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I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have an 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 know linear algebra and basic abstract algebra.

I am looking for some textbooks which provide a lot of examples (more computations using linear algebra and calculus). Actually, I am also looking for some textbooks based on very basic mathematics but which talk a little bit about a modern view point.

Thanks in advance!

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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.

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    $\begingroup$ That's Karen Smith and three other authors: Lauri Kahanpaa, Pekka Kekalainen, and William Traves. $\endgroup$
    – Todd Trimble
    Commented Sep 4, 2017 at 2:23
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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.

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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.

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    $\begingroup$ I forgot about this one. It is excellent. $\endgroup$
    – Wanderer
    Commented Feb 6, 2010 at 14:56
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    $\begingroup$ I second Miranda--it nice to actually have some geometric examples and ways to think about things when moving on to non-complex algebraic geometry. $\endgroup$
    – Ryan Eberhart
    Commented Feb 6, 2010 at 16:23
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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.)

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    $\begingroup$ I just read that last (or so) section. It's nothing I didn't expect, but the straw-man remarks make me a little sad nonetheless. $\endgroup$
    – Todd Trimble
    Commented Jun 27, 2014 at 12:27
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The first volume of Kenji Ueno's "Algebraic Geometry" is a really nice undergraduate book.

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  • $\begingroup$ There is currently a seminar being lectured from this exact book at the OP's university! $\endgroup$
    – B. Bischof
    Commented Feb 7, 2010 at 0:32
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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.

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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.

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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".]

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  • $\begingroup$ can you tell me the name of this book? I can not find it when googling $\endgroup$
    – Shizhuo Zhang
    Commented Feb 6, 2010 at 11:40
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    $\begingroup$ I think Amnon has quite smart undergrad students! He wrote in the introduction of his paper The K-theory of triangulated category that his undergrad students have little trouble understanding it! $\endgroup$
    – Hailong Dao
    Commented Feb 6, 2010 at 17:25
  • $\begingroup$ 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) $\endgroup$
    – babubba
    Commented Feb 7, 2010 at 1:00
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Introduction to Algebraic Curves by Phillip A. Griffiths

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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.

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

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    $\begingroup$ A very interesting thread, true, but I don't think the answers given there will be helpful for the above question. There is a difference between learning stacks etc and introducing algebraic geometry to undergraduate students who don't know commutative algebra. $\endgroup$
    – Wanderer
    Commented Feb 6, 2010 at 12:24
  • $\begingroup$ perhaps a course in commutative alebra should be a prerequisite for a course in algebraic geometry ... I've made the experience that otherwise the students can't really work on their own. $\endgroup$
    – Martin Brandenburg
    Commented Feb 6, 2010 at 13:45
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I think Complex Algebraic Curves by Frances Clare Kirwan is a great introduction. In the first chapter,she speaks examples of brieskorn sphere. Abel's theorem is treated quite nicely. and much more!

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