Introduction to Algebraic Geometry
(Hebrew: מבוא לגיאומטריה אלגברית.)
The official course website is on the Moodle system.
Lecture notes (in Hebrew)
Disclaimer: I am quite happy with the material I managed to communicate, but there are changes which I would definitely
make the next time giving the course. For instance, smoothness and general varieties should be their own chapter,
and
I would change the order of some sections. Also, the final lectures are more sketchy than I would have liked, partially
because lack of time, but also because the material should be rearranged, possibly putting more focus on open affines and how
to utilize them.
- Lecture 1. [Chapter 1.] Commutative rings, polynomial rings, algebras.
- Lecture 2.
Algebras, algebraically closed fields. [Chapter 2.] algebraic sets in the affine space, algebraic sets in the projective space.
- Lecture 3.
Algebraic sets in the projective space, embedding the affine space in the projective space.
- Lecture 4. Morphisms, birational maps.
- Lecture 5. Birational maps, example of a non-rational variety.
- Lecture 6. Example of a non-rational variety
(appendix), smoothness: first steps.
[Chapter 3.] the algebra of functions of an affine algebraic set.
- Lecture 7.
The algebra of functions of an affine an algebraic set, the ideal associated to an affine algebraic set.
- Lecture 8.
The ideal associated to an affine algebraic set, Hilbert's Nullstellensatz.
- Lecture 9.
Hilbert's Nullstellensatz, the function algebra determines its affine algebraic set, bijection between affine algebraic sets and polynomial ideals
- Lecture 10.
Bijection between affine algebraic sets and polynomial ideals, irreducible algebraic sets and prime ideals.
- Lecture 11.
Irreducible algebraic sets and prime ideals, Zariski topology and the Zariski closure in A^n.
- Lecture 12.
Zariski topology and the Zariski closure in A^n, Zariski topology of an affine algebraic set.
- Lecture 13.
Zariski topology of an affine algebraic set, reconstructing an affine algebraic from its algebra of functions
- Lecture 14.
Reconstructing an affine algebraic from its algebra of functions.
- Lecture 15.
Reconstructing an affine algebraic from its algebra of functions, dimension (the affine case).
- Lecture 16.
Dimension (the affine case). [Chapter 4.] Zariski open sets.
- Lecture 17.
Zariski open sets, regular functions.
- Lecture 18.
Regular functions, making (some) Zariski open subsets into affine algebraic sets.
- Lecture 19.
The local ring of an algebraic set at a point, derivaties and smoothness.
- Lecture 20.
Derivaties and smoothness.
- Lecture 21.
Derivaties and smoothness.
- Lecture 22.
Derivaties and smoothness.
- Lecture 23.
Derivaties and smoothness, quasi-affine sets.
- Lecture 24.
General algebraic sets.
- Lecture 25.
General algebraic sets.
- Lecture 26.
General algebraic sets.
- Lecture 26.
General algebraic sets, irreducible subsets and dimension.
- Final Work.
Further exploration into topics from the course:
Example of a regular function which cannot be written as a rational function, projective varieties, product of irreducible affine varieties is irreducible,
separatedness and the definition of general varieties.
Exercises (in Hebrew)