**MTH ****617: Introduction to Algebraic Geometry**** ****(4)**

*Learning Objectives*:

This course aims to provide an introduction to some of the basic objects and techniques and objects of algebraic geometry with minimal prerequisites. The main emphasis is on geometrical ideas and so most of the treatment will be over algebraically closed fields. Results from commutative algebra will introduced and proved as required and so no prior experience with commutative algebra will be assumed. After introducing the basic objects and techniques, they will be illustrated by application to the theory of algebraic curves.

*Course Contents*:

Closed subsets of affine space, coordinate rings, correspondence between ideals and closed subsets, affine varieties, regular maps, rational functions, Hilbert's nullstellensatz

Projective and quasi-projective varieties, regular and rational functions on projective varieties, products and maps of quasi-projective varieties

Dimension of varieties, examples and applications

Local ring of a point, tangent and cotangent space, local parameters, non-singular points and non-singular varieties

Birational maps, blowups, disingularization of curves

Intersection numbers for plane curves, divisors on curves, Bezout's theorem, Riemann-Roch theorem for curves, Residue theorem, Riemann-Hurwitz formula

*Suggested Books*:

- W. Fulton, Algebraic curves: An introduction to algebraic geometry, 2008 ed. (available online).
- R. Shafarevich, Basic Algebraic Geometry, Vol. 1, Third Edition, Springer, Heidelberg, 2013.
- S. Abhyankar, Algebraic geometry for scientists and engineers, Mathematical Surveys and Monographs 35, American Mathematical Society, 1990.
- K. Smith et al, An invitation to algebraic geometry, Springer, 2004.

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