Question

I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have interest in this subject.

I wonder whether there are some basic algebraic geometry texts considering the level of undergraduate students(who have not learnt commutative algebra or homological algebra, they just knew linear algebra and basic abstract algebra)

I am also looking for some texts book which provided a lot of examples(more computations using linear algebra and calculus). Actually, I am still looking for some texts book based on the very basic mathematics but talked a little bit modern view point.

Thanks in advance!

Answer 1

An invitation to algebraic geometry by Karen Smith is excellent; it is very intuitive, and does everything over the complex numbers. For absolute newcomers, this is probably the best introduction.

Algebraic curves by William Fulton is a classic, quite easily readable for beginners, and free available online in pdf! (He recently published the third edition on his site.)

The books by Reid, Miranda and Hulek are also good. Reid does many explicit examples.

Answer 2

Miranda's "Algebraic Curves and Riemann Surfaces" assumes that you're familiar with a bit of complex analysis in one variable, but that's it. One of my favorites, aside from the ones that everyone else posted.

Answer 3

I can't recall exactly how much background it assumes, but I found Reid's Undergraduate Algebraic Geometry quite accessible. (The forthright views in its last section can be taken either as a blemish or a bonus depending on one's POV.)

Answer 4

Cox Little and O'Shea's "Ideals varieties and algorithms" (http://www.amazon.com/Ideals-Varieties-Algorithms-Computational-Undergraduate/dp/0387356509/ref=sr_1_1?ie=UTF8&s=books&qid=1265456210&sr=1-1) is very accessible, assumes almost no background in commutative algebra, and has many examples. The emphasis is on computational algebraic geometry (including Groebner bases).

BTW, Milne's "Algebraic Geometry" (http://jmilne.org/math/CourseNotes/AG.pdf) includes an "Annotated Bibliography" Appendix with an "Elementary Algebraic Geometry" section, and perhaps this is a good place to start the search.

Answer 5

The first volume of Kenji Ueno's "Algebraic Geometry" is a really nice undergraduate book.

Answer 6

I recommend Harris' book. It has a nice pace, it is not very technical and has plenty of examples which can be worked out by simple linear algebra.

If your students have a good grasp in analysis you may also consider some parts from Griffiths-Harris, but that is probably too advanced.

Answer 7

I think Hulek's book is a nice introduction that does the commutative algebra as needed. I think it is a little more demanding than some of the other suggestions though.

Answer 8

Amnon Neeman's book does GAGA for projective space at the undergraduate level, or so he tells me.

[Edit: The book's called "Algebraic and Analytic Geometry".]

Answer 9

Introduction to Algebraic Curves by Phillip A. Griffiths

Answer 10

I suggest "Introduction to Algebraic Geometry" by Brendan Hassett. It is a nice, down-to-earth introduction to algebraic geometry, and it also spends a lot of time on computational topics like Grobner bases. When I was an undergraduate, I took a class from Brendan using an early version of this book (in fact, I think he was writing it as he taught), and it was completely accessible despite the fact that at time I did not know much commutative algebra.

Answer 11

You can also take a look at the question A learning roadmap for algebraic geometry.

Answer 12

As an introduction to Algebraic Geometry for Undergraduates, Amnon Neeman's book "Algebraic and Analytical Geometry" is really great. It goes directly into schemes, but works over C and gives motivation for everything he does. It also has an introduction for teachers intending to use the book. (Sorry for the duplicate answer, but I can't make comments yet)

Answer 13

REALLY hard to pick past Ried's Undergraduate Algebraic Geometry or O'Shea et.al.'s Ideals,Vareties And Algorithms for a first course in AG. In fact,using them both with create an extraordinarily deep course for undergraduates with just a basic abstract algebra course. Ried's book supplies a great theoretical foundation while IVAs gives much more hands-on examples as well as computer experiences. Great choices both.