Already Rene Descartes criticized the separation of algebra and geometry and
believed that I could borrow all that was best both in geometrical analysis and in algebra, and correct all the defects of the one by help of the other
greatly contributing to both by their synthesis in his analytic geometry.
In 19th century after the works of Poncelet and subsequent developments of Abel, Cayley, Salmon and others the new discipline emerged and flourished in the Italian school of late 19 - early 20th century.
The aim of the course is to give a taste of projective and affine algebraic geometry with minimal pre-requisites (linear algebra and basic algebra of polynomial rings) and to show some classical results accessible with pre-WWII machinery.
I copy an abstract of the textbook of Gorodentsev
This is a geometric introduction to the algebraic geometry. I hope to acquaint the readers with some basic figures underlying the modern algebraic technique and show how to translate things from the infinitely rich (but quite intuitive) world of figures to the scanty and finite (but very explicit) language of formulas. These lecture notes contain a lot of exercises crucial for understanding the subject.
Projective (and affine) varieties. Key examples. Smooth vs. Singular points and dimension. Plane Cubic Curves. Projection and blowup. Cubic Surfaces.
All of the following books introduce the subject without the new techniques of schemes/sheaves/cohomology, showing instead how far one can go with geometric methods. Harris provides a lot of interesting examples and does not even discuss a genus of a curve!
http://gorod.bogomolov-lab.ru/ps/stud/projgeom/1718/list.html (2017.11.24)
https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf (2013.10.20)
https://dept.math.lsa.umich.edu/~wfulton/CurveBook.pdf (2008.01.28)
https://bookstore.ams.org/stml-20
https://link.springer.com/book/10.1007/978-1-4757-2189-8
The course is suitable to either undergraduate or early post-graduate students.
Required: Proficiency with polynomials of many (e.g 3) variables.
Desirable: Knowledge how to compute a rank of a rectangular matrix with real or complex entries.
I copy pre-requisites from the authors of the reference textbooks.
Harris: just some linear and multilinear algebra and a basic background in abstract algebra (definitions and basic properties of groups, rings, fields, etc.)
Shafarevich (Part 1): in addition to an undergraduate algebra course, we assume known basic material from field theory: finite and transcendental extensions (but not Galois theory), and from ring theory: ideals and quotient rings.
Fulton: some basic properties of rings, ideals, and polynomials, such as is often covered in a one-semester course in modern algebra
Beltrametti et al: basic elements of projective geometry and abstract algebra, e.g. some knowledge of the geometry of subspaces and properties of fields
Reid is more explicit:
- Algebra: Quadratic forms, easy properties of commutative rings and their ideals, principal ideal domains and unique factorisation.
- Galois Theory: Fields, polynomial rings, finite extensions, algebraic versus transcendental extensions, separability.
- Topology and geometry: Definition of topological space, projective space $\mathbf{P}^n$ (but I’ll go through it again in detail).
- Calculus in $\mathbf{R}^n$: Partial derivatives, implicit function theorem (but I’ll remind you of what I need when we get there).
- Commutative algebra: Other experience with commutative rings is desirable, but not essential.
Garrity et al:
- C1 (conics) - appropriate for first-year college students (and many high school students).
- C2-3 (cubic and higher degree curves) - appropriate for people who have taken multivariable calculus and linear algebra.
- C4-5 (higher dimensional varieties) - abstract algebra now plays a critical role, making a first course in abstract algebra necessary from this point on.
Hulek: standard courses on algebra and complex analysis (in principle, some analytic parts could be replaced by other material)
https://bookstore.ams.org/stml-66 “This text consists of a series of exercises, plus some background information and explanations, starting with conics and ending with sheaves and cohomology.”
https://www.mathematik.uni-kl.de/~gathmann/de/alggeom.php
http://www.math.lsa.umich.edu/~idolga/631.pdf (2013.08.19)
http://www.math.lsa.umich.edu/~idolga/CAG.21.pdf (2021)
http://www.mat.ucm.es/~arrondo/projvar.pdf (2017.11.26) “The scope of these notes is to present a soft and practical introduction to algebraic geometry, i.e. with very few algebraic requirements but arriving soon to deep results and concrete examples that can be obtained “by hand”… My approach consists of avoiding all the algebraic preliminaries that a standard algebraic geometry course uses for affine varieties and thus start directly with projective varieties (which are the varieties that have good properties)….”