AFFINE AND PROJECTIVE VARIETIES: Noetherian rings and modules;
Emmy Noether's theorem and Hilbert's Basissatz; Hilbert's
Nullstellensatz; Affine and Projective algebraic sets; Krull's
Hauptidealsatz; topological irreducibility, Noetherian decomposition; local
ring, function field, transcendence degree and dimension theory; QuasiCompactness and Hausdorffness; Prime and maximal spectra; Example:
linear varieties, hypersurfaces, curves.
MORPHISMS: Morphisms in the category of commutative algebras over
a commutative ring; behaviour under localization; morphisms of local
rings; tensor products; Product varieties; standard embeddings like the
segreand the d-uple embedding.
RATIONAL MAPS: Relevance to function fields and birational
classification; Example:
Classification of curves; blowing-up.
NONSINGULAR VARIETIES: Nonsingularity; Jacobian Criterion; singular
locus; Regular local rings; Normal rings; normal varieties; Normalization;
concept of desingularisation and its relevance to Classification Problems;
Jacobian Conjecture; relationships between a ring and its completion;
nonsingular curves.
INTERSECTIONS IN PROJECTIVE SPACE: Notions of multiplicity and
intersection with examples.
- Teacher: tevbal VENKATA BALAJI T E