### MA6180 INTRODUCTION TO ALGEBRAIC GEOMETRY

#### Course Details

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.

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.

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.

#### Course References:

Books:

1. Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics (GTM) Vol.52, Corr. 8th Printing, 1997, Springer-Verlag .

2. C. Musili, Algebraic Geometry for Beginners, Texts and Readings in Mathematics 20, Hindustan Book Agency, India, 2001