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 $(k[x_1, \ldots, x_n]/I)_{\mathfrak{m}} \cong k[[x_1, \ldots, x_n]]/Ik[[x_1, \ldots, x_n]]$, 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

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

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

• Exercise IV.6.1: http://math.stackexchange.com/questions/1004277/rational-quartic-in-mathbbp3

Chapter V: Surfaces

Section V.1: Geometry on a Surface

• Page 357: This implies, by the way, that C and D are each nonsingular at P:

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.