that takes a curve to its Jacobian injective?" and the Schottky Prob- lem is "Which abelian varieties come from curves?" (b) Hodge Conjecture: "Given an algebraic variety, describe the subva- rieties via cohomology." (c) Class Field Theory/Geometric Langlands Program: (Curves, Vector Bundles, moduli, etc) Complicated to state the conjectures. (d) Classification in Dim ≥ 3/Mori Program (We will not be talking much about this) 1 (e) Classification and Study of Calabi-Yau Manifolds (f) Lots of problems on moduli spaces i.
    [Show full text]
  • COMPLEX GEOMETRY NOTES 1. Characteristic Numbers Of
    COMPLEX GEOMETRY NOTES JEFF A. VIACLOVSKY 1. Characteristic numbers of hypersurfaces Let V ⊂ Pn be a smooth complex hypersurface. We know that the line bundle [V ] = O(d) for some d ≥ 1. We have the exact sequence (1,0) (1,0) 2 (1.1) 0 → T (V ) → T P V → NV → 0. The adjunction formula says that (1.2) NV = O(d) V . We have the smooth splitting of (1.1), (1,0) n (1,0) (1.3) T P V = T (V ) ⊕ O(d) V . Taking Chern classes, (1,0) n (1,0) (1.4) c(T P V ) = c(T (V )) · c(O(d) V ). From the Euler sequence [GH78, page 409], ⊕(n+1) (1,0) n (1.5) 0 → C → O(1) → T P → 0, it follows that (1,0) n n+1 (1.6) c(T P ) = (1 + c1(O(1))) . Note that for any divisor D, (1.7) c1([D]) = ηD, where ηD is the Poincar´edual to D. That is Z Z (1.8) ξ = ξ ∧ ηD, n D P 2n−2 for all ξ ∈ H (P), see [GH78, page 141]. So in particular c1(O(1)) = ω, where ω is the Poincar´edual of a hyperplane in Pn (note that ω is integral, and is some multiple of the Fubini-Study metric). Therefore (1,0) n n+1 (1.9) c(T P ) = (1 + ω) . ⊗d Also c1(O(d)) = d · ω, since O(d) = O(1) . The formula (1.4) is then n+1 (1.10) (1 + ω) V = (1 + c1 + c2 + ..
    [Show full text]
  • Geometry of Algebraic Curves
    Geometry of Algebraic Curves Lectures delivered by Joe Harris Notes by Akhil Mathew Fall 2011, Harvard Contents Lecture 1 9/2 x1 Introduction 5 x2 Topics 5 x3 Basics 6 x4 Homework 11 Lecture 2 9/7 x1 Riemann surfaces associated to a polynomial 11 x2 IOUs from last time: the degree of KX , the Riemann-Hurwitz relation 13 x3 Maps to projective space 15 x4 Trefoils 16 Lecture 3 9/9 x1 The criterion for very ampleness 17 x2 Hyperelliptic curves 18 x3 Properties of projective varieties 19 x4 The adjunction formula 20 x5 Starting the course proper 21 Lecture 4 9/12 x1 Motivation 23 x2 A really horrible answer 24 x3 Plane curves birational to a given curve 25 x4 Statement of the result 26 Lecture 5 9/16 x1 Homework 27 x2 Abel's theorem 27 x3 Consequences of Abel's theorem 29 x4 Curves of genus one 31 x5 Genus two, beginnings 32 Lecture 6 9/21 x1 Differentials on smooth plane curves 34 x2 The more general problem 36 x3 Differentials on general curves 37 x4 Finding L(D) on a general curve 39 Lecture 7 9/23 x1 More on L(D) 40 x2 Riemann-Roch 41 x3 Sheaf cohomology 43 Lecture 8 9/28 x1 Divisors for g = 3; hyperelliptic curves 46 x2 g = 4 48 x3 g = 5 50 1 Lecture 9 9/30 x1 Low genus examples 51 x2 The Hurwitz bound 52 2.1 Step 1 . 53 2.2 Step 10 ................................. 54 2.3 Step 100 ................................ 54 2.4 Step 2 .
    [Show full text]
  • Some Classical Formulae for Curves and Surfaces Thomas Dedieu
    Some classical formulae for curves and surfaces Thomas Dedieu To cite this version: Thomas Dedieu. Some classical formulae for curves and surfaces. 2020. hal-02914653 HAL Id: hal-02914653 https://hal.archives-ouvertes.fr/hal-02914653 Preprint submitted on 12 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some classical formulæ for curves and surfaces Thomas Dedieu Abstract. These notes have been taken on the occasion of the seminar Degenerazioni e enumerazione di curve su una superficie run at Roma Tor Vergata 2015–2017. THIS IS ONLY A PRELIMINARY VERSION, still largely to be completed; in particular many references shall be added. Contents 1 Background material 2 1.1 Projectiveduality............................... ..... 2 1.2 Projectivedualityforhypersurfaces. ........... 3 1.3 Plückerformulaeforplanecurves . ........ 4 2 Double curves of the dual to a surface in P3 5 2.1 Local geometry of a surface in P3 anditsdual ................... 5 2.2 Degreesofthedoublecurves. ...... 6 3 Zero-dimensional strata of the dual surface 8 3.1 Localmodels ..................................... 8 3.2 Polarconesandtheircuspidaledges . ........ 9 3.3 Application to the case X = S∨ ............................ 13 4 Hessian and node-couple developables 15 4.1 TheHessiandevelopablesurface . ....... 17 4.2 Thenode-coupledevelopablesurface .
    [Show full text]
  • CHOW RING of a BLOW-UP 1. Introduction in This Document, We
    CHOW RING OF A BLOW-UP JESSE LEO KASS 1. Introduction In this document, we compute the Chow ring of the blow-up of P3 along the twisted cubic. Our goal is to illustrate the main results of §6.7 of [Ful84] with a concrete example. We begin by fixing notation. (1) X denotes projective space P3, (2) Z denotes the twisted cubic curve in X, (3) X˜ denotes the blow-up of X along Z, (4) Z˜ denotes the exceptional divisor of the blow-up. If we forget about the embedding in X, then Z is just the projective line. These three schemes fit into the following Cartesian diagram: j Z˜ −−−−→ X˜ (1.1) g f Zy −−−−→i Xy ∗ Our goal is to give a complete description of the Chow ring Ch (X˜). ∗ Following Fulton, we describe the additive structure of Ch (X˜) in terms of the ∗ ∗ ∗ groups Ch (X), Ch (Z˜), and Ch (Z), together with the Chern classes of the normal bundle N := NZ/X . Having described the additive structure, we then compute the multiplicative structure using some standard intersection theory tools. The Chow rings of X and Z are well-known. They are given by (1.2) Ch(X)= Z[h]/(h4), (1.3) Ch(Z)= Z[w]/(w2). Here h is the class of a hyperplane and w is the class of a point. The maps i∗ and i∗ are also easy to describe. The subvariety Z is the image of a map P1 → X given by cubic polynomials, and it is easy to deduce the following equations from that fact: 2 (1.4) i∗(1) = 3h , 3 (1.5) i∗(w)= h , ∗ (1.6) i (h)=3w.
    [Show full text]