Lecture 1: "Ideals, Varieties and Algorithms" Chapters 8.1 and 8.2: projective plane, projective space and projective varieties
Lecture 2: "Ideals, Varieties and Algorithms" Chapters 8.3 and 8.4: homogeneous ideals, Nullstellensätze and projective ideal-variety correspondence
Lecture 3: "Ideals, Varieties and Algorithms" Chapters 8.5 and 8.6: Projective Elimination Theorem, projective equivalence, quadric hypersurfaces
Lecture 4: Serge map, Plücker coordinates, Bezout’s theorem, dimension of monomial ideals
Lecture 5: correspondence between coordinate subspaces of the variety of a monomial ideal and coordinate subspaces of standard monomials, counting standard monomials of total degree at most s, Hilbert function, Hilbert polynomial
Lecture 6: discussion of homework on projective varieties, dimension of an affine and projective variety
No lectures on November 18 and 20. First page of the course project due on November 25.
Lecture 7: discussion of homework on dimension, properties of dimension
Lecture 8: tangent spaces, dimension of a variety at a point, singular locus
Lecture 9: polynomial mappings, affine coordinate ring
Lecture 10 (planned): isomorphic varieties, rational functions on a variety, binational equivalence of varieties
Course project presentations on 12.12 at 10!