Algebraic Geometry by Robin Hartshorne Full Solutions

I. Varieties I.1 x Ex, I.1.1 g x b. x g Ex, I.1.2 x g Ex, I.1.3 x g Ex, I.1.4 x g Ex, I.1.5 x Ex, I.1.6 x g Ex, I.1.7 x b. x (c) x. d x Ex, I.1.10 x (b). x g and I.2.7.a c. x d. x e. x I.2 x I.2.1 x homogenous nullstellensatz I.2.2 projective containments x I.2.3.a. containments. x b. x c. x d. x e. x b. x g and below c. x g I.2.5 (a) x g (b). x I.2.7 a. x g b. x I.2.8 x g I.2.9 x I.2.10 x b. x c. x g I.2.11 x g (use p2) b. x g x. c. x I.2.12 x g and below .b. x g and above and below (part d) c. x d. x g and above I.2.13 x g I.2.14 x g I.2.15 x g b. x c. x g I.2.16 x g b. x g I.2.17 x g complete intersections and below b. x g starred. I.3 x I.3.1 x g b. x g c. x g d. x g e. x g I.3.2 g bijective, bicontinuous but not isomorphism. x b. frobenius not isomorphism. x g I.3.3a. x b. x c. x I.3.4 x g I.3.5 x g I.3.6 x g I.3.7 a x. g b. x I.3.8 x g I.3.9 x g I.3.10 x I.3.11 x I.3.12 x g I.3.13 x g I.3.14 x g (and below) projection from point b. x g I.3.15 x b. x c. x d. x g I.3.16 x g b. x I.3.17 x. g b. x g c. x g d. x e. x. I.3.18.a x b. x g c. x I.3.19 x I.3.20 x g and below b. x g I.3.21a. x b. x c. x d. x e. x I.4.1 x g I.4.2 x I.4.3 x g and below b x. g I.4.4 x g all parts b. x g c. x g I.4.5 x g I.4.6 x g and below b. x g c. x I.4.7 x I.4.8 x g b. x g I.4.9 x g I.4.10 x g I.5 x I.5.1.a x I.5.1.b. x g what kind of singularities c. x d. x g what kind of singularity I.5.2a. x pinch point x b. x conical double point c. x I.5.3 x g b. x g I.5.4 x g b. x g c. x I.5.5 x I.5.6 x g b. x g c. x g d. x g I.5.7 x b. (important) x g c. x I.5.8 x I.5.9 x I.5.10 x g:a,b,c b. x c. x I.5.11 x I.5.12 x b. x c. x d. x. I.5.13 x I.5.14 x g b. x c. x d (starred) I.5.15 x g:a,b b. x I.6 x I.6.1 x g:a,b,c b. x c. x I.6.2 x g b. x c. x d. x e. x I.6.3 x I.6.4 x g I.6.5 x I.6.6 x g:a,c b. x c. x I.6.7 x g I.7 x I.7.1 x g b. x g I.7.2 x g arithmetic genus of projective space. b. x g c. x g important d. g x e. x I.7.3 x g I.7.4 x I.7.5 x g:a,b upper bound on multiplicity x b. x I.7.6 x Linear Varieties x I.7.7 x b. x I.7.8 x contained in linear subspace. x II Schemes II.1 x x II.1.1 Constant presheaf II.1.2 x g b. x c. x. II.1.3x surjective condition. x 2.1.3.b. x g Surjective not on stalks x II.1.4 x b. x x II.1.5 II.1.6.a x map to quotient is surjective II.1.6.b x II.1.7 x II.1.7.b x II.1.8 x g II.1.9 x Direct sum of sheaves x g II.1.10 x Direct Limits x II.1.11 x II.1.12 x II.1.13 Espace Etale x II.1.14 x g II.1.15 Sheaf Hom x: g II.1.16 g Flasque Sheaves x b. x c. x d. x g e. x sheaf of discontinuous sections II.1.17 x g skyscraper sheaves (important) II.1.18 x g Adjoint property of f^(-1) II.1.19 x g Extending a Sheaf by Zero (important?) b. g x extending by zero c. x II.1.20 x Subsheaf with Supports b. x II.1.21 g sheaf of ideals x b. x g c. x g d. x g e. x II.1.22 x g Glueing sheaves (important) II.2 II.2.1 x II.2.2 x g induced scheme structure. II.2.3 a. x reduced is stalk local x b. reduced scheme x c. x II.2.4 x II.2.5 x g II.2.6 x g II.2.7 x g II.2.8 x g Dual numbers + Zariski Tangent Space II.2.9 x g Unique Generic Point (Important) II.2.10 x g II.2.11 x g (Spec Fp important) II.2.12 x g Glueing Lemma II.2.13 x quasicompact vs noetherian. a b. x c. x d. x II.2.14 x b. x c. x d. x II.2.15x g (important) b. x c. x II.2.16 x b. x c. x d. x II.2.17 Criterion for affineness x b. x II.2.18 g x b. g x c. x g (important) d. g x II.2.19 x g II.3 x II.3.1 x II.3.2 x II.3.3 x b. x c. x II.3.4 x II.3.5 x g b. x g c. x II.3.6 x g Function Field II.3.7 x II.3.8 x Normalization II.3.9 x g Topological Space of a Product b. g x II.3.10 x g Fibres of a morphism b. x II.3.11 x g Closed subschemes b. (Starred) c. x d. x g scheme-theoretic image II.3.12 x Closed subschemes of Proj S b. x II.3.13 x g Properties of Morphisms of Finite Type b. x g c. x d. x e. x f. x g. x II.3.14 x g II.3.15 x b. x c. x II.3.16 x g Noetherian Induction II.3.17 x Zariski Spaces b. x g c. x g d. x g e. x g specialization f. x II.3.18x Constructible Sets b. x c. x d. x II.3.19 x b (starred) c. x d. x g II.3.20 x g Dimension b. x g c. x d. x g e. x g f. x II.3.21 x II.3.22* (Starred) II.3.23 x II.4 x stopped g'ing here II.4.1 x g Nice example valuative crit II.4.2 x II.4.3 x g II.4.4 x II.4.5 x g center is unique by valuative criterion b. x (starred) d. x II.4.6 x g II.4.7 x R-scheme b. x c. x d. x e. x II.4.8 x e. x f. x II.4.9 x g important - used stein factorization II.4.10 Chow's Lemma (starred) b part of starred c. part of starred d. part of starred II.4.11 x b. x II.4.12 x Examples of Valuation Rings b. (1) x (2) x (3) x II.5 x II.5.1 g x (b) g x (c) x g (d) x g Projection Formula II.5.2 (a) x (b) x II.5.3 x g II.5.4 x II.5.5 (a) x g b. x g closed immersion is finite. (c) x g II.5.6 (a) x g (b) x g (c) x g (d) x (e) x II.5.7 x b. x (c) x II.5.8 x (b) x (c) x II.5.9 x (b) x (c) x II.5.10 x (b) x (c) x (d) x II.5.11 x g II.5.12 x g (b) x g II.5.13 x II.5.14 x (b) x (c) x (d) x II.5.15 x Extension of Coherent Sheaves (b) x (c) x (d) x e. x II.5.16 xc a. g Tensor Operations on Sheaves (b) xc g (c) cx (d) x (e) xc II.5.17 x Affine Morphisms (b) x g (c) x g (used for stein factorization) d. x e. x II.5.18 x Vector Bundles b. x c. x d. x II.6 x Divisors II.6.1 x g II.6.2 (starred) Varieties in Projective Space II.6.4 x II.6.6 x g (b) x g (c) x g (d). x II.6.7 (starred) II.6.8 (a) x g (use easier method) (b) x (c) x II.6.9 (starred) Singular Curves (starred) II.6.10 x g The Grothendieck Group K(X) (b) x (c) x II.6.12 x g:1st paragraph II.7 x Projective Morphisms II.7.1 x g II.7.2 x II.7.3 x b. x II.7.4 a. x g b. x II.7.5 x g b. x g c. x g d. x g x g ample large multiple is very ample. x II.7.6 x g The Riemann Roch Problem b. x II.7.7 x g Some Rational Surfaces b. x c. x II.7.8 x sections vs quotient invertible sheaves II.7.9 x g b. x II.7.10 x P^n Bundles over a Scheme b. x g d. x II.7.11 x b. x c. x II.7.12 x g II.7.13 A Complete Nonprojective Variety * II.7.14 x g b. x II.8 x Differentials II.8.1 x b. x c. x d. x g II.8.2 x II.8.3 x g Product Schemes b. x g c. x g II.8.4 x Complete Intersections in Pn b. x g c. x g d. x g e. x g f. x g g. x g II.8.5 x g Relative Canonicals Important! b. x g II.8.6 x Infinitesimal Lifting Property b. x c. x II.8.7 x II.8.8 x II.9 Formal Schemes - skip III Cohomology III.1 III.2 x III.2.1.a x g III.2.2 x flasque resolution g III.2.3 x Cohomology with Supports b. x Flasque Global sections are exact III.2.2.c x x x excision x II.2.4 x Mayer-Vietoris III.2.5 x III.2.6 x x III.2.7a g Cohomology of circle b. x III.3 x Cohomology of a Noetherian Affine Scheme III.3.1 x III.3.2 x III.3.3 a is left exact. x b. x x III.3.4 x Cohomological Interpretation of Depth b. x III.3.5 x III.3.6 x part c. x III.3.7 x b. x III.3.8 x Localization not injective non noetherian. III.4 x Cech Cohomology III.4.1 x g pushforward cohomology affine morphism III.4.2 x b. x c. x III.4.3 g nice x III.4.4 x b. x c. x III.4.5 x III.4.6 x III.4.7 x III.4.8 x cohomological dimension b. x c x. e. x III.4.9 x III.4.10 (starred) III.4.11 x III.5 x Cohomology_Of_Projective_Space III.5.1 x g III.5.2 x b. x g III.5.3a. x Arithmetic genus III.5.3.b. x c. x g Important genus is birational invariant for curves!! b. x III.5.5 x g b. x g c. x g complete intersection cohomology. d. x g III.5.6 x curves on a nonsingular quadric b. x c. x III.5.7 x g x g ample iff red is ample c. x g d. x g finite pullbacks ampleness III.5.8.a x g b. x c. x g d. x III.5.9 x g Nonprojective Scheme III.5.10 x g III.6 x Ext Groups and Sheaves III.6.1x III.6.2.a. x b. x III.6.3.a. x b. x III.6.4 x III.6.5 x b. x c. x III.6.6.a x b. x III.6.7 x III.6.8 x b. x III.6.9 x b. x III.6.10 x Duality for Finite Flat Morphism b. x c. x d. x III.7 x Serre Duality Theorem III.7.1 x g Special Case Kodaira Vanishing III.7.2 x III.7.3 x Cohomology of differentials on Pn III.7.4 (starred) III.8 x Higher Direct Images of Sheaves III.8.1 x g Leray Degenerate Case III.8.2 x g III.8.3 x g Projection Formula derived III.8.4 x b. x c. x d. x e. x III.9 x Flat Morphisms III.9.1 x III.9.2 x twisted cubic III.9.3 x g b. x c. x III.9.4 x open nature of flatness III.9.5 x Very Flat Families b. x c. x d. x III.9.6 x III.9.7 x III.9.9 x rigid example III.9.10 x g b. x III.9.11 x interesting g III.10 x Smooth Morphisms III.10.1 x regular != smooth always III.10.2 x g III.10.3 x III.10.4 x III.10.5 x etale neighborhood x III.10.6 x g Etale Cover of degree 2. x III.10.7 x Serre's linear system with moving singularities b. x III.11 x Theorem On Formal Functions III.11.1 x g higher derived cohomology of plane minus origin. III.11.2 x g III.11.3 x III.11.4 x Principle of Connectedness III.12 x Semicontinuity III.12.1 x g upper semi-continuous tangent dimension III.12.2 x III.12.3 x Rational Normal Quartic III.12.4x III.12.5 x Picard Group of projective bundle IV Curves IV.1 x Riemann_Roch_Theorem x IV.1.1 g Regular except at a point x IV.1.2 g Regular Except pole at Points x IV.1.3 g Nonproper Curve is affine IV.1.4 x IV.1.5 x g Dimension less than degree IV.1.6 x g finite morphism to P1 IV.1.7 x g b. x g IV.1.8 x g arithmetic genus of a singular curve (b) x g Genus 0 is nonsingular. x g difference of very amples. x g Invertible sheaves are L(D) x. Alternative riemann-roch IV.1.10 g x IV.2 x Hurwitz Theorem IV.2.1x g projective space simply connected IV.2.2 x g classification of genus 2 curves b. x g c. x .x conclusion. x IV.2.3 x inflection points gauss map b. x multiple tangents. c. x g d. x g e. x g f. x g g. x g h. x IV.2.4 x g Funny curve in characteristic p IV.2.5 x Automorphisms f a curve in genus >= 2 b. x IV.2.6 x g pushforward of divisors b. x g c. x d. x branch divisor IV.2.7 x Etale Covers degree 2 b. x c. x IV.3 Embeddings In Projective Space IV.3.1 x g IV.3.2 x g :a,b,c b. x g c. x g IV.3.3 x g IV.3.4 x g b. x g c. x d. x g IV.3.5 x g b. x g c. x IV.3.6 x g Curves of Degree 4 b. x g IV.3.7 x IV.3.8 x b. x g No strange curves in char 0!!!! IV.3.9 x g IV.3.10 x g IV.3.11 x g b. x g IV.3.12 x g (just explain the advanced method) IV.4 Elliptic Curves IV.4.1 x g IV.4.2 x IV.4.3 x g IV.4.4 x IV.4.5 x b. x c. x d. x IV.4.6.a. x g b. x osculating hyperplanes c. x g IV.4.7 x g Dual of a morphism b. x c. x g e. x f. x IV.4.8 x Algebraic Fundamental Group IV.4.9 x g isogeny is equivalence relation. b. x g IV.4.10 x picard of product on genus 1 IV.4.11 x g b. x c. x IV.4.12.a x b. x IV.4.13 x IV.4.14 x Fermat Curve and Dirichlet's Theorem IV.4.15 x IV.4.16 x b.x g kernel of frobenius c. x d. x Hasse's Riemann Hypothesis for Elliptic Curves e. x IV.4.17 a. x b. x IV.4.18 x IV.4.19 x IV.4.20 x g Slight issue? c. x d. x IV.4.21 x skip - not algebraic geometry IV.5 Canonical Embedding IV.5.1 x g complete intersect is nonhyperelliptic IV.5.2 x g Aut X is finite. IV.5.3 x g Moduli of Curves of Genus 4 IV.5.4 x g x g IV.5.5 x g Curves of Genus 5 b. x g IV.5.6 x g IV.5.7.a x g IV.6 Curves In P3 IV.6.1 x g IV.6.2 x g IV.6.3 x g IV.6.4 x g IV.6.5 x g complete intersection doesn't lie on small degree surface IV.6.6 x g Projectively normal curves not in a plane IV.6.7 x g IV.6.8 x g IV.6.9 (starred) V Surfaces V.1 Geometry On A Surface V.1.1 x g Intersection Via Euler Characteristic V.1.2 x g Degree via hypersurface V.1.3 x g:a,b adjunction computational formula b. x g: with above c. x g V.1.4 x g Self intersection of rational curve on surface b. x. V.1.5 x g Canonical for a surface in P3 b. x g V.1.6 x g b. x g V.1.7 x Algebraic Equivalence of Divisors b. x c. x V.1.8 x g cohomology class of a divisor b. x g V.1.9 x g Hodge inequality b. x g V.1.10 x g Weil Riemann Hypothesis for Curves V.1.11 x g b. x g V.1.12 x g Very Ample not numerically equiv V.2 Ruled Surfaces V.2.1 x g V.2.2 x V.2.3 x x g tangent sheaf not extension of invertibles V.2.4 a x b. x V.2.5 x g b. x c. x d. x V.2.6 x g Grothendieck's Theorem V.2.7 x V.2.8.a x g decomposable is never stable b.x g c. x g V.2.9 x g Curves on Quadric Cone V.2.10 x V.2.11 x b. x V.2.12 x b. x b. x b. x c. x V.2.16 x V.2.17* (starred) V.3 Monoidal Transformations V.3.1 x g V.3.2 x g V.3.3 x g V.3.4 x Multiplicity of local ring b. x c. x d. x e. x V.3.5 x g hyperelliptic every genus V.3.6 x V.3.7 x V.3.8a,b x V.4 Cubic Surface V.4.1 x g P2 blown at 2 points V.4.2 x g V.4.3 x V.4.4 x g important? b. x V.4.5 x g Pascal's Theorem V.4.6 x g V.4.7 x V.4.9 x genus bound for cubic surface. V.4.10 x V.4.11 x Weyl Groups b. x V.4.12 x g kodaira vanishing for cubic surface b. x g V.4.14 x V.4.15 x admissible transformation b. x c. x g d. x V.4.16 x Fermat Cubic V.5 Birational Transformations V.5.1 x g Resolving singularities of f V.5.2 x g Castelnuovo Lookalike V.5.3 x g hodge numbers excercise V.5.4 x g hodge index theorem corollary x b. x g Hodge Index negative definite V.5.5 x g V.5.6 x V.5.7 x V.5.8 x A surface Singularity V.5.8.b. x Surface singularity V.6 Classification Of Surfaces V.6.1 x g V.6.2 x g