Notes on Hypersurfaces

As wiki said,

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface. Equivalently, the codimension of a hypersurface is 1.

In algebraic geometry, a hypersurface in projective space of dimension n is an algebraic set that is purely of dimension n − 1. It is then defined by a single equation F = 0, a homogeneous polynomial in the homogeneous coordinates. It may have singularities, so not in fact be a submanifold in the strict sense.

In the sense of Fulton, Algebraic Curves, if F is not a constant in the ring of polynomials in n variables over R, the set of zeros of F is called the hypersurface defined by F, and is denoted by V(F).

For example, F(X,Y) = (X² + Y²)² + 3X² Y - Y³ = 0 in R[X,Y] is a beautiful curve.

Q: Why do we study V(F) first?