Algebraic Geometry

Syllabus Details

• Unit I: (Commutative algebra) Localization, dimension theorem for Noetherian rings (without proof), Rings of dimension $0$, $1$, normal Noetherian rings of dimension $1$ (i.e., Dedekind domains), normalization lemma and Hilbert’s null stellensatz (without proof), transcendental extensions and Luroth’s theorem.
• Unit II: (Geometric concepts) Prime spectrum of a commutative ring with Zariski topology, irreducible algebraic sets and affine algebraic varieties, criterion for connectedness of affine algebraic sets; Noetherian topological spaces, principle of Noetherian induction and application to algebraic varieties (decomposing an algebraic variety into a finite union of irreducible components); projective spectrum of a polynomial ring and projective varieties.
• Unit III: (Affine plane curves) Classification of algebraic subsets of the plane, degree of a plane curve, intersection of curves (via dimension theorem and also via elimination theory) and weak form of Bezout’s theorem; regular and rational functions on a curve.
• Unit IV: (Rational curves) Rational and non-rational curves, conics, a characterization of rational curves in terms of the field of rational functions on the curve, birational isomorphisms. Projective plane curves: Projective completion of an affine curve, homogenization and dehomogenization, resultant of homogeneous polynomials.
• Unit V: (Analysis of singularities) Order of contact of a line with a curve at a point, multiplicity of a point, smooth and multiple points, tangent space and tangent cone at a point, simple and multiple tangents, ordinary multiple points, finiteness of the singular locus of a curve; characterization of smooth points and r-fold points, Bezout’s theorem.

Course Learning Objective

1. Understand the basic theory in Algebraic geometry.
2. Understand and solve few basic problems.
3. Apply algebraic concepts to understand geometry.

Recommended Books

1. Miles, Reid, Undergraduate Algebraic Geometry, Student Text Books, London Mathematical Society Student Texts (Book 12), Cambridge University Press, 1989.
2. Walker, R. J., Algebraic Curves, Springer-Verlag, Berlin-New York, 1978.
3. Hartshorne, R., Algebraic Geometry, Issue 52 of Graduate Texts in Mathematics, Lecture notes in mathematics, Volume 687, Springer, 1977.
4. Shafarevich, I. R., Basic Algebraic Geometry, GTM, Springer, second revised and expanded edition, 1994.
5. Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Addison-Wesley Publishing Co., 1969.
6. Fulton, W., Algebraic Curves, An Introduction to Algebraic Geometry, Notes written with the collaboration of Richard Weiss, Mathematics Lecture Notes Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969.