Course

Unit-I

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; Quasi-Compactness and Hausdorffness; Prime and maximal spectra; Example: linear varieties, hypersurfaces, curves.

Unit-II

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 segre- and the d-uple embedding.

Unit-III

RATIONAL MAPS: Relevance to function fields and birational classification; Example:
Classification of curves; blowing-up.

Unit-IV

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.

Unit-V

INTERSECTIONS IN PROJECTIVE SPACE: Notions of multiplicity and intersection with examples.

CO 1: To understand the various structures introduced in Algebraic geometry and to prove the standard theorems due to Hilbert/Krull/Noether, which give correspondence between algebraic varieties and ideals, rings and fields.
CO 2: To understands properties of morphisms and its applications
CO 3: To familiarize the concept of rational maps
CO 4: To identify non-singualrity through various criteria and understand the process of desingularisation
CO 5: To familiarize the idea of multiplicity and intersection with examples

1. Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics (GTM)
   8th Printing, Springer, 1997.
2. C. Musili, Algebraic Geometry for Beginners, Texts and Readings in Mathematics 20, Hindustan Book Agency, 2001.