# Algebraic Geometry Approach To Study The Surfaces Given The Intersection Curve

I'm NOT a Mathematician and I'm totally new to the field of Algebraic Geometry.

A friend of mine told me that one thing which is studied in this field is to consider a curve as a set of points in n-D space and try to find (n+1)-D hyper-planes whose intersection is that curve and then study those planes. (finding polynomials whose zeros are curve points and study them)

I'm interested in this topic and want to know more but searching algebraic geometry literature in general is to broad so I'd really appreciate if somebody can give me some hints and keywords (names of algorithms and theorems) to narrow my search.

• You can try Projective Geometry. – KittyL Mar 25 '15 at 8:45
• You can see these notes on Projective Geometry by N. Hitchin: people.maths.ox.ac.uk/hitchin/hitchinnotes/hitchinnotes.html – Krish Mar 25 '15 at 9:26
• Thank you both very much! seems so close to what I need. But may I ask if you know any point of view to this problem which comes from "Algebraic Geometry"? Actually I wanna know what my friend told me is what you recommended or there is another specific study of this topic through algeb.geo? or maybe they are the same?!! @Krish – Kasra Manshaei Mar 25 '15 at 13:41

So you mean "hypersurfaces," not "hyperplanes," since a curve which is the intersection of hyperplanes is just a line. Also, a hypersurface in N-dimensional space is (N-1)-dimensional, not (N+1)-dimensional.

Of course, these hypersurfaces are just the loci given by individual equations cutting out the curve. For instance, if you have the twisted cubic curve

$$x(t) = t,$$ $$y(t) = t^2,$$ $$z(t) = t^3,$$

this is cut out of three-dimensional affine space by the equations

$$x z - y^2 = 0,$$ $$y - x^2 = 0,$$ $$z - x y = 0$$

which you can consider as three surfaces whose intersection is this curve.

This sort of thing is the very first thing you learn in algebraic geometry. Here are some good introductory sources:

• Perrin, Algebraic Geometry: an Introduction
• Smith et al, An Invitation to Algebraic Geometry
• Cox, Donal, and O'Shea, Ideals, Varieties, and Algorithms
• Nice to meet you Daniel! :))))) Thanks a lot my friend. Clear and comprehensive. Cheers – Kasra Manshaei Mar 25 '15 at 17:26
• I can only add some more references to this nice answer: math.stackexchange.com/questions/255063/… math.stackexchange.com/questions/269384/… math.stackexchange.com/questions/285201/… – Krish Mar 25 '15 at 17:58
• Excuse me @DanielMcLaury, Shouldn't x(t) and z(t) be the other way around?!! and thanks for refrences. I found them and they are great, but could you please narrow my search by recommending a certain chapter regarding this specific question? – Kasra Manshaei Mar 26 '15 at 11:14
• Thanks, I fixed the implicit equations. As for the books, just start at the beginning. Loci of systems of polynomial equations (and, eventually, a generalization of this) is exactly what algebraic geometry studies. – Daniel McLaury Mar 26 '15 at 15:46