Hartshorne - Algebraic Geometry



Springer GTM 52.

Algebraic geometry

"This book provides an introduction to abstract algebraic geometry using the methods of schemes and cohomology."

Exercise Solutions Available:
 * www.lomont.org/Math/Solutions.pdf
 * http://algebraicgeometry.blogspot.com/
 * http://www.math.mcgill.ca/bcais/CourseNotes/AlgGeom04/Hartshorne_Solutions.pdf
 * www.math.jhu.edu/.../Hartshorne%20Algebraic%20Geometry%20Solutions.pdf
 * http://mathsci.kaist.ac.kr/~jinhyun/sol2/hart.html
 * nicf.net/math/rh-IV12.pdf
 * http://en.wikibooks.org/wiki/Solutions_to_Hartshorne%27s_Algebraic_Geometry

= Chapter I: Varieties =

Section I.1: Affine Varieties

 * Height of a prime ideal : Height of a prime ideal is like codimension of a subvariety.


 * Proposition I.1.10: $$\overline{Z_i} \neq \overline{Z_{i+1}}$$ because $$\overline{Z_i} \cap Y = Z_i$$ and $$Z_i \neq Z_{i+1}$$.
 * Exercise I.1.1:http://math.stackexchange.com/questions/69015/exercise-in-hartshorne,http://math.stackexchange.com/questions/100906/hartshorne-exercise-1-1-a

Section I.2: Projective Varieties

 * Exercise I.2.14: http://math.stackexchange.com/questions/353846/hartshorne-problem-1-2-14-on-segre-embedding

Section I.3: Morphisms

 * Page 18: http://math.stackexchange.com/questions/209279/localization-notation-in-hartshorne


 * Exercise I.3.7: http://math.stackexchange.com/questions/699433/exercise-3-7-hartshorne
 * Exercise I.3.8: http://math.stackexchange.com/questions/860577/hartshorne-exercise-1-3-8
 * Exercise I.3.13: http://math.stackexchange.com/questions/124422/local-ring-of-a-subvariety-problem-1-3-13-in-hartshorne

Section I.4: Rational Maps

 * Exercise I.4.8: http://mathoverflow.net/questions/76174/hartshornes-exercise-i-4-8-any-two-curves-over-k-homeomorphic

Section I.5: Nonsingular Varieties

 * One can show easily that [the Jacobian] definition of nonsingularity is independent of the choice of generators of the ideal : The point is that what we're really calculating is the dimension of the space spanned by the vectors $$((\partial f / \partial x_1)(P), \ldots, (\partial f / \partial x_n)(P))$$ for each $$f \in I$$.  Suppose that $$I = (f_1, \ldots, f_m)$$ and $$f \in I$$; then $$f = \sum_i p_i f_i$$ for some $$p_i \in k[x_1, \ldots, x_n]$$, and so we have $$\frac{\partial f}{\partial x_i}(P) = \sum_i \frac{\partial f_i}{\partial x_i}(P) p_i(P) + f_i(P) \frac{\partial p_i}{\partial x_i}(P)$$.  Since $$f_i \in I$$ and $$P \in V(I)$$, we have $$f_i(P) = 0$$ by definition, so this reduces to $$\frac{\partial f}{\partial x_i}(P) = \sum_i \frac{\partial f_i}{\partial x_i}(P) p_i(P)$$.  Clearly, then, the vector $$((\partial f / \partial x_1)(P), \ldots, (\partial f / \partial x_n)(P))$$ lies in the span of the vectors $$((\partial f_j / \partial x_1)(P), \ldots, (\partial f_j / \partial x_n)(P))$$ for $$j = 1, \ldots, m$$.


 * Example I.5.6: We need an additional commutative algebra result not found in the chapter to make sense of this, namely that $$(k[x_1, \ldots, x_n]/I)_{\mathfrak{m}} \cong kx_1, \ldots, x_n/Ikx_1, \ldots, x_n$$, where $$\mathfrak{m} = (x_1, \ldots, x_n)$$. See Chapter 7 of Eisenbud - Commutative Algebra - with a View Toward Algebraic Geometry.


 * Exercise I.5.1: http://math.stackexchange.com/questions/331876/do-the-pictures-in-hartshorne-ex-1-5-1-make-sense
 * Exercise I.5.2: http://math.stackexchange.com/questions/961889/definition-of-intersection-multiplicity-in-hartshorne-vs-fulton-for-plane-curves

Section I.8: What is Algebraic Geometry?
= Chapter II: Schemes =

Section II.1: Sheaves

 * Exercise II.1.16(b): http://math.stackexchange.com/questions/1025910/hartshorne-ex-ii-1-16-b-flasque-sheaves-and-exact-sequences
 * Exercise II.1.18: http://math.stackexchange.com/questions/1000044/maps-between-direct-limits-and-functoriality-of-f-1shvy-rightarrow-shvx
 * Exercise II.1.21: http://math.stackexchange.com/questions/35490/hartshorne-exercise-about-sheaves-on-mathbbp1

Section II.2: Schemes

 * Proposition II.2.6: http://math.stackexchange.com/questions/240477/question-of-hartshorne-books-proposion-ii-2-6


 * Exercise II.2.13: http://math.stackexchange.com/questions/57024/quasi-compact-and-compact-in-algebraic-geometry


 * Exercise II.2.18(b): http://math.stackexchange.com/questions/424884/possible-mistake-in-exercise-in-hartshorne-exercise-ii-2-18b

Section II.3: First Properties of Schemes

 * Proposition II.3.1: Then $$ \mathcal{O}(U_1 \cup U_2) = \mathcal{O}(U_1) \times \mathcal{O}(U_2)$$ which is not an integral domain. A product of nontrivial rings can never be an integral domain: (a, 0) * (0, b) = (0, 0), so (a, 0) and (0, b) are zero divisors. (In some sense it seems like Proposition 3.1 is a partial converse to this.)


 * Proposition II.3.2: http://math.stackexchange.com/questions/55825/an-isomorphism-problem-in-scheme-while-reading-hartshorne
 * Exercise II.3.1: http://math.stackexchange.com/questions/803636/question-on-morphism-locally-of-finite-type


 * Example II.3.2.4: http://math.stackexchange.com/questions/297123/some-question-of-schemehartshorne-exampleii-3-2-4


 * Example II.3.5: http://math.stackexchange.com/questions/422922/gluing-schemes-hartshorne-example


 * Exercise II.3.18: http://math.stackexchange.com/questions/1001237/hartshorne-exercise-3-18-chapter-2
 * Exercise II.3.19: http://math.stackexchange.com/questions/259609/hartshorne-exercise-ii-3-19-a, http://math.stackexchange.com/questions/259117/hartshorne-exercise-ii-3-19-b, http://math.stackexchange.com/questions/261114/hartshorne-exercise-ii-3-19-c
 * Exercise II.3.20: http://math.stackexchange.com/questions/921764/exercise-3-20-in-hartshorne-on-dimension-of-integral-schemes-of-finite-type-over

Section II.5: Sheaves of Modules

 * Proposition II.5.4: http://math.stackexchange.com/questions/52856/is-noetherian-condition-always-needed-when-speaking-of-a-coherent-sheaf


 * Proposition II.5.13: http://math.stackexchange.com/questions/313752/hartshorne-book-proposition-ii-5-13
 * Exercise II.5.7: http://math.stackexchange.com/questions/967012/hartshorne-chapter-ii-exercise-5-7-on-invertible-sheaves


 * Exercise II.5.9: http://mathoverflow.net/questions/19105/question-on-an-exercise-in-hartshorne-equivalence-of-categories


 * Exercise II.5.12: http://math.stackexchange.com/questions/85688/hartshorne-exercise-ii-5-12b


 * Exercise II.5.13: http://math.stackexchange.com/questions/58470/proj-of-graded-rings
 * Exercise II.5.18(d): http://math.stackexchange.com/questions/822731/exercise-5-18d-chapter-2-hartshorne

Section II.6: Divisors

 * Page 129: For each line L in P^2, we consider $$L \cap C$$ which is a finite set of points on C. Note that the "points" mentioned here are intrinsic points of C, not points of P^2 on the embedded copy of C.
 * Proof of Proposition II.6.2: It is well-known that a UFD is integrally closed. This is a consequence of the rational root theorem.


 * Example II.6.5.2: http://math.stackexchange.com/questions/241666/example-6-5-2-p-133-in-hartshorne, http://math.stackexchange.com/questions/326509/a-question-on-generic-point-and-a-question-on-hartshorne


 * Lemma II.6.1: http://math.stackexchange.com/questions/417793/hartshorne-lemmaii-6-1


 * Proposition II.6.5: http://math.stackexchange.com/questions/388724/hartshorne-proposition-ii-6-5


 * Page 141: http://math.stackexchange.com/questions/223773/functional-sheaf-hartshorne-cartier-divisors
 * Exercises II.6.11-12: http://math.stackexchange.com/questions/869796/torsion-sheaves-on-a-curve

Section II.7: Projective Morphisms

 * http://math.stackexchange.com/questions/112926/global-sections-of-mathcalo-1-and-mathcalo1-understanding-structu

Section II.8: Differentials

 * Page 172: http://math.stackexchange.com/questions/410881/sheaf-of-differential-forms-tangent-sheaf-hartshorne, http://mathoverflow.net/questions/132718/sheaf-of-differential-forms-tangent-sheaf-hartshorne
 * Theorem II.8.13: http://math.stackexchange.com/questions/1094953/e-to-s-surjective-in-degrees-geq-1-implies-widetildee-to-widetildes


 * Proposition II.8.20: http://math.stackexchange.com/questions/417798/hartshorne-proof-of-adjunction-formula-proposition-ii-8-20


 * Example II.8.20.3: http://math.stackexchange.com/questions/83227/image-type-of-the-canonical-divisor-under-the-isomorphism-mathrmpic-mathbb

Section II.9: Formal Schemes
= Chapter III: Cohomology =

Section III.2: Cohomology of Sheaves

 * Proposition III.2.6: http://math.stackexchange.com/questions/447220/hartshornes-weird-definition-of-right-derived-functors-and-prop-iii-2-6/447234#447234


 * Exercise III.2.1(a): http://math.stackexchange.com/questions/74641/hartshorne-exercise-iii-2-1a
 * Exericse III.2.1(b): http://math.stackexchange.com/questions/910385/showing-grothendiecks-vanishing-theorem-provides-a-strict-bound

Section III.6: Ext Groups and Sheaves

 * Exercise III.6.2(a): http://math.stackexchange.com/questions/897880/hartshorne-exercise-iii-6-2-a

Section III.9: Flat Morphisms

 * III.9.5: http://math.stackexchange.com/questions/430948/base-step-of-induction-in-hartshorne-iii-9-5


 * Proposition III.9.8: http://math.stackexchange.com/questions/430766/hartshorne-iii-9-8-understanding-associated-points-and-extensions
 * Theorem III.9.9.: https://math.stackexchange.com/questions/2368912/proof-of-theorem-9-9-in-part-iii-of-hartshornes-algebraic-geometry/2370490#2370490

Section III.10: Smooth Morphisms

 * Exercise III.10.2: http://math.stackexchange.com/questions/830565/hartshorne-ex-iii-10-2-on-smooth-morphisms

Section III.11: The Theorem on Formal Functions

 * http://math.stackexchange.com/questions/47378/base-change-for-proper-morphism-in-hartshorne

Section III.12: The Semicontinuity Theorem

 * http://math.stackexchange.com/questions/47378/base-change-for-proper-morphism-in-hartshorne


 * Exercise III.12.3: http://math.stackexchange.com/questions/116601/hartshorne-exercise-iii-12-3

= Chapter IV: Curves =

Section IV.1: Riemann-Roch Theorem

 * Proof of Theorem IV.1.3: we have an exact sequence $$ 0 \to \mathcal{L}(-P) \to \mathcal{O}_X \to k(P) \to 0$$

This is an issue that comes up all over the place. It makes sense to talk about k(P), a k-valued skyscraper sheaf supported at P, only when we're thinking about k(P) as a sheaf of rings, like when we're thinking of it as the structure sheaf of P. In the exact sequence above, though, we're regarding it not as a sheaf of rings but as a sheaf of $$\mathcal{O}_X$$-modules, and in this case writing k(P) isn't that helpful because it doesn't really specify the $$\mathcal{O}_X$$-module structure. In this case, $$\mathcal{O}_X$$ acts on $$k(P)$$ by $$f \cdot \alpha = \alpha f(P)$$. This clarifies why $$\mathcal{O}_X \otimes k(P) = k(P)$$: we can simply move functions on the left over to the right side of the tensor product.


 * Proof of Theorem IV.1.3: $$\chi(k(P)) = 1$$ : Recall that k(P) here really means $$j_\ast \mathcal{O}_P$$, where j denotes the inclusion $$j : P \hookrightarrow X$$. By Lemma III.2.10, $$H^i(X, j_\ast \mathcal{O}_P) = H^i(P, \mathcal{O}_P)$$, so it follows that $$\chi(j_\ast \mathcal{O}_P) = \chi(\mathcal{O}_P)$$.  By Grothendieck vanishing (Theorem III.2.7), a sheaf on the zero-dimensional space P only has zeroth cohomology, so $$\chi(\mathcal{O}_P) = h^0(P, \mathcal{O}_P) = 1$$.


 * Example IV.1.8: http://math.stackexchange.com/questions/381585/length-of-a-quotient-in-hartshorne-iv-ex-1-8

Section IV.6: Classification of Curves in P^n
= Chapter V: Surfaces =
 * Exercise IV.6.1: http://math.stackexchange.com/questions/1004277/rational-quartic-in-mathbbp3

Section V.1: Geometry on a Surface
Since the maximal ideal of $$\mathcal{O}_{D, P}$$ is generated by f, {f} is a regular system of parameters. Since its cardinality is equal to the Krull dimension $$\dim \mathcal{O}_{D, P} = 1$$, $$\mathcal{O}_{D, P}$$ is a regular local ring.
 * Page 357: This implies, by the way, that C and D are each nonsingular at P :

Section V.6: Classification of Surfaces
= Appendix A: Intersection Theory = = Appendix B: Transcendental Methods = = Appendix C: The Weil Conjectures =