I'm looking at algebraic plane curves of the form $F(x,y)=0$ and trying to figure out why for points on the curve such that $\frac{\partial F}{\partial x} = \frac{\partial F}{\partial y}=0$, the plane curve has a singularity.
I've been looking at the surface $z=y^2-x^3$ and comparing it to $z=y-x^2$ and trying my darndest to understand what about having a flat tangent plane causes the corresponding curve to have a singularity right at the point where the surface 'dips' into the $z=0$ plane. So not being able to figure out geometrically/intuitively why one thing causes the other, I started looking for a proof of this fact hoping that it would show me the 'why' of it, but I can't find one online or in any of my books.
Can someone explain intuitively why this is, and maybe point me in the direction of a proof as well?
I’ve applauded someone else here at MSE for wanting to have an intuition different from others’, and I’ll say the same to you: differing intuitions lead to more interesting mathematics. But yet, I don’t think it’s at all productive to think of a curve $F(x,y)=0$ in the plane as the intersection of the surface $z=F(x,y)$ with the $(x,y)$-plane.
If you aren’t willing to accept the condition $\partial F/\partial x=\partial F/\partial y=0$ at a point $P$ as the definition of a singular point, then you need an independent definition of what it means for a point to be singular.
First, let’s look at cases where we agree that the origin is a singular or nonsingular point of some curve. If the origin is a nonsingular point, then any line passing very near to the origin in a direction not parallel to the curve’s tangent at the origin cuts the curve in only one point. If we move toward the origin with a line parallel to the tangent, though, the points of intersection are not comparably close (think of a horizontal line $y=\varepsilon$ and the parabola $y=x^2$) but when we hit, ¡plink!, the contact is multiple rather than single.
Compare this behavior with what happens at the origin with the curve $y^2=x^2(x+1)$, in other words $x^2+x^3-y^2=0$, which you and I agree has a clearly visible singularity at $(0,0)$, a node. Any line coming close to the origin, of no matter what slope, has two intersections with the curve comparably close to the distance $\varepsilon$ from the origin to the line. And in case the slope is $\pm1$, at the origin, ¡plink!, the intersection multiplicity there is three.
So here’s a proposed definition of singularity for point of a plane curve, independent of the standard one. A point $P$ on the locus of $F(x,y)=0$ is nonsingular if all but one of the straight lines passing through $P$ has only a single intersection with the locus in the immediate neighborhood of $P$. And $P$ is singular if all lines through $P$ have at least a double intersection with the locus, in the immediate neighborhood of $P$.
Now how does this work out in practice? Let’s again restrict our view to curves passing through the origin, so that the polynomial $F(x,y)$ has no constant term. And then we can write $$ F(x,y)=a_{10}x + a_{01}y+a_{20}x^2+a_{11}xy+a_{02}y^2+\cdots\>, $$ where the ellipsis represents monomials of degree $3$ and greater. Now all is clear. The partial-derivative condition for singularity is exactly that the two linear coefficients $a_{10}$ and $a_{01}$ should vanish. And I hope that you see that this vanishing is exactly the condition that any line $\alpha x+\beta y=0$ through the origin should have at least a double intersection with the curve at the origin. I hope also that you’ll go through all this for the singular curve $y^2=x^3$, and see how the concepts fit together there.