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