How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Question

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on their own on the books given on the answer to that question

Also, i think it is too much and incomplete to try on your own to search on the whole subject of plane curve. What is a fast and comprehensive way to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Answer 1

For something I was working on about a year ago, I checked out a library copy of Bix's text Conics and Cubics for a couple of months. I liked the text. Indeed, speaking just for myself, if I wanted to begin learning algebraic geometry, I think I would begin with Bix's text. However, it seemed to me that Bix's text is at the upper level you stated (advanced undergraduate level), and is not really a text that would bridge the gap between high school precalculus and advanced undergraduate level work. Thus, I think the suggestions by Gerry Myerson would be more useful to you.

In any event, perhaps some of the material in the following notes by I. R. Shafarevich could be of use, although they are probably more algebraical/analytical and less geometrical than what would be ideal for your purposes. The last reference is a paper by Joerg Meyer that might also be of interest to you.

Selected Chapters From Algebra, I (22 pages)

http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=tm&rbr=1

Selected Chapters From Algebra, II (30 pages)

http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=tm&rbr=2

Selected Chapters from Algebra, III/1 (16 pages)

http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=tm&rbr=3

Selected Chapters from Algebra, III/2 (26 pages)

http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=tm&rbr=4

Selected Chapters from Algebra, IV. Primes (20 pages)

http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=tm&rbr=5

Selected Chapters From Algebra, V. Real Numbers and Polynomials (34 pages)

http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=tm&rbr=6

Selected Chapters From Algebra, VI. Infinite Sets (34 pages)

http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=tm&rbr=7

Selected Chapters From Algebra, VII. Power Series (36 pages)

http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=tm&rbr=8

Projections of the twisted cubic by Joerg Meyer (12 pages)

http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=tm&rbr=18

Answer 2

Try Bix, Conics and cubics: a concrete introduction to algebraic curves.

Answer 3

I'd say that to go from high school to plane curves you need two semesters of calculus, a semester of linear algebra, a semester of groups-rings-fields, and a semester of geometry (especially projective geometry).