Algebraic geometry is the study of solutions to systems of polynomial equations. It is a central and very active area of modern mathematics with deep connections to commutative algebra, complex analysis, number theory, combinatorics, and topology. It has applications in physics, robotics, coding theory, optimization theory, and more recently in biology and statistics.
We will begin by developing the standard "dictionary" for translating between algebraic properties of polynomials and geometric properties of their solution sets. In the first part of the course we shall present classical algebraic geometry, in order to develop intuition and give a feel for the field. The second part of the course will discuss schemes, to introduce this important concept of modern mathematics.
Tuesday 2:30-3:30pm, or by appointment.
Affine space, algebraic sets, Zariski topology, irreducible algebraic sets, affine varieties. The affine coordinate ring of an affine variety. Regular functions. The dimension of an affine variety.
Projective space, algebraic sets, Zariski topology, projective varieties. Relations between projective space and affine space. The homogeneous coordinate ring of a projective variety. Regular functions.
Morphisms of varieties. The d-uple embedding and the Segre embedding. The ring of germs of regular functions, the field of rational functions of a variety. The product of two affine varieties. The product of two projective varieties.
Rational maps, birational maps, birational equivalence. The blow-up of an affine variety at a point.
Singular and nonsingular points, regular local rings. (Zariski) tangent and cotangent spaces.
Finite morphisms: definition and main properties (the fibers of a finite morphisms are finite, a finite morphism is surjective, a finite morphism is a closed map, finiteness is a local property, projections are finite morphisms).
Presheaves and sheaves of abelian groups. Morphisms of sheaves. The sheaf associated to a presheaf. Kernel, cokernel and image of a morphism of sheaves. Subsheaves and quotient sheaves. Direct images and inverse images of sheaves. Ringed spaces, locally ringed spaces.
The spectrum of a ring, affine schemes, distinguished open subsets. Morphisms of affine schemes. Schemes, morphisms of schemes. Glueing of schemes, glueing of morphisms of schemes. The Proj of a graded ring. Subschemes: open immersions, open subschemes, closed immersions, closed subschemes.
Basic properties of schemes and morphisms: connected schemes, reduced schemes, irreducible schemes, integral schemes, (locally) noetherian schemes. Morphisms locally of finite type, of finite type, finite. Fibred product of schemes, fibres of a morphism of schemes, families of schemes. Normal schemes: the normalization of an integral scheme. Separated schemes, separated morphisms. The valuative criterion of separatedness.
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During the lectures some homework will be assigned. This homework has to be handed in the next week. While cooperation is encouraged when thinking about the problems, students are required to write up and hand in the homework individually. Copying from others is not cooperation and will not be condoned. Late homework is not accepted.
At the end of the course, each student is required to take an oral exam. This will consist in a lecture on a topic chosen by the student, suject to agreement of the instructor. The student's lecture will be followed by some questions on the material covered in the course. For further information contact the instructor.
The final grade will be determined as follows: 15% homework, 85% final exam.
