ementas

Algebraic Geometry [MAT2255]

Geometria Algébrica

Syllabus

• Affine space.
• Closed algebraic subsets.
• Zariski topology.
• Regular functions.
• Sheaves.
• Algebraic varieties.
• Morphisms.
• Projective varieties.
• Properness theorem.
• Irreducible components.
• Rational functions.
• Finite morphisms.
• Dimension:
  • Krull dimension,
  • transcendence degree,
  • Zariski tangent space.
• Krull’s lemma.
• Local properties.
• Smooth points.
• Rational maps.
• Blow-up.
• Normalization.
• Dimension of the fibers.
• Bertini’s theorem.
• Vector bundles.
• Canonical line bundle.
• Adjunction formula.
• Divisors, inversible sheaves, canonical divisor.
• Linear systems.
• Ample line bundles, immersions in the projective space.
• Coherent sheaves.
• Riemann-Roch for curves. Applications.
• Intersection numbers.
• Hodge index theorem.
• Birational maps of surfaces.

Bibliography

Basic bibliography:

1. Igor Shafarevich: Basic Algebraic Geometry.

2. Basic Algebraic Geometry 1: Varieties in Projective Space.
3. Basic Algebraic Geometry 2: Schemes and Complex Manifolds.
4. third tome (about complex manifolds) won’t be used in this course 3rd English edition: 2013, Springer–Verlag, ISBN 978-3-642-37955-0 (tome 1) and 978-3-642-38009-9 (tome 2). MR3100243.
5. George Kempf: Algebraic Varieties. Cambridge University Press, 1993.
6. Karen E. Smith, Lauri Kahampää, Pekka Kekäläinen, William Traves. An Invitation to Algebraic Geometry. Springer-Verlag, 2000. doi:10.1007/978-1-4757-4497-2. Also see a comment on author’s site.

Free bibliography:

• James S. Milne: Algebraic Geometry, v6.03 (Nov 2023). 223 pages. Available at jmilne.org/math
• Igor Dolgachev: Introduction to Algebraic Geometry, August 19, 2013. 194 pages. Available at math.lsa.umich.edu/~idolga
• Ravi Vakil: The Rising Sea: Foundations of Algebraic Geometry, Feb 21, 2024. 850 pages. Available at math.stanford.edu/~vakil/216blog

Software: PARI.math.u-bordeaux.fr, SageMath.org.

Prefaces.

Shafarevich:

This book is a general introduction to algebraic geometry. Its aim is a treatment of the subject as a whole, including the widest possible spectrum of topics… The nature of the book requires the algebraic apparatus to be kept to a minimum. In addition to an undergraduate algebra course, we assume known basic material from field theory: finite and transcendental extensions (but not Galois theory), and from ring theory: ideals and quotient rings.

Kempf:

Algebraic geometry is a mixture of the ideas of two Mediterranean cultures. It is the superposition of the Arab science of the lightning calculation of the solutions of equations over the Greek art of position and shape. This tapestry was originally woven on European soil and is still being refined under the influence of international fashion. Algebraic geometry studies the delicate balance between the geometrically plausible and the algebraically possible. Whenever one side of this mathematical teeter totter outweighs the other, one immediately loses interest and runs off in search of a more exciting amusement… In this book we present from a modern point of view the basic theory of algebraic varieties and their coherent cohomology. The local part of the study includes dimension and smoothness. I have tried to keep the commutative algebra down to minimum while putting the geometry close to the algebra as part of the exposition. The basic tools in algebraic geometry are sheaves and their cohomology. This material is presented from the beginning. I have included the basic discussion of curves to illustrate the theory… The main battle was to teach the reader to think globally in sheaf theoretic language.