Algebraic Geometry: A First Geometric Look
- Students will be competent in basic constructions of algebraic geometry that will allow them to start studying more advanced courses in algebraic geometry or apply their knowledge to study the courses like algebraic groups, curves and surfaces.
- Students will lean basis constructions, theorems of algebraic geometry. Also they will gain sufficient package of examples of algebraic varieties and the method of their study.
- Projective spaces. Geometry of projective quadrics. Spaces of quadrics
- Lines, conics. Rational curves and Veronese curves. Plane cubic curves. Additive law on the points of cubic curve
- Grassmannians, Veronese’s, and Segre’s varieties. Examples of projective maps coming from tensor algebra
- Integer elements in ring extensions, finitely generated algebras over a field, transcendence generators, Hilbert’s theorems on basis and on the set of zeros
- Affine Algebraic Geometry from the viewpoint of Commutative Algebra. Maximal spectrum, pullback morphisms, Zariski topology, geometry of ring homomorphisms
- Algebraic manifolds, separateness. Irreducible decomposition. Projective manifolds, properness. Rational functions and maps.
- Dimension. Dimensions of subvarieties and fibers of regular maps. Dimensions of projective varieties.
- Linear spaces on quadrics. Lines on cubic surface. Chow varieties.
- Vector bundles and their sheaves of sections. Vector bundles on the projective line. Linear systems, invertible sheaves, and divisors. The Picard group
- Tangent and normal spaces and cones, smoothness, blowup. The Euler exact sequence on a projective space and Grassmannian
- Dolgachev, I. (2012). Classical Algebraic Geometry : A Modern View. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=473170
- Hartshorne, R., & American Mathematical Society. (1975). Algebraic Geometry, Arcata 1974 : [proceedings]. Providence: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=772699