Algebraic curves. Conics (or quadrics), their euclidean and affine classification over the fields of real and complex numbers. Projective plane, homogeneous coordinates, projective transformations. Lines in projective plane. Projective classification of conics. Parametrization of nondegenerate conics.
Homogeneous polynomials or forms. Roots of polynomials and their multiplicities. Bézout's Theorem, proof when one of the curve is a line or a quadric. Factorization of forms vanishing along lines and nondegenerate conics. Spaces of d-forms vanishing at certain points and their dimensions. Applications to quadrics passing through 5 points and cubics passing through 9 points. Pascal's Theorem.
Nodal and cuspidal cubics, their parametrization. Tangent lines. Group law on a cubic.
Affine algebraic sets, their ideals. Noetherian rings. Hilbert Basis Theorem. Algebraic sets defined by ideals, their properties. Zariski topology. Termination of descending chains of algebraic sets. Irreducible algebraic sets, their relation with prime ideals. Unique decomposition of algebraic set into irreducible components.
Nullstellensatz (Hilber Zero Theorem) and Weak Nullstellensatz. Proof of the Nullstellensatz assuming Weak Nullstellensatz.
Polynomial functions on affine algebraic sets. Coordinate ring. Polynomials maps between affine algebraic sets. Relation between polynomial maps and coordinate ring homomorphisms. Polynomial isomorphisms. Affine varieties. Rational functions on affine algebraic sets. Regular points of rational functions. Rational maps. Dominant maps.
Projective algebraic sets. Homogeneous ideals and correspondence between them and projective algebraic sets. The affine cone over a projective algebraic set. Rational functions on projective algebraic sets, rational maps between them. Regular points of rational functions and maps. Morphisms and isomorphisms. Segre embedding of the product of two projective spaces into another projective space. Finite unions, finite products, and arbitrary intersections of projective algebraic sets are again projective algebraic.
For exam-related problems look in TCD past examination papers and Mathematics department examination papers.
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