The degree of a curve is the degree of the defining polynomial. For example, a line is a curve of degree .
Let be a curve in , and . Then will be called a simple point if or . What does this condition really imply? It means that the function doesn’t locally stop changing as we move away from and . It is not a local maximum, or minimum (it’s not an extremal point). In this case the line is called the tangent line. Why do we get tangent lines whose equations look like this? This is analogous to the dot product rule for writing down the equation to a plane (or any hyperplane in general). The line is the hyperplane in .
Take an equation , and write it down as , where . Here each is a homogeneous component. Then is the multiplicity of at . What does this mean? As we approach , the higher degree terms approach much faster than . Hence, “looks like” near . An analogy would be . Near , the function “looks like” . Hence, it makes sense to call the multiplicity of the curve at .
But how do we define the multiplicity of the curve at an arbitrary point on it? Why only ? We can create new variables and , and then re-write the equation. The new equation contains the point . Now we can follow the same procedure.
As is a function of two variables in this case, it can be factorized into a product of linear polynomials, raised to various exponents. Is it weird that the function looks like a bunch of lines near ? No. Locally, every polynomial, and in fact every function, looks like a linear near a point. Here, because we’re talking about varieties, and hence possibly a product of polynomials, the local description of a polynomial is a bunch of lines. Nothing weird in it at all.
Why are the factors of tangents at ? Are we taking the “local” argument too far? No. It’s a revelation though. I have never seen it before. It makes complete mathematical sense. Say we take . Then the tangent at is , which is in fact true.