5
votes
1answer
219 views

Determine if a conic is degenerate with the determinant.

There is a natural bijection between conics (written as homogeneous quadratic) and 3x3 matrices: $$C=aX^2+2bXY+cY^2+2dXZ+2eYZ+fZ^2\Leftrightarrow \left(\begin{array}{ccc} a&b&d\\ ...
4
votes
1answer
96 views

A zero for a homogeneous polynomial is a zero for the associated inhomogeneous polynomial

I am trying to prove a simple statement from Reid, Undergraduate Algebraic Geometry, pg 16. Let $F(U,V)$ be a nonzero homogeneous polynomial of degree $d$: ...
23
votes
3answers
1k views

Why the emphasis on Projective Space in Algebraic Geometry?

I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what little ...
4
votes
3answers
445 views

The elementary coordinate geometry of polynomials? Of rational expressions? Of radicals?

With a few colleagues, we're trying to design an (intermediate) algebra course (US terminology) where we stress the interplay between algebra and geometry. The algebraic topics we would like to cover ...