Algebraic Geometry
Hartshorne, Algebraic Geometry
Algebraic geometry shows tremendous applicability in fields other than mathematics and mathematics.
This is a new perspective on geometry and requires sufficient prior knowledge.
1. Varieties
1.1 Affine Varieties
1.2 Projective Varieties
1.3 Morphisms
1.4 Rational Maps
1.5 Nonsingular Varieties
1.6 Nonsingular Curves
1.7 Intersections in Projective Space
1.8 What is Algebraic Geometry?
2. Schemes
2.1 Sheaves
2.2 Schemes
2.3 First Properties of Schemes
2.4 Separated and Proper Morphisms
2.5 Divisors
2.6 Projecive Morphisms
2.8 Differentials
2.9 Formal Schemes
3. Cohomology
3.1 Derived Functors
3.2 Cohomology of Sheaves
3.3 Cohomology of a Noetherian Affine Scheme
3.4 Cech Cohomolopgy
3.5 The Cohomology of Projective Space
3.6 Ext Groups and Sheaves
3.7 The Serre Duality Theorem
3.8 Higher Direct Images of Sheaves
3.9 Flat Morphisms
3.10 Smooth Morphisms
3.11 The Theorem on Formal Functions
3.12 The Semicontinuity Theorem
4. Curves
4.1 Riemann-Roch Theorem
4.2 Hurwitz's Theorem
4.3 Embeddings in Projective Space
4.4 Elliptic Curves
4.5 The Canonical Embedding
4.6 Classification of Curves in $\mathbf{P}^3$
5. Surfaces
5.1 Geometry on a Surface
5.2 Ruled Surfaces
5.3 Monoidal Transformations
5.4 The Cubic Surface in $\mathbf{P}^3$
5.5 Birational Transformations
5.6 Classification of Surfaces
A. Intersection Theory
A.1 Intersection Theory
A.2 Properties of the Chow Ring
A.3 Chern Classes
A.4 The Riemann-Roch Theorem
A.5 Complements and Generalizations
B. Transcendental Methods
B.1 The Associated Complex Analytic Space
B.2 Comparison of the Algebric and Analytic Categories
B.3 When is a Compact Complex Manifold Algebric?
B.4 Kahler Manifolds
B.5 The Exponential Sequence
C. The Weil Conjectures
C.1 The Zeta Function and the Weil Conjectures
C.2 History of Work on the Weil Conjectures
C.3 The $l$-adic Cohomology
C.4 Cohomological Interpretation of the Weil Conjectures
Use
This is one part of ULA.
Algebraic Geometry is understand geometrical structure of algebraic system.