# Real Algebraic Geometry

Publisher:
Springer
Publication Date:
2013
Number of Pages:
100
Format:
Paperback
Series:
Unitext 66
Price:
39.95
ISBN:
9783642362422
Category:
Monograph
[Reviewed by
Fernando Q. Gouvêa
, on
08/15/2013
]

It all starts from such a simple idea: let’s study the curves and surfaces we get by writing down algebraic equations. Nevertheless, algebraic geometry has achieved a rather fearsome reputation for being difficult and arcane. Part of it is the result of its long history, but the truth is that the massive arsenal required for fruitful attacks on some of the main problems really does pose an effective barrier to the non-participating fan.

Enter this little book, which contains notes from lectures by Vladimir Arnold on some very elementary algebraic geometry. This is mathematics that anyone can enjoy, requiring only the rudiments of coordinate geometry and algebra.

The title seems to be intentionally ambiguous. Most of the lectures do deal with the geometry of curves and surfaces defined over $$\mathbb{R}$$, so it is technically correct. The editors suggest, however, that Arnold might also have been quietly insisting that this kind of algebraic geometry is a bit more real than the super-abstract kind.

Arnold begins the book by stating Hilbert’s problem on the topological structure of real algebraic curves, but the book becomes much more elementary after that, with chapters on conics (both their geometry and their role in physics) and on projective geometry. The next lecture breaks the boundary suggested by the title to discuss complex algebraic curves (just the very basics). The book concludes with a very interesting lecture on an elementary problem: if you delete n lines from the plane, how many connected pieces are you left with? A research article about this last question (but for the projective plane) is appended to the lectures, followed by notes from the editors.

In summary, a neat little book that your students will understand, but that still contains meaty ideas and difficult problems for them to work on.

Fernando Q. Gouvêa is planning to ask a group of first-year students about lines on a plane, just to see what they can do with the problem.