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HartshorneAlgebraicGeometry

Springer GTM 52.

Algebraic geometry

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

Exercise Solutions Available:

Chapter I: Varieties Edit

Section I.1: Affine Varieties Edit

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

Section I.2: Projective Varieties Edit

Section I.3: Morphisms Edit

Section I.4: Rational Maps Edit

Section I.5: Nonsingular Varieties Edit

  • 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.

Section I.6: Nonsingular Curves Edit

Section I.7: Intersections in Projective Space Edit

Section I.8: What is Algebraic Geometry? Edit

Chapter II: Schemes Edit

Section II.1: Sheaves Edit

Section II.2: Schemes Edit

Section II.3: First Properties of Schemes Edit

  • 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.)

Section II.4: Separated and Proper Morphisms Edit

Section II.5: Sheaves of Modules Edit

Section II.6: Divisors Edit

  • 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.

Section II.7: Projective Morphisms Edit

Section II.8: Differentials Edit

Section II.9: Formal Schemes Edit

Chapter III: Cohomology Edit

Section III.1: Derived FunctorsEdit

Section III.2: Cohomology of SheavesEdit

Section III.3: Cohomology of a Noetherian Affine SchemeEdit

Section III.4: Cech CohomologyEdit

Section III.5: The Cohomology of Projective SpaceEdit

Section III.6: Ext Groups and SheavesEdit

Section III.7: The Serre Duality TheoremEdit

Section III.8: Higher Direct Images of SheavesEdit

Section III.9: Flat MorphismsEdit

Section III.10: Smooth MorphismsEdit

Section III.11: The Theorem on Formal FunctionsEdit

Section III.12: The Semicontinuity TheoremEdit

Chapter IV: Curves Edit

Section IV.1: Riemann-Roch TheoremEdit

  • 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 $.

Section IV.2: Hurwitz's TheoremEdit

Section IV.3: Embeddings in Projective SpaceEdit

Section IV.4: Elliptic CurvesEdit

Section IV.5: The Canonical EmbeddingEdit

Section IV.6: Classification of Curves in P^nEdit

Chapter V: Surfaces Edit

Section V.1: Geometry on a SurfaceEdit

  • 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.

Section V.2: Ruled SurfacesEdit

Section V.3: Monoidal TransformationsEdit

Section V.4: The Cubic Surface in P^nEdit

Section V.5: Birational TransformationsEdit

Section V.6: Classification of SurfacesEdit

Appendix A: Intersection Theory Edit

Appendix B: Transcendental Methods Edit

Appendix C: The Weil Conjectures Edit