Algebraic geometry studies geometric loci defined by polynomial equations, for example, the complex plane curve \(f(x,y)=0\). It plays an important role at both elementary and sophisticated levels in many areas of mathematics and theoretical physics, and provides the most visual and elegant tools to express all aspects of the interaction between different branches of mathematical knowledge. The course gives the flavor of the subject by presenting examples and applications of the ideas of algebraic geometry, as well as a first discussion of its technical apparatus.
Prerequisites
Advanced Linear Algebra (tensor products, multilinear maps), basic ideas of commutative algebra (polynomial rings and their ideals). Some experience in Analysis on Manifolds, Topology I and Topology II: Introduction to Homology and Cohomology Theory (projective spaces, metric and topological spaces, simplicial complexes and homology groups) is desirable but not essential.
Curriculum:
- Projective spaces.
- Projective conics and \(\operatorname{PGL}(2)\).
- Geometry of projective quadrics. Spaces of quadrics.
- Grassmannians.
- Examples of projective maps: Plücker, Segre, Veronese.
- Integral ring extensions, polynomial ideals, affine algebraic geometry and Hilbert’s theorems.
- Algebraic varieties, Zariski topology, schemes, geometry of ring homomorphisms.
- Irreducible varieties. Dimension.
- Plane projective algebraic curves: point multiplicities, intersection numbers, Bézout’s theorem.
- Plane projective algebraic curves: singularities, duality, Plücker formulas.
- Rational curves. Veronese curve. Cubic curves.
- Curves on surfaces. The 27 lines on a smooth cubic surface.
- Vector bundles and their section sheaves. Vector bundles on the projective line.
- Linear systems and invertible sheaves, the Picard group, line bundles on affine and projective spaces.
- Tangent, cotangent, normal and conormal bundles. The Euler exact sequence.
- Singularities and tangent cone. Blow-up.
- Complex projective curves: canonical class, genus, Serre duality and Riemann-Roch theorem.
- If time allows: Poncelet’s porism; quadrics through a canonical curve; Clebsch and Lüroth problems, and so on.
Textbooks
- J. Harris, Algebraic Geometry. A First Course., Springer.
- M. Reid, Undergraduate algebraic geometry, Cambridge: Cambridge Univ. Press, 1988.