I see that the Cayley-Salmon Theorem states that a smooth cubic surface over an algebraically closed field contains exactly 27 lines. But what exactly does "smooth" mean here? Alternatively, what would be an example of a homogeneous cubic polynomial in four variables which does not generate a "smooth" surface? My background is very elementary, and if answers could try to restrict the discussion to varieties and keep the scheme-theoretic language to a minimum, it would be much appreciated.
What exactly does “smooth” mean in the sense of the Cayley-Salmon “27 Lines on Cubic” Theorem
algebraic-geometry
Related Solutions
A compact Riemann surface $X$ can always be embedded holomorphically into $\mathbb P^3( \mathbb C)$ . This is a highly non-trivial theorem.
Claim: It cannot in general be embedded into $\mathbb P^2( \mathbb C)$.
The simplest argument to support this claim is to recall that a smooth complex curve of degree $d$ in $\mathbb P^2( \mathbb C)$ has genus $g=\frac{(d-1)(d-2)}{2}$.
Since there exist Riemann surfaces of any genus $g\geq 0$ on one hand, and since on the other hand most integers are not of the form $\frac{(d-1)(d-2)}{2}$ , the claim is proved.
However any Riemann surface can be immersed, but not necessarily injectively, into $\mathbb P^2( \mathbb C)$ by skilfully (or skillfully if you prefer American English) composing an embedding into $\mathbb P^3( \mathbb C)$ with a projection onto a plane, so that the immersed curve will have nodal singularities at worst.
Bibliography
An excellent reference for these questions is Miranda's Algebraic Curves and Riemann Surfaces.
If you want to see the complete proofs of the embedding theorem of compact Riemann surfaces into $\mathbb P^3( \mathbb C)$ (which implies algebraicity of said Riemann surfaces), look at Forster's Lectures on Riemann Surfaces (Springer) or at Narasimhan's Compact Riemann Surfaces (Birkhäuser).
(Caution: the analysis used in these two books is not for the faint-hearted!)
For your question about projective varieties and Proj, the answer is yes.
Spec of a ring, and Proj of a graded ring, are always separated, and being separated is inherited by open subschemes, so non-separated schemes give examples of schemes that are neither quasi-affine (i.e. open in an affine scheme) nor open in a Proj. [As an aside, note that for varieties, quasi-affine (open in an affine variety) implies quasi-projective (open in a projective variety), because affine space itself is open in projective space of the same dimension, while for any ring $A$, if we make $A[T]$ a graded ring by putting $A$ in degree $0$ and $T$ in degree $1$, then Proj $A[T] = $ Spec $A$, so being open in a Proj subsumes the possibility of being open in an affine scheme.]
As you wrote, Weil in fact introduced the concept of "abstract algebraic variety", which is an object which is locally an affine variety. In Hartshorne Ch. II, he similarly defines a variety over $k$ to be a separated integral finite type $k$-scheme, so quasi-projectivity is not assumed. It is convenient in Ch. I to impose quasi-projectivity just because it lets you get to some non-trivial examples and theorems without having to spend forever on the foundations. Also, it takes some effort to write down non-quasi-projective varieties; you don't tend to stumble upon them in beginners' excercises and constructions. [Weil introduced the concept because from his original definition/construction of the Jacobian of a curve over $k$, it wasn't clear that the Jacobian was projective.]
Best Answer
I will assume varieties to be irreducible. One way to define smoothness is the following (for affine varieties, but you can define it similarly for projective varieties):
Let $X \subset \mathbb{A}^n$ be a variety and let $I(X) = (f_1,\dots,f_r) \in k[x_1,\dots,x_n]$. A point $x \in X$ is smooth if $\text{rank}(J(a)) \geq n - \text{dim}(X) = \text{codim}(X)$, where $J(a) = \left( \frac{\partial f_i}{\partial x_j}(a)\right)_{i,j}$ denotes the Jacobian matrix. If every point $x \in X$ is smooth, we say that $X$ is smooth.
If you allow varieties to be reducible you need to replace $\text{dim}(X)$ in this definition. For more information check for example this link.
You can take $X = V(xyz + xyw + xzw + yzw) \subset \mathbb{P}^3$ to get an example of a non-smooth cubic hypersurface with singular points $[1:0:0:0],\dots,[0:0:0:1] \in X$.