Algebraic Geometry Approach To Study The Surfaces Given The Intersection Curve

Question

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.

Thank you in advance,

Answer 1

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